Quasisymmetric Groups

Quasisymmetric GroupsAuthor(s): Vladimir MarkovicSource: Journal of the American Mathematical Society, Vol. 19, No. 3 (Jul., 2006), pp. 673-715Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/20161295 .

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JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 19, Number 3, Pages 673-715 S 0894-0347(06)00518-2 Article electronically published on January 25, 2006

?2006 American Mathematical Society Reverts to public domain 28 years from publication

QUASISYMMETRIC GROUPS

VLADIMIR MARKOVIC

1. Introduction

1.1. The main results. Let T denote the unit circle and D the unit disc. Suppose that / : T ?? T is a homeomorphism. Let / : D ?> D be a homeomorphism too.

We say that / extends / if / and / agree on T. All mappings in this paper are sense preserving (see the remark at the end of introduction).

Definition 1.1. We say that a homeomorphism / : T ? T is if-quasisymmetric if there exists a if-quasiconformal map / : D ?> D that extends /.

This is one of a number of equivalent ways to define quasisymmetric maps of the unit circle T. Let S be a Riemann surface and let / be an element of the mapping class group of the surface S. We say that / is If-quasisymmetric if there exists a

if-quasiconformal map / : S ?> S that represents /. If S = D, then this agrees with the above definition.

Definition 1.2. Let ? be a subgroup of the group of homeomorphisms of T. We

say that G is a If-quasisymmetric group if every element of G is if-quasisymmetric.

We will also say that a subgroup G of the mapping class group of a Riemann surface S is if-quasisymmetric if every element from G can be represented by a

if-quasiconformal map of S onto itself.

Remark. Unless specified differently a quasiconformal map (or a quasiisometry; see Section 2) is assumed to be a selfmap of D.

By Ai we denote the Lie group of M?bius transformations that preserve the unit disc D (therefore, those transformations preserve T as well). If u E M, we consider wasa homeomorphism of T. The corresponding group that acts on D is denoted

by M, and the corresponding element is denoted by u. A subgroup T of M. is called a M?bius group. If T is discrete, we say that T is a Fuchsian group. The

corresponding group that acts on D is denoted by T. The following are the main results of this paper.

Theorem 1.1. Let G he a discrete K-quasisymmetric group. Then there exists a

K\-quasisymmetric map ip : T ?> T and a Fuchsian group T such that G = tyT(?~x. The constant K\ is a function of K; that is, K\ =

ifi(if).

Hinkkanen proved (see [14], [16] and Proposition 1.2 below) that the same is true for quasisymmetric groups that are not discrete. This gives the next theorem.

Received by the editors December 15, 2004.

2000 Mathematics Subject Classification. Primary 20H10, 37F30.

673



674 VLADIMIR MARKOVIC

Theorem 1.2. Let G be a K-quasisymmetric group. Then there exists a if 2

quasisymmetric map (p : T ?> T and a M?bius group T such that G = ^pT^p~x. The constant if2 is a function of if; that is, if2 = if2 (if).

It is a classical result of Sullivan and Tukia (see [23], [25]) that every quasicon formal group that acts on the Riemann sphere S2 is quasiconformally conjugated to a M?bius group (subgroup of the group of M?bius transformations of S2). Tukia showed (see [26]) that this is no longer true for higher dimensional spheres. Martin

[19] and Freedman and Skora [7] gave various important examples of this nature.

Theorem 1.2 settles in positive the case of the one dimensional sphere T. The notion of convergence groups was introduced by Gehring and Martin (see

the Gehring and Palka [11] for the origins of the theory of quasiconformal groups). The theory of quasiconformal groups is closely related to the theory of convergence groups (see [9], [10], [8], [27]). For example, see [2] for connections with the geomet ric group theory and for further references. In particular, quasiconformal groups are convergence groups. One of the central results in geometric group theory is that

every convergence group of the circle homeomorphism is a conjugate of a M?bius

group. This theorem was proved by Gabai [8]. Prior to that Tukia [27] proved this result for many cases. Hinkkanen [15] proved the result for non-discrete groups. This theorem was independently proved by Casson and Jungreis [3] by different methods (see [3], [8], [27] for references to other important papers on this subject). We have the following proposition.

Proposition 1.1. Let G be a K-quasisymmetric group. Then there exists a home

omorphism (p : T ?? T and a M?bius group T such that G = <pT<p~~x.

To prove Theorem 1.2, one needs to show that in Proposition 1.1 one can find a M?bius group and a homeomorphism (p so that (p is quasiconformal. The meth ods used in [8], [25] to produce (p are constructive and explicit. Therefore, one can suspect that by repeating and modifying their construction while keeping in

mind that G is a quasisymmetric group, the resulting homeomorphism would be

quasisymmetric. However, this does not appear to be the case. Nevertheless we

will make frequent use of Proposition 1.1. It follows from the theorem on uniformly quasiconformal groups that Theorem

1.2 is true if and only if every if-quasisymmetric group can be extended to a if 1

quasiconformal group of D, ifi = ifi(if). In [5] it was shown that there is no

general algorithm that would produce such an extension. The proof of Theorem 1.1 is also explicit and for a given discrete quasisymmetric group we construct such a quasiconformal extension (this extension can be recovered easily from our proof).

We will assume that the reader is familiar with some elementary facts from

the theory of convergence and quasiconformal groups. In particular, we will freely use the notion of hyperbolic, parabolic, and elliptic elements of a quasisymmetric group. The order of an elliptic element e G G is the smallest integer n G N, such that en = id G G.

1.2. Elementary and non-discrete quasisymmetric groups. Let (p : T ?> T be a homeomorphism. Let a, b G T, a ^ b, and let I be the geodesic that connects

them. Let V be the geodesic that connects the points (p(a) and (p(b). We say that V is a push forward of the the geodesic I and write ipj,

= V.



QUASISYMMETRIC GROUPS 675

Hinkkanen proved Theorem 1.2 for various cases. In particular, the following proposition is a subcollection of results proved in [14] and [16] that are going to be used in this paper.

Proposition 1.2. Let G be a K-quasisymmetric group and suppose that G is either

(1) a discrete elementary group or

(2) a non-discrete group.

Then there exist K = if (if ) and a K-quasisymmetric map (p such that (p conjugates G to a Fuchsian group.

In fact, Hinkkanen proves the above proposition for Abelian and non-discrete

groups. But this readily implies the case of discrete elementary groups as follows. We know that every discrete elementary quasisymmetric group is conjugated to a Fuchsian group. The list of discrete elementary Fuchsian groups is short (see [17]). Up to a conjugacy (in M) the only non-Abelian discrete elementary group is generated by a hyperbolic element u ? M (with the fixed points i,

? i) and the

elliptic transformation eo G M, eo(z) = ?z, z G T. Note that u and eo satisfy the

relation

(1.1) u~x = eo ouoe0.

Every element in this group is either hyperbolic (from the same cyclic group gen erated by u) or it is an elliptic transformation of order two that permutes the two fixed points of u (there are infinitely many of these elliptic elements).

By Proposition 1.1 any elementary discrete quasisymmetric group G (that is not cyclic) is generated by a hyperbolic element h and the corresponding elliptic element e (that permutes the fixed points of h). By conjugating ft by a suitable

quasisymmetric map, we may assume that h = u e M. So the group G is generated by u and e, where u is a M?bius map and e is some quasisymmetric map of order two. We can assume that u fixes the points i,

? i. Since the group G is topologically conjugated to a M?bius group, we have the relation

(1.2) u~l = eouoe.

Denote by R the interval (i, ?i) and by L the interval (?i, i) (we take the standard counterclockwise orientation on T). By replacing e if necessary by a map of the form

u~koeouk, k G Z, we can assume that e(?l) belongs to the interval (u~x(l), u(l)) C R. Let q : T ?> T be the earthquake map that is the identity on L and such that

q(l) =

e(?1) (such q is unique and it is quasisymmetric). The map q commutes with u (which means that it conjugates u to itself). By replacing e if necessary by q~x

o e o ?, we can assume that e(?1) = 1.

Since the maps u, e, eo satisfy (1.1) and (1.2), we conclude that e = eo on 0(1) and 0(?1), where 0(1) is the orbit of the point 1 under the action of the cyclic group generated by u (similarly for 0(?1)). Set f(z)

= (eo o e)(z) for z G R, and

set f(z) = z?ovzEL. We have that / fixes every point from 0(1) and 0(?1).

This readily implies that / is quasisymmetric (we already know that / is locally quasisymmetric on both R and L). It follows that / conjugates the group G to the M?bius group generated by u and eo.

We also note the following elementary proposition.

Proposition 1.3. Let G be a K-quasisymmetric group and suppose that every

finitely generated subgroup of G is K-quasiconformally conjugated to a Fuchsian

group. Then so is G




From now on in this paper we assume that every quasisymmetric group is discrete

(unless specified otherwise).

1.3. A brief outline. The proof of Theorem 1.1 is divided into several steps. However, there are two main intermediate results which constitute the heart of this

paper. The first is Theorem 1.3. This theorem "takes care" of elliptic elements of order three or more.

Remark. This case turned out to be combinatorially very complicated in the proof of Proposition 1.1 about convergence groups. Note that so-called triangle groups must contain at least one elliptic element of order at least three.

Theorem 1.3 will be repeated and proved as Theorem 7.1 in Section 7.

Theorem 1.3. For an arbitrary K-quasisymmetric group G there exists a K\

quasisymmetric group Gi, if i = ifi(if), with the following properties.

(1) G\ does not contain elliptic elements of order three or more.

(2) IfGi is K' -quasisymmetrically conjugated to a Fuchsian group, K1 ? K'(K),

then there exists K" = if" (if) such that G is K"-quasisymmetrically con

jugated to a Fuchsian group.

Let J7 be a Fuchsian group and ip a homeomorphism such that ^pT^p~x = G>

Denote by E' c D the set of fixed points of all elliptic elements of T that are of order three or more. Let (p denote the barycentric extension of (p and set E =

(p(E')\ S = D ? E. Set G\

= (pT(p~x. Then the group G[ is a if i-quasisymmetric group

on S, K\ = ifi(if), which means that every element of G[ is isotopic as a map

of S (rel dS) to a if i-quasiconformal map. By covering the surface S by the unit

disc, we can lift G\ to the group G\ that is a if i -quasisymmetric group on D. The

group G\ is not isomorphic to G[ but it naturally projects to G[. The kernel of this projection is the group of covering transformations of S. This is the outline of the proof of Theorem 1.3. Note that we put no restriction on the choice of the

homeomorphism (p (in particular, it does not have to be quasisymmetric). The key to proving this theorem is certain analytical properties of the barycentric extension of Douady and Earle (see [4], [1]) that are of a different nature than the standard conformai naturality requirement this extension satisfies (which is also important of course).

The second main intermediate result is Lemma 4.2. Once we eliminate elliptic elements of order three or more (by Theorem 1.3), we can prove Lemma 4.2. This

lemma shows that the subgroup Gz of G that is generated by small elements with

respect to a point ? G D is cyclic (see the definition in Section 4). Here by small

elements we mean elements that are close to the identity (in C? topology) when seen

from the point z G D. This is a quasisymmetric version of the corresponding results

about small elements of Fuchsian groups. These results for the Fuchsian case are

corollaries of theorems like the Jorgensen inequality or the Margulis lemma (which of course holds in the context of discrete lattices in Lie groups).

Remark. Note that in the Fuchsian case in order to prove the Jorgensen inequality or the Margulis lemma, one does not have to assume that a given Fuchsian group does not have elliptic elements of order three or more. It is possible to prove Lemma 4.2 (the quasisymmetric case) without that assumption as well, but that

would require a more delicate proof that would involve the commutator trick (see




Chapter 4 in [24]) that is used to prove the Margulis lemma in the context of Lie

groups. Although quasiconformal maps do not make a Lie group, they still can be endowed with a manifold structure, and one can modify the ideas from the proof of the Margulis lemma to this case. It is interesting to try to investigate this for

quasiconformal groups in higher dimensions, which are not necessarily conjugates of M?bius groups. In the general case, the corresponding group Gz should be almost Abelian.

We use the following rules in connection with the notation. If C and D stand for some abstract constants, then the labeling C =

C(D) means that the constant C is a function of D. Sometimes C is a function that is itself a function of D; that is, we

say that C depends on D. In other words, for a fixed D one can choose such a C. What is important here is that saying that C =

C(D) also says that the constant C does not depend on any other parameter. Typically, the constant C will depend on the constant D in a concrete way. Usually it will be bounded above or below in some way that depends on D. This will always be clear from the context but we

will not necessarily make a note of it, since we do not need it. From Section 4 onwards if will always stand for the quasisymmetry constant of a

if-quasisymmetric group. Recall from Proposition 1.2 that if is the quasisymmetry constant of a map i? that conjugates an elementary if-quasisymmetric group to a

Fuchsian group. We allow if to be as large as necessary (but always as a function

of if) so that we can choose appropriate if-quasiconformal extensions of the map

i?. From Section 4 onwards, the constant if will have this meaning. All other constants will be valid throughout the subsection in which they were introduced. If we refer to a particular constant from a previous section, we will do this in a clear

way and no confusion should arise. In Sections 2 and 3 we prove various technical lemmas about quasiisometries,

quasiisometric groups, quasiconformal maps, and their geometry in D. Some of these results might be known. In Section 4 we study small elements and show how to remove them. In Sections 5 and 6 we prove Theorem 1.1 for torsion-free groups. In Section 7 we show how to eliminate elliptic elements of order three or more. In Section 8 we deal with groups whose only elliptic elements are of order two.

After reading the introduction, the reader can go straight to the very end of this

paper to consult the (very short) subsection where we give the summary of the

proof of Theorem 1.1. This gives clear guidelines of what is the logical order of the

proof.

Remark. It is a part of the definition of quasisymmetric (and quasiconformal maps in general) that they are sense-preserving. Originally, quasiconformal groups are defined to be sense-preserving (see [11], [14]). However, one can naturally extend this definition to sense-reversing maps. A sense-reversing map is quasiconformal if its complex conjugate is quasiconformal in the ordinary sense. Hinkkanen (in [16]) considered these generalized quasisymmetric groups, and his results are valid in this case as well.

We do not state Theorem 1.2 for generalized quasisymmetric groups, but we make the following observation. It appears that all methods that we use go through for sense-reversing maps as well. The only place where we use that our maps are

sense-preserving is when listing elementary Fuchsian groups. If one allows sense

reversing M?bius transformations, then this list would have a few more members.




Consequently this implies that there are a few more cases of elementary discrete

generalized quasisymmetric groups. Hinkkanen has dealt with this issue in [16], and it seems that his work covers all technical aspects that arise in dealing with these additional elementary groups.

2. QUASIISOMETRIES OF THE UNIT DISC

2.1. Quasiisometric continuation. In this subsection we state several results about quasiisometries of the unit disc. Some of these results are classical. In this paper, d stands for the hyperbolic metric on D, and d(z, w) always denotes the hyperbolic distance between the points z,w G D. By A(z,r) we denote the

hyperbolic disc centered at z G D and with the hyperbolic radius r > 0.

Definition 2.1. Let / : D ?> D be a homeomorphism. We say that / is a (L,a) quasiisometry if

L~xd(z,w) -a < d(f(z), f(w)) < Ld(z,w) + a, for some L, a > 0.

Remark. The assumption that / is a homeomorphism of D is often weakened in the

literature by assuming that / is only a surjective map of D onto itself. To simplify the notation, we will say that / is an L-quasiisometry if L = max{L, a}.

It is well known that every L-quasiisometry has a continuous extension to D. Let / : T ?> T be the corresponding map (the restriction of the extended map

/). There exists if (L) such that / is a if (L)-quasisymmetric map. On the other

hand, if / : D ?> D is a if-quasiconformal map, there is a(if ) > 0 such that / is a

(if, a(if))-quasiisometry (see [6]). Let I be a geodesic in D and let 7 :1 ?-> D be a map such that

L~1d(z,w) < d(i(z),i(w)) < Ld(z,w), for every z,w G / and some L > 0. We say that the map 7 is an L bilipschitz quasi geodesic. Sometimes we will say that the corresponding curve 7(Z) is a bilipschitz quasigeodesic if it is clear what the mapping is. Let l\ be the geodesic with the same endpoints as 7(Z). The main property of 7 is that there is D(L) > 0 such that

d(z,h) < D(L), for every z G 7(Z) Let / be a L-quasiisometry and / C D a geodesic. The restriction of / on Z

does not have to be a bilipschitz quasigeodesic. Nevertheless, it is easy to construct an L'(L) bilipschitz quasigeodesic 7 : I ?> D such that for some Dq =

A)(L) we

have d(f(z),j(z)) < Do, for every z G I. This implies that every point of f(l) is within a bounded hyperbolic distance of the corresponding geodesic with the same endpoints. This observation is the key ingredient of the proof of the following

well-known proposition.

Proposition 2.1. Let f : D ?? D be a homeomorphism. Let f : T ?> T be the

restriction of f. We have the following.

(1) Suppose that f is the identity and that f is an L-quasiisometry. There

exists D = D(L) > 0 such that d(f(z), z) < D, for every z G D.

(2) Suppose that for every zq G D; there exists an L-quasiisometry fZo which

extends f and such that d(fZo(zq), f (zq)) < D, for some D = D(L) > 0.

Then there exists V = Lf(L) such that f is V -quasiisometry.




Definition 2.2. Let (M, di) and (iV, (I2) be two metric spaces. Let 6 : R+ ? R+ be a function, e ?> ?(e). We say that a map F : M ?> N is ?(e)-continuous if

d2(F(z),F(ti;)) < e, whenever di(z, w) < 6(e), for every 2,w G M. We also say that such F is uniformly continuous. A homeomorphism F is 5(e)-continuous if both F and F_1 are.

If we say that a map defined on a subset of the unit disc is uniformly continu

ous, that always refers (unless specified otherwise) to the corresponding hyperbolic metric on D.

Let x, y, z G T, and let S be the geodesic triangle with vertices x, y, z. Denote the geodesies that represent the sides of S by ax,y,ay,z,az,x. Suppose that / is a

if-quasisymmetric map. Suppose that there is an L\ = Li(if) with the following

properties. There exist Li-quasiisometries fx^y, fy^z, fZiX that extend / and such

tnat Jx,y\^x,y) =

J*^x,y) Jy,z\0Jy,z) =

J*Ojy,Z') Jz,x\^z,x) =

J*^z,X'

Lemma 2.1. With the assumptions as above, the following holds. There exist

L[ =

L[(K) and an L[-quasiisometry f that extends f such that the restriction of

f on ax,y, ay,z, and aZfX agrees with fx,y, fy,z, and fz,x, respectively. Moreover,

suppose that the restriction of f is 5(e)-continuous on the sides of the triangle S. Then there exists a function ?\ : R+ ?> R+ (8\(e) depends only on 0(e)), such that

f is 61 (e) -continuous on S.

Proof By pre-composing and post-composing / by M?bius transformations, we

may assume that x = i,y = ?1,z = 1 and that f(i) = i, /(?1)

= ?1,/(l)

= 1. On each halfspace determined by one of the geodesies ax,y, ay^z, and az?x that does not

contain the triangle S, set / equal to the corresponding quasiisometry. It remains to define f on S (see Figure 1).

Let rai,ra2,ra3 be the middle points (in the Euclidean sense) of the geodesic

ax,y, ay,z, and az,x, respectively. Denote by So the corresponding geodesic triangle with vertices 7711,7712,7713, and denote by s?,s_i,si the sides of So that face the

points i,?l,l, respectively. Let s[,s'__i,?i denote the geodesic arcs which connect the points that are the images of the endpoints of s?, s_i, s\ under the correspond ing maps fx^y, fy,z, fz,x. Let Sf0 be the corresponding triangle (note that s?, s'_i, s[ also face the points i,?l,l, respectively). We define / on each of the sides of 50 so that / maps each side of S to the corresponding side of Sf0 and so that / is the restriction of the unique M?bius transformation (of C) that maps the middle

points of Si, 5-1, s 1 onto the corresponding middle points of s[, s'__x, s[, respectively.

We define / inside So to be any homeomorphism that extends the values of / on

dSo (this homeomorphism maps So onto Sf0). We can arrange that this homeo

morphism is ?1 (e)-continuous for some function ?i that is a function of 6 (after supplying the regions So and Sf0 with their own Poincar? metrics, one can take this

homeomorphism to be either the Euclidean harmonic extension or the barycentric extension or any other classical extension that is a homeomorphism). Note that the hyperbolic diameters of both So, S? are bounded above by a constant that is a function of if.

Denote by S?, S_i, Si the corresponding subtriangles of S such that S is a disjoint union of So, S?,S_i, and Si. For a point w G S? let a be the geodesic arc that contains w, where a is parallel to s? and a connects the two sides (other than

Si) of the triangle S?. Two geodesic arcs are parallel if they are subarcs of two



680 VLADIMIR MARKO VIC

Figure 1

concentric circles in C. By o? we denote the geodesic arc that connects the points that are the images of the endpoints of a under two of the corresponding maps

/x,2/5 fy,zi fzyX- We define / on a so that / maps a to o? and so that / is the restriction of the unique M?bius transformation (of C) that maps the (Euclidean) middle point of a onto the corresponding middle point of a''. We repeat the same

process for the remaining points in S. It follows from this construction that / is a

Li-quasiisometry, for some L[ =

L[(K). Also, it follows from the geometric nature of our extension that for a fixed function S(e) there exists a function 8\ : R+ ? R+

such that / is 8\-continuous. D

Let A be a countable collection of geodesies in D that is locally finite and so that no two geodesies intersect transversally in D (they may have a common endpoint in T). Clearly A is a closed subset of D. Such a collection A is also called a

discrete geodesic lamination. Suppose that / is if-quasisymmetric and that fj G

A, /~1Z G A, for each Z G A. In addition, suppose that for each Z G A, there

exists an L2-quasiisometry f\ that extends / and such that fi(l) = fj G A. Here

L2 = L2(if).

Lemma 2.2. With the assumptions as above, the following holds. There exist

Ls = Ls(K) and an L^-quasiisometry f that extends f such that the restriction of

f on each l G A agrees with the restriction of f\. Let ?i, i G [1, N], N = N(K) G N,

be mutually disjoint geodesic arcs that do not intersect A transversally but whose

endpoints belong to certain geodesies from A. Then we can choose the above map f so that f(?i) is a geodesic arc.




In addition, assume that the above f is 6(e)-continuous on A (here f agrees with

fi). Then f is 8 \(e)-continuous, where the function 8\ is itself a function of 6(e) and if.

Proof. Each geodesic lamination can be completed to a maximal geodesic lami nation. Let A' be a maximal geodesic lamination that contains A. Then D is a

disjoint union of A' and a collection of open geodesic triangles whose sides are in A'

(if A' foliates the whole unit disc, then these triangles do not exist). Let sn, n G N, be the sequence of geodesies from A' ? A so that the closure of the union of A and

UnGN Sn is e(mal to ̂ ''

We first define / on si. For z G si let p be the geodesic that is orthogonal to si. Let a be the maximal geodesic subarc of p that contains z and that does not intersect A. There are three possibilities: (i) the endpoints of a belong to two different geodesies from A; (ii) one endpoint belongs to a geodesic from A and the other is in T; (iii) both endpoints belong to T. In any case the value

of / is determined at these two endpoints by the maps //, Z G A. Denote by o/ the corresponding geodesic arc whose endpoints are determined by the images of the endpoints of a. Then o? does not transversally intersect A. Define f(z) to be the intersection between o? and /*Si. This is well defined because / is a

homeomorphism and /*A = A. Now, repeat this process for s2, but instead of A

we use A U 5i, and so on. By this we define / on every sn, and it follows from

the construction that / is well defined on A'. On the remaining geodesic triangles we define / by using Lemma 2.1. It follows from Proposition 2.1 that / is an L2 quasiisometry, Lr2

= L2(K). Because of the geometric construction we made, it

follows that / is ?i(e)-continuous on D. One can directly compute 8\(e) in terms of

8(e) and the Holder continuity constants of the map / (which are explicit functions of if by the theorem of Mori).

Let ?i, i G [l,iV], be a geodesic arc from the statement of this lemma. Let ?[ be the geodesic arc with the same endpoints as the curve f(?). We have that ?[ and f(?) are a finite hyperbolic distance apart (this upper bound depends only on

if) since / is a quasiisometry. Because of that, one can find an L2-quasiisometry / : D ?> D (Lf2

= L2' (if )) which extends the identity map of T, so that I pointwise

fixes each Z G A and /(/(/?)) = ?f. The map I o / is an L3-quasiisometry, L3 =

L$(K). Since / is 8\(e)-continuous, we see that if ? is a very short geodesic arc, then f(?) is very close to the corresponding arc ?'. This implies the existence of

?1 (e) such that / is 8\ (e)-continuous. Moreover, 8\ depends only on 8.

Repeat this process N = N(K) times for all the arcs ?i to obtain the resulting

quasiisometry. D

2.2. Quasiisometric groups. Let G be an L-quasiisometric group on D. This means that G is a group whose elements are homeomorphisms of D, each of them

being L-quasiisometric. The restrictions of elements from G on the unit circle T form a group which we denote by G- Clearly, G is a if-quasisymmetric group, if = if (L). Our aim is to show that there is if' =

Kf(L) such that G is if'

quasisymmetrically conjugated to a Fuchsian group. We will show this under extra

assumptions. Suppose that there exist p > 0 and a function 8 : R+ ?> R+ with the

following properties. For every / G G \id we have that d(z,f(z)) > p, for every z G D. Also, each / is ?(e)-continuous. These assumptions on the group G are




valid throughout this subsection. Our aim is to show that there is a quasiconformal group G that extends G- Note that these assumptions imply that G (and therefore

G as well) does not have any elliptic elements. Let po > 0. We define a discrete set E =

E(p0) C D as follows. For any point z G D denote by O(z) its orbit under ?7. Let S denote the Fuchsian group that is the covering group of a closed Riemann surface of genus at least two. Let wn, n G N, denote the orbit of the point w\ = 0 under E. Set z\ = w\ = 0, and let

Ei = O(zi). Choose the smallest no > 1 so that d(O(zi),O(wn0)) > p0. If such n0

does not exist, then set E = Ei. Otherwise, let z2 = wno and set E2 = Ei U 0(z2). Similarly, let ni be the smallest number so that d(0(wni),E2) > p0. If such ni does not exist, then set E = E2. Otherwise, let z% = wni and Es = E2U O(zs), and so on. Let E =

(J Ei. The main properties of the set E are as follows.

(1) E is invariant under G>

(2) There exists pi = pi(po,p,8(e)) such that in every geodesic ball of radius

pi there is at least one point from E.

(3) Every geodesic ball of radius ^ contains at most one point from E.

Remark. The one and only reason we choose ? to be the covering group of a closed surface is to achieve that the sequence wn is well distributed in D, which then yields condition (2) above.

If we let po ?* 0, then we can choose pi so that pi ? 0 (this comes from the fact

that f is ?(e)-continuous). Because of that we can choose po = po(p,8(e)) small

enough so that the corresponding Pi < joq. We choose such po and pi (see the remark below).

Set S = D ? E. Denote by ds the hyperbolic metric on S. It follows from the

assumptions that the metrics d and ds are comparable, except on a microscale very near the points from E. This means that if we take the corresponding densities

dens(d) and dens(ds) that define the metrics d and ds, respectively, then there exists a constant X > 0 so that for z G S = D ? E we have

1_ < dens(d)(z) <x X dens(ds)(z) and the constant X depends only on the d(z,S). For z,w G E and a simple curve

(simple arc) 7 C S with the endpoints z, w, let a(z, w, 7) denote the geodesic (with respect to ds) homotopic to 7 (homotopic in S). We say that 7 is ?i (e)-continuous if there is a ?i (e)-continuous homeomorphism between 7 and a(z,w,j) (here the uniform continuity is with respect to the metric d). We assume that the inverse of this homeomorphism is also ?i (e)-continuous. Here 81 : (0,00) ?> (0,00).

Let z,w G E and let 7 C S be a curve that connects them. We say that the

homotopy class [7] (of 7) is admissible if for every / G G we have that the length of the curve cx(f(z),f(w),f^) is at most P2 = ^- Here we measure the length of a(f(z),f(w),f*j) in the hyperbolic metric d of the unit disc (the length of

a(f(z),f(w),f*j) in the metric ds is infinite). Numerate points in E by zn, n G N. Fix i,jeN. Suppose that [7] is an admis

sible homotopy class (with the corresponding endpoints Zi,Zj) and let a(zi,Zj,^y) be the corresponding geodesic. Then a(z{,Zj,j) has length at most p2- Let Fijn be the orbit of a(z,w,^) under G> By F?? we denote the union of all i^j,7, where




[7] ranges over all corresponding admissible homotopy classes (we note that there are only finitely many such homotopy classes for a fixed pair i,j). The union of all

the sets F?? is denoted by F.

Remark. By taking p0 and pi a little bit smaller (but keeping P2 as it is), we can arrange that every connected component of the set D ? F is a polygon whose diameter is bounded above by a constant that is a function of 8(e) and p. This is the final choice of po and pi.

By definition each curve in F is ?(e)-continuous (8(e) is the function that was

fixed at the beginning of the subsection). Each curve from F is homotopic to a

geodesic (in metric ds) whose d length is less than p2- From ?(e)-continuity of curves from F, it follows that there cannot be very many curves from F ending at a given point of E. This implies that there is N =

N(p,8(e)) G N such that for each point z G E there are at most N curves from F that end at z. Because of

that, the fact the each curve in F is uniformly continuous, and since each f EC is

uniformly continuous, we conclude that there exists iVi = Ni(p, 8(e)) G N such that in each hyperbolic disc of radius p0 there are at most Ni points that are intersection

points between different curves from F. Now, we slightly deform the curves (within their homotopy classes in S) from F so that the hyperbolic distance (distance d) between any two intersection points is greater than p'

= p'(p,8(e)) and such that

every intersection point is contained in exactly two curves from F. In addition, we can arrange that the new set of curves (which we also denote by F) is also invariant

under G> This is because P2 was chosen to be small enough so that the distance between a curve from F and any other curve in its orbit is bounded below by a

positive constant that is a function of our original parameters p and 8(e). Denote by Ei the union of E and the set of all intersection points (after the

deformation). It follows from the construction that each connected component of D ? F is a polygon with at most ?V2 = N2(p,8(e)) sides. We can now add new curves (that connect points from Ei) to the set F (the new set is also called F), so that every component of D ? F is a triangle. We can do so by retaining the uniform continuity properties of the curves from the original F.

Let Si = D ? Ei. Replace each curve from F by the corresponding geodesic

(geodesic with respect to the hyperbolic metric on Si). The new set of curves is denoted by Fi. Clearly, Fi retains the same essential properties of F, except that

i*i is no longer invariant under ?/. However, for each 7 G Fi there is a unique curve

in Fi that is homotopic to /*7

Remark. Note that if T is a triangle from this partition, then /*T is well defined. If /*T = T, then / is the identity. This follows from the fact that / has no fixed

points inside the unit disc. We will not make any use of this fact.

Because the distance between any two points in Ei is bounded below by p', there are constants P3,P4 > 0 that are functions of p and 8(e) such that the hyperbolic length ld(l) of 7 G Fi satisfies P4 < ld(l) < P3- Here Id denotes the length with

respect to d.

We now define the new group G of homeomorphisms of D. For / G G we denote

by / the homeomorphism from G that extends /. We define / to be equal to / on the set Ei. On each curve 7 from Fi we define / to be the affine map with

respect to the natural parameters on each of the two curves 7 and /*7 = f(^).




The natural parameter is taken with respect to the metric d (both curves have

finite d length). It also follows that / respects the composition (the composition of two affine maps is an affine map). It remains to define / on the interior of the

corresponding triangles. We denote the set of all triangles by T. Let X?, i G I, be a list of all mutually non-conjugated triangles from T. Here I is either a finite set or I = N. The equivalence class of X? is denoted by [T?]. For each / G G let

/ : T ?? f*T E T be a homeomorphism that extends the already-defined values of

/ on dT. Let ps be the minimal distance between points from Ei divided by 10.

Then, it follows from our construction that this homeomorphism / can be chosen to be if i-quasiconformal on D, except at the points that are p$ close (in the hyperbolic metric d) to the points from E\. Moreover, ifi =

K$p,5(e),ps). This is readily seen by passing onto the universal cover of Si. Then the triangles T and /*T can

be seen as geodesic (ideal) triangles. We can take / to be the barycentric extension

(Douady-Earle extension and barycentric extension refer to the same thing) of the

map already assigned on the boundary of these triangles. The only place where this

map may fail to be uniformly quasiconformal is near the cusps (that correspond to

points from E$. Fix a single representative, say T?, from each class. Let T[ E [T?] be any triangle

and let %T? =

T?''. Let g', g" E G be such that &7i =

T? and ̂ T? =

T?' (such

g',g" are unique, since elements from G have no fixed points in D). Set

f(z) = (rog'-1)(z), for z ET[. Here g' and g" are the above-defined maps on T? that correspond to g' and g", respectively. By repeating this process for every i El, we have defined the

extension / for every f E G> Denote the corresponding group by G> By the same

arguments as above, for every z E Ei we can choose a small disc Dz (not necessarily geodesic) of the hyperbolic diameter ~ ps such that the set S2 = D ?

\JzeEl Dz is a

connected Riemann surface homotopic to Si. Moreover, we can arrange that S2 is

invariant under the action of G. We now fix such ps = Ps(p, 8(e)). The fact that G is a group follows readily from the construction. Clearly, G is a if 1-quasiconformal group on S2, ifi =

ifi(p, 8(e)). Therefore, there exist a Riemann surface S2, a

if2-quasiconformal map i? : S2 ?> S2, if2 = if2(ifi) =

K2(p,8(e)), and a discrete

conformai group T that acts on S2, such that G = i?Ti?'1. Note that the restriction

of the map i? to T is a quasisymmetric map (see the remark below). Since one can realize S2 as the unit disc minus many (but countably many) small

topological discs that lie inside D, we can apply the results from [12] and conclude that there exists a conformai map tt : S2 -^ S2, where S2 is obtained as the unit disc

minus countably many geometric discs (these discs are now hyperbolic discs in the unit disc). The map 7r extends continuously to the quasisymmetric map of the unit

circle (see the remark below). Using [12] again, we know that every conformai map of S2 onto itself must be a restriction of a M?bius transformation. This shows that

7rj^7r-1 acts as a Fuchsian group on the unit circle. This implies that the restriction

of T on the unit circle is a quasisymmetric conjugate of a Fuchsian group.

Remark. Here we use the following result of Kozlovski, Shen, and van Strien about

quasiconformal mappings (see appendix in [18] for the formulation and proof). This

theorem was inspired by the work of Heinonen and Koskela [13] (also see [22]). Let




F be a homeomorphism of D that is ii-quasiconformal on the set M. The set M is

of the form M ? D ? [j Di, where Di is a collection of topological discs in D with

the following properties. The hyperbolic distance between Di and Dj is bounded below by C > 0, for every i ̂ j. The diameter of each Di is bounded above by Ci > 0. Then the restriction of F on T (which has also been proved to exist) is

if (C, Ci, K ) -quasisymmetric.

Lemma 2.3. With the assumptions and notation as above, we have that the group G is K2-quasisymmetrically conjugated to a Fuchsian group. Here K2 ?

K2(p, 8(e)).

3. Quasiconformal continuation and barycentric extensions

3.1. Quasiconformal continuation. Suppose that / : T ?> T is a if-quasisym metric map. Let S =

{Si}, i E N, be a collection of subsets of D with the following properties. Each Si is either a Jordan region or a Jordan curve. Each S? is a

closed connected subset of D such that there are no bounded (in D) components of

D ? Si. This implies that the interior (if any) of Si is simply connected. Also, each

Si touches the unit circle at, at most two points. In addition, there exists R > 0 such that

(3.1) d(Si,Sj)>R, ijij. Let K1 ?

if'(if). Suppose that for each i E N, there is a if'-quasiconformal map

fi that extends / and such that fi(Si) E S.

Lemma 3.1. With the assumption as above, the following holds. There exists (large enough) R =

R(K), so that if (3.1) holds for R, then there exist ifi = ifi (if) and

a K\-quasiconformal map f that extends f and such that f =

fi on each Si.

Remark. By a more involved argument, one can prove that in the above lemma we can always take R = 1. Also, the assumption that each S? touches the unit circle at, at most two points is not essential. However, this will be the case in all

applications of this lemma.

Proof. Let / be any if-quasiconformal extension of /. Let fi(Si) =

Sj for some

j E N. The distance d(f(z),fi(z)), z E Si, is bounded by a constant depending only on if. Choose R ?

R(K) large enough so that the following holds. For each

i E N, there exists an open connected set Ui C D that contains both Sj and /(S?) and such that Ui H Uj is an empty set, i ̂ j. Moreover, every point in dUi is at the

hyperbolic distance at least 1 from any point from Sj U fi(Si). In addition, we can

choose the set Ui so that Ui touches the unit circle at the same points that Si does and so that the boundary of ?7? is a Jordan curve. Note that the above assumptions imply that Ui is simply connected.

Now, one can construct a if "-quasiconformal map i? : D ? D, if" =

K"(K), so that li o f

= fi on Si and i? is the identity on D ?

?/?. To do this, let fi? be one

of at most two components of the open set Ui ?

/(S?). In either case, ??? has two

boundary components (these two components meet at T). One of them is a piece of the boundary of /(S?), and another one is a piece of the boundary of E/?. Let ?2? be the corresponding component of the set Ui

? Sj, so that <9f?? contains the same

piece of the boundary of ?/? as does dVti. Let g : dQi ?? ofy be the homeomorphism that is the identity on the part of the boundary coming from ?/? and g = fi o f~x




on the other part. One can use the standard rescaling-compactness argument to show that there exists if{

= if((if) such that the map g is if (-quasisymmetric on

dSli. Repeat the same process for the other component of Ui ?

Si (if it exists). Set

Ii equal to g on Ui ?

/(S?), and set U = fio f~x on /(S?). This shows the existence of the map i?. Since the boundary of S? is a Jordan curve (or Si is a Jordan curve

itself), we conclude that Ii is quasiconformal. Set / = lim Ii o ... o Ii o f. We have that for some constant ifi ? ifi (if), / is

i??oo a ifi-quasiconformal map. D

Definition 3.1. Let E C D be a discrete set, and let p > 0. We say that F is a

p-discrete set if the hyperbolic distance between any two points in E is at least p.

Lemma 3.2. Let E be a p-discrete set. Let C : E ?> D, where E' ? C(E) is a

p-discrete set, and d(C(z), z) < D, for some D > 0, and every z E E. Then, there exist if2 = if2(p, D) and a K2-quasiconformal map I that extends the identity such that I(z)

= C(z), ZEE.

Proof. Suppose first that the constant D = Do satisfies 0 < Do < ?. Let az be the

geodesic arc that connects z and C(z). Since d(C(z),z) < DQ, no two such arcs would intersect. The positive orientation on az is from z toward C(z). Let Xz be the vector field on az so that for w E az we have that Xz(w) is the positive tangent vector to az at w. We take Xz(w) to have the length

ld(<*z)

Do '

where ld(oLz) is the length with respect to the hyperbolic metric d. Choose Do =

Do(p) small enough such that we can define a vector field X on D which agrees with each Xz and satisfies the following. We can choose X so that there exists the

one-parameter family of diffeomorphism It : D ?? D, t E [0, Do], such that

For this it is enough to arrange that X is a smooth vector field and uniformly Lipschitz in D (which we can do since E is p-discrete and for Dq small enough). Moreover, since E is p-discrete, we can choose X uniformly (uniform on D) smooth, so that the map Id0 is if^-quasiconformal, for some K2

= K2(p). This conclusion

comes from the standard estimates on the solutions of ODE's. Note that we can choose such X for some jDo > 0, where A) does not depend on the particular set E.

We have Dq = A)(p)- It follows from the construction that the time Do map Id0

satisfies the properties stated in the lemma. Note that the family It is continuous in the variable t as well.

The general case goes as follows. We can join z and C(z) by a C?? curve *)z, whose length is equal to some fixed L(p) > 0 for each z E E. We can also arrange the following. Let zt be the point on jz such that the length of the piece of ^z between z and zt is equal to t G [0,L(p)]. Let Et be the set of all zt. We can choose the curves jz so that each set Et is pi-discrete, pi =

pi(p). Now, chop up the interval [0, L(p)] into N small intervals, N =

N(p, D) E N, such that on each of them we can apply the above construction. This proves the lemma. D

Let F be a p-discrete set, and let ft : D ?? D, t E [0, ko], be a continuous family

(in t) of homeomorphisms (of D) with the following properties. For every t, ft is




the identity on T. We have /o = id. For every e > 0 there exists 8(e)

= 8 > 0 such

that for every t,s E [0, fco] and z E E, we have that d(ft(z), fs(z)) < e, whenever

\t ?

s\ < 8. In addition, we assume that the set Et = ft(E) is p-discrete for every t.

Set St = D - Et.

Lemma 3.3. With the above assumptions, we have that there exists a K^-quasi

conformal map ft : So ?

St, where ft is isotopic to ft (isotopic in So) for every t E [0, fco]. The constant if3 is a function of p, the function 8(e), and k0.

Proof. First, we break the interval [0, fco] into small enough intervals so that on each of these small intervals we use the construction from the first part of the proof of the previous lemma. Precisely, let n E N, n ? n(p,8(e),k0), be such that for each

interval [s?,s?+i] we can construct a if2-quasiconformal map fi, 1 < i < n, that

maps ESi to ESi+1 (the map fi is from the first part of the proof of the previous

lemma). Here s? = ^Q-. We can do that since ft is ?(e)-continuous. Also, the map

fi is very close to the identity in the C? sense uniformly in the metric d on D.

In fact, we can choose n large enough so that the maps fi and fSi+1 o jF"1 are

isotopic. This can be proved by contradiction as follows. Suppose that for every n E N we can produce an example where the above conclusion fails. Then because all the maps involved are uniformly continuous and the set F is p-discrete for some

fixed p > 0, we can pass onto the limit. The limit of the maps f8i+1 o jf"1 o f~l is

the identity map. This is a contradiction since we assumed that none of the maps in the sequence are homotopic to the identity.

By doing this, we produce a continuous family of if3-quasiconformal mappings

ft, t E [0,fco], such that for every z E E, we have ft(z) =

ft(z)> Moreover,

ft(z) o (ft)'1 is isotopic to the identity on S. D

Lemma 3.4. Let E C D be a p-discrete set and suppose that 0 G E. Set S = D?E.

For x G T let sx be the geodesic in D with the endpoints x and ?x. Let f : D ? D be a homeomorphism that extends the identity map and that fixes every point from

E. Suppose that there is L > 0 such that the restriction of f on sx is homotopic (in S) to a L-bilipschitz quasigeodesic 7^ : sx ?> D. Then there exists if4 = K?(p, L) such that f is isotopic (in S) to a K4-quasiconformal map.

Proof. Denote by ds the hyperbolic distance on S. Since F is p-discrete, we have that ds is comparable with d, except on a microscale near the points from E. Let

ax be the geodesic in the metric ds that is homotopic to sx in S. Here ax is really a union of geodesies in S that connect the corresponding points from F. By ?x we denote the geodesic (with respect to ds) that is homotopic to f(sx). From the

assumption of the lemma and the comparability between d and ds, we conclude that both curves ax and ?x can be parametrized as Li bilipschitz quasigeodesics in D, Li =

Li(p,L). This implies that for every point z E az, there is a point w E ?x such that ds(z, w) < M, M ?

M(p, L). Suppose that w E ?x is such that

ds(z,w) =

ds(z,?x). Define / : S ? S by setting f(z) = w. The restriction of /

on each ax is the so-called nearest point retraction map.

Fixing an (arbitrarily small) neighborhood of the set F, it is easy to show that

there exists Li = Li(p,L) such that / is an Li bilipschitz map (with respect to

the metric ds) outside that neighborhood. We will not compute this, but we will

suggest how to do it. An easy way to see this is by passing onto the universal




cover of S. There (meaning in the universal cover = unit disc), for each z and

the corresponding w ? f(z), ax and ?x can be realized as two non-intersecting

geodesies and the distance between the corresponding lifts of z and w is less than M. Since those two geodesies do not intersect, the bilipschitz constant of the nearest point retraction map (away from the cusps) depends only on the distance

M. Since / is bilipschitz on the unit disc minus a fixed neighborhood of the set

E, it is quasiconformal as well on the set. Now, one can change the map / in the

neighborhood of the set F, so that the new map is isotopic to / and quasiconformal as well on D. D

The following lemma is elementary.

Lemma 3.5. Let f be a K-quasisymmetric map and let I be a geodesic in D. There

exists a K-quasiconformal and L-bilipschitz map f which extends f,K = if (if), L = L(K), such that f(l) = fj.

Proof. Let / be the barycentric extension of /. Then / is L-bilipschitz. Let 7 = f(l) and V = fj. We have that 7 is a bilipschitz quasigeodesic with the same endpoints as V. Denote by f2i and Q2 the two regions obtained by removing 7 from D. By Hi and H2 we denote the corresponding halfspaces obtained by removing V. Let

p : 7 ?> Z be any bilipschitz map such that d(p(z), z) < P, P = P(K). Denote by

qi the barycentric extension of the map qi : dQi ?> dHi (here one has to map fii and Hi onto the unit by the Riemann maps to define the barycentric extensions). The map qi is defined to be the identity on the T part of the boundary of f?i (this part of the boundary also borders Hi). We set ?i

= p on 7. We do the analogous thing for Q2 to obtain the map q2. Define g : D ?> D so that g = qi on Oi and

g = q2 on Q2. It follows from standard estimates about the boundary behavior of conformai maps and standard estimates on the barycentric extensions that the

map f = go f satisfies the properties stated in the lemma. D

3.2. The barycentric extension. Let T be a homeomorphism. The

barycentric extension of Douady and Earle is a homeomorphism (p : D ?> D that extends (p and which maps the point 0 to the barycenter <J?(0) of the corresponding probability measure <p*(oo). Here <jo is the normalized Lebesgue measure on T

and <p*(cto) means the push forward of 00 by (p. It is also customary to write

??(0) =

Bar((p*ao)- By requiring that <p be conformally natural, the above uniquely determines this extension. See [4], [1], [21] for definitions and properties of the

barycentric extension.

If ip is quasisymmetric, then (p is known to be quasiconformal. There are many

conformally natural extensions that have this property. One important thing about

this particular extension is that one can say what happens with (p when <p degen erates. The theory of barycentric extensions can be developed not only for homeo

morphisms of T but also for limits of homeomorphisms and for certain probability measures on T. This case is of interest to us. In fact, if p is a probability measure

on T that does not contain strong atoms, then the barycenter p(0) G D is well

defined. An atom for a probability measure is said to be strong if it has mass at

least \. However, the only results we need in this direction are known and they all follow

from [1] (they were hinted in [4]). One of the results that will be used later is the

following. Let pn be a sequence of probability measures on T such that none of



QUASISYMMETRIC groups 689

these measures has atoms. Suppose that p is a probability measure that has no

atoms of mass > \. If pn ?> p (in the sense of measures), then the barycenters

p^(0) ?? p(0) G D. This observation was already stated in [4], but it was developed in detail in Sections 3 and 4 of [1].

Another result that we need is to show that under certain assumptions the

barycentric extension (p is quasiconformal in a neighborhood of 0, even though the

homeomorphism (or a limit of homeomorphisms) <p is not. The following lemma

follows from the so-called four sines law, and both the statement and the proof of

this lemma can be found in [1].

Lemma 3.6. Let R be an

increasing function such that <p(elt) =

elu(<t\ where t E R. Suppose that there are

0 < so < tt and C > 0, so that u(t + s) ?

u(t) > C, for every 0 < so < s < tt

and every t E R. Then, there exist K = K(so, C) and D =

D(so, C) such that the

barycentric extension (p is K-quasiconformal in the hyperbolic disc A(0, D).

Proof. Following [21], we have

(3.2) l-|?eZt($5)(0)|2>^. Here Belt((p)(0) is the complex dilatation of (p at 0, and

7T / 7T X

(3.3) 8 = ?? / sin(s) I / v(t,s)dt j ds ? ?^ \ sin(s)v(t,s)dsdt. 0X0/ [0,tt]2

Here

v(t, s) = sin(0i(t, s)) + sin(62(t, s)) + sin(6$(t, s)) + sin(6?(t, s)),

for6i(t,s) =

u(t+s)-u(t); 62(t,s) =

u(t+27r)-u(t+s+ir); 0s(t,s) =u(t+s+n)

u(t+ir)', 64(t, s) =

u(t+ir)-u(t+s). Note that 6x(t, s)+02(t, s)+03(t, s)+<94(t, s) =

2?r, and 6i(t, s)>0,i = l, 2,3,4.

It follows (see Lemma 4.4 in [1]) from basic trigonometry that

(3.4) v(t, s) = Asm-?-?-sin-?-sin-?-?- > 0.

The above formula holds for any choice of three out of the four #?'s. Let si = tt ? sq. For 0 < s < si, we have that both 62(t, s) and 6?(t, s) are greater or equal to C.

Let Po = {s,t : 0 < s < si,0 < t < tt}. Let s2 ? ^. Since every interval of length

at least tt ? si = so is mapped by u onto a set that contains some interval of length

C, we conclude that for some to E (^, tt ?

^), we have

C Oi(t0,s2) > ?,

where TV = N(so) is the minimal positive integer such that s2 > j??. Set S3 = ^-. Then for every (t, s) E Pi =

{(t, s) : 0 < t0-s3 < t < t0 and s2 + s3 < |f < s < tt}, we have

(3.5) 0i(M)>?. Since the area of Pi is greater than ^s3 and since 62(t, s), 6?(t, s) > C, we conclude

from (3.3), (3.4), and (3.5) that there is 80 = 80(s0,C) > 0, such that S > S0. The

rest follows from (3.2).




We need to show that (p is quasiconformal at the points close to the origin. We do that by bringing them back to the origin (by M?bius transformations) and then

using the same argument as above. Let az E M be such that az(0) = z and az

preserves the geodesic that contains both 0 and z. Let (pz = 0 small enough such that for z E A(0, D) the map <pz satisfies the assumption of this lemma, where instead of so we take so + 2L-^ra. By repeating the same argument, we conclude the proof of the lemma. D

4. Small elements in discrete quasisymmetric groups

4.1. Discrete groups generated by small elements are cyclic. Let G be a

if-quasisymmetric group. For each / G G, let i? be a if-quasisymmetric map and

let u E M, such that f =

i?ouoi?~1. Let i? denote a if-quasiconformal extension

of i?, K = if (if ). Set / = i? o u o i?-1. The / is a if2-quasiconformal map that extends /. For z ET> and for / G G, if / is not elliptic, let

Pf(z,$)=d$-\z),U$-1(z))). Set

Pf(z) =

supjPf(z,i?), where the supremum is being taken with respect to all such if-quasiconformal maps that fix l,z,?l, and all corresponding u E Ai. Because of the compactness, there are u E A4 and i? such that the supremum above is attained. Since we only consider

Pf for non-elliptic / G G, it is proper to say that Pf(z) is the K distance (or just the distance) between / and the identity when seen from the point z.

Remark. Here we consider only non-elliptic elements because of the nature of the

subsequent applications. However, one can naturally define the notion of being small for elliptic elements.

Recall that if u is a hyperbolic M?bius transformation, then we define its length as

mind(z,u(z)). zED

This minimum is attained for every point z E D that belongs to the axis of

the transformation u. Let / G G be hyperbolic. Let Lf(i?) be the length of

the corresponding hyperbolic transformation u. Set Lf =

sup^Lf(i?), where the

supremum is being taken with respect to all such if-quasiconformal maps and all

corresponding u E M. We say that Lf is the if length (or just the length). For e > 0 we say that / is e-small if Lf < e.

Remark. When we say that an element / G G is e-small, that implies that we are

talking about a hyperbolic element.

Let z ET>. We say that z is moved by the hyperbolic distance d > 0 under u if

d(z, u(z)) = d. Suppose that / is either hyperbolic or parabolic (the same is true for

the corresponding u E M that is a conjugate of /). Then the set of points in D that are moved for some fixed hyperbolic distance under u is either the geodesic (or an

equidistant line) that connects the fixed points of u in the case when u is hyperbolic or it is a horocircle if u is parabolic. If i? is a fixed if-quasiconformal extension of a map i? that conjugates / to some u E M, then we will call the image (under




i?) of the corresponding set, respectively, the quasigeodesic, the quasiequidistant, the quasihorocircle, all of them with respect to i?. Let fibea symmetric crescent

around the axis of a hyperbolic u E M. Set U = i?(R). We say that U is a

quasicrescent. If I is the quasiequidistant that borders U from either side, then we

say that ?7 is determined by Z. Similarly, let H be a horoball that touches Tatx, which is a fixed point of the parabolic transformation u. Set U =

i?(H). We say that U is a quasihoroball. If Z is the quasihorocircle that borders U, then we say that U is determined by Z.

Lemma 4.1. With the notation as above, the following holds. Let f EG such that

f is non-elliptic.

(1) For v E M set Gv = vQv~l. Then Pf(z) =

Pg(v(z)) and Lf =

Lg, where

g ?

y o f

O V_1.

(2) Let I be a quasiequidistant (quasihorocircle) and z E I such that Pf(z) < e.

Then there is a function c(e) =

c(K)(e) > 0, c(e) ?> 0 when e ?? 0, such that for any w EU (U is the quasicrescent (quasihoroball) that corresponds to I), we have Pf(w) < c(e).

(3) There exists ci(e) =

ci(if)(e) > 0 such that ci(e) ?> oo when e ?? 0 and such that if Pf(z) < e, then Pf(w) < 2e for every w ET>, with d(z,w) <

ci(e). (4) Let f E G, Lf < e. Then there is c2(e)

= c2(K)(e) > 0, c2(e)

? 0 when e ? 0, such that if g is a conjugate of f in G, then Lg < c2(e).

(5) Let f E G be parabolic. Then for every e > 0 there is a quasihoroball U

for f such that Pf(z) < e, for z EU. Also, for every r > 0, there exists a

quasihoroball U such that Pf(z) > r, z G D ? U.

Proof. Item (1) follows from the definitions of Pf and Lf. We prove (2). Let U be

a quasicrescent with respect to a if-quasiconformal map i?i, and let i?i G M be the

corresponding hyperbolic transformation such that U = i?i (R) for some symmetric

crescent R that corresponds to ui. Let w E U. Let u0 E M and ô be such that

Pf(w,i?o) =

d(i?Q1(w),uo(i??1(^))) =

Pf(w)- Let R' be the symmetric crescent with respect to i?o such that w belongs to one of the two boundary equidistant lines of R'. Then, if V^1^) E Rf,we have that i??1^) is a bounded hyperbolic distance

away (the upper bound depends only on if because the distance between i?i and

V>o depends only on if) from the equidistant that contains ^^(w). This shows that

Pf(w)<qPf(z,i?o)<qPf(z), for some q = q(K) > 0. If i?^1(z) does not belong to Rl', we have Pf(w) < Pf(z). Either way, this yields (2).

Items (3), (4), and (5) are proved in a very similar way, and they, as does (2), all follow from the basic estimates on the distortion of the hyperbolic metric under

if-quasiconformal maps. We omit the details. D

For a given if-quasisymmetric group G and e > 0, let Gz(e) be the group gener ated by all / G G (f is not elliptic) such that Pf(z) < e.

Lemma 4.2. There exists e(K) > 0 with the following properties. Let G be a if

quasisymmetric group so that every elliptic element in G (if any) is of order two. Then the group ?z(e(if)) is either cyclic or it contains only the identity.




Remark. In fact, our proof works even if we assume that the order of any elliptic element in G is less than some fixed constant C > 2. However, then the constant

e(if) would depend on that constant C as well. Although Qz is generated by hyperbolic and/or parabolic elements, the group Gz could contain elliptic elements. In our case, the order of every such element is two.

Proof. We show that the group Gz is Abelian. Note that every Abelian Fuchsian

group is cyclic. We give a proof by contradiction. Assume that there is a sequence of if-quasisymmetric groups Gn such that for some sequence zn E D we have that neither of the groups GZn 1S Abelian. Here n E N. Therefore, the group GZn has at least two generators fn,gn E G that do not commute (here we used the fact that

every Abelian Fuchsian group, and therefore every quasisymmetric group as well, must be cyclic). We have

as n ?> oo, and neither fn nor gn is elliptic. Set gn? fn? 9n~1 =

fnEGn- Since fn and gn do not commute, we have that f'n and fn do not commute. Also, by Lemma 4.1 we have that Pfn(zn)

?* 0 as n ?> oo. By replacing gn by f'n if necessary, we can assume that both fn and gn are either hyperbolic or both are parabolic. Denote by Gn C GZn the group generated by fn^9n- Since Gn is a sequence of if

quasisymmetric groups, we have that the geometric limit G (of the sequence Gn) is a if-quasiconformal group.

Remark. Here by the geometric limit we mean the following. A homeomorphism h : T ?> T belongs to G if for every n G N we can choose hn E Gn such that

hn ?? h in the C? sense on T. Clearly G is a if-quasisymmetric group.

Our aim is to show that the geometric limit of Gn is a non-elementary and non-discrete if-quasisymmetric group. However, since this limit cannot contain

elliptic elements of arbitrarily high order, we will obtain a contradiction (see [17]). Here we use the fact that a non-discrete if-quasisymmetric group is a conjugate of a

subgroup of M.. For each n E N fix if-quasisymmetric normalized maps i?n and (?n such that there are un, vn G M., so that fn

= i?n?Un0^?n~1 and gn

? 4>n?^n?(?ri~1

By i?n and (?n we denote a choice of if-quasiconformal extensions of i?n and (?n, respectively.

First we consider the case when both fnign are hyperbolic. By Zn and sn we

denote the corresponding quasigeodesics of fn and gn, with respect to i?n and (?n, respectively. Suppose first that there is do > 0 such that 0 < d(ln,sn) < do, for

every n (this includes the case when ln and sn intersect transversally). By (xn, yn) and (xn,yn) we denote the ordered pairs of the repelling and the attracting fixed

points of fn and gn, respectively. Since they do not commute, we have that either

x'n or yn is disjoint from the set {xn,yn}.

Remark. In fact, since the group G is discrete, it is known (see [20]) that non

commuting hyperbolic elements cannot fix the same point.

From Lemma 4.1 we have that the lengths of both fn and gn tend to 0 as n ?? oo. Conjugating the whole group Gn by a M?bius transformation (which does not change the if lengths of hyperbolic elements nor the if distance from the identity), we can assume that xn = ?i, yn = h and either that a;^

= -1 or

x'n ? l. Choose xn

= ?1. Since d(Zn,sn) < do, we have that y'n, after passing to a subsequence if necessary, tends to a point y ET, and y ̂ x'n

? ? 1. Denote this




subsequence by ran. Let G denote the geometric limit of the sequence Grnn, and let F be a Fuchsian group that is conjugate to G. We have that the cyclic group

generated by fmn tends to a one-parameter, hyperbolic, if-quasisymmetric group with the fixed points ?i,i. We have a similar conclusion for gmn. Since F contains two one-parameter, hyperbolic groups that do not have the same fixed points, we

have that F is neither discrete nor elementary. But the only elliptic elements F

may contain are of order two, which is a contradiction.

Now, suppose that d(Zn,sn) ?> oo. Let pn be the quasiequidistant (for fn and

with respect to i?n) that contains zn. If the hyperbolic distance between pn and sn

stays bounded (including the case when pn and sn intersect), asnôo, then we choose a quasiequidistant p'n (for fn) with the following properties: p'n is between pn and Zn, the distances d(pn, ln) and d(p'n, sn) both tend to oo, and for every z E pfn we have that Pgn(z)

?> 0, n ?> oo. We can make this choice because d(Zn, sn) ?? oo and by Lemma 4.1. Note that Pfn (z)

? 0 for every z E p'n, because p'n is between pn and ln (this follows from Lemma 4.1). Now, let qn be a quasiequidistant (for gn and

with respect to <fin) such that pn C\qn is non-empty and such that the interior of the

quasicrescents that correspond to p'n and qn, respectively, have empty intersection. Let wn be any point in p'n D qn. We have that both Pfn(wn) and P9n(wn) tend to 0. By Lemma 4.1 we can assume that wn ? 0.

Let n ?? oo. Since d(Zn, 0) ?> oo, we conclude that both fixed points of fn tend to a single point x E T. Similarly, both fixed points for gn tend to a single point y E T. We want to show that x ^ y. By passing to a subsequence if necessary, we may assume that i?n and c?n converge to if-quasiconformal maps i? and <?. The crescents that correspond to pn and qn converge to quasihorocircles, with respect to i? and <j>, respectively. Note that both of these horocircles contain the origin 0. Since the interiors of the corresponding quasihoroballs have no points in common, we conclude that x ^ y. Similarly as above, after choosing a proper subsequence, we conclude that the geometric limit of Gn (or a properly chosen subsequence), contains two one-parameter parabolic groups that do not have the same fixed point. This implies that this limit group is neither discrete nor elementary. But the only elliptic elements it may contain are of order two, which is a contradiction.

The case when both fn and gn are both parabolic is almost identical to the case when d(Zn, sn) ?> oo above. The only difference is that instead of quasiequidistants we use quasihorocircles.

We have shown that there exists e(if ) > 0 such that the corresponding group Gz is cyclic. D

The above results are analogues of the corresponding results for Fuchsian groups which state that two non-commuting (hyperbolic or parabolic) elements of a Fuch sian group cannot both move a given point for a very small distance. Some of the main results of this sort for Fuchsian groups are the Jorgensen inequality and the Margulis lemma. This lemma is one of the central results in the theory of Lie

groups, and it generalizes these results for Fuchsian groups to discrete lattices in Lie groups (see [24]).

Lemma 4.3. Let J7 be a Fuchsian group. Suppose that u,v E T are elliptic ele ments of order at least three. Then there is e > 0 such that for z,w E D, z ^ w, that are the fixed points ofu and v, respectively, we have d(z, w) > e. The constant e is universal (it does not depend on the choice of T or the elements u,v E T).




Proof This proof follows the same line of argument as the proof of the fact that

every Fuchsian group that contains only elliptic elements must be cyclic. Suppose that z ? 0 (z is the fixed point of u), and denote by g = u~l o v~x o u o v the commutator of u and v. Assume first that the trace of g is greater than two. Then

g is a hyperbolic transformation. Moreover, the length of g is less than 8, where <5 ?> 0 when e ?> 0. Set / = i?_1 o g ou. Since u is not of order two, we have that

/ and g are hyperbolic elements that do not commute. Taking e small enough, we obtain a contradiction from the Jorgensen inequality (or any other result of that

sort). Therefore, we have that the trace of g is equal to two. But this yields that

z = w = 0, which is a contradiction. D

4.2. Removing small hyperbolic elements.

Theorem 4.1. There exists e(K) > 0 with the following properties. For an arbi

trary K-quasisymmetric group G that does not contain any elliptic elements of order three or more, there exist Ki-quasisymmetric groups Gi, i G N; ifi = ifi (if), such that the following hold.

(1) None of the groups Gi contain any e(K)-small hyperbolic elements nor any elliptic elements of order three or more.

(2) If every Gi is if'-quasisymmetrically conjugated to a Fuchsian group, K' =

K'(K), then there exists if" = if" (if) such that G is K" -quasisymmetri cally conjugated to a Fuchsian group.

Proof. Let G be an arbitrary if-quasisymmetric group. We will construct the

groups Gi- Providing that these groups are quasisymmetrically conjugated to the

corresponding Fuchsian groups, we construct a homomorphism E of G into the

group of quasiconformal selfmaps of D such that E is an isomorphism onto its

image.

Let e > 0, and let hi,.., h^... E G, i E I, be the list of all primitive, mutually non

conjugate, e-small hyperbolic elements. Here I is either the set 1,2, ...,n, for some

n E N, or I = N, depending on whether there are finitely or infinitely many hi. Recall that hi is a primitive element means that hi is not a power of another element in G- Set [hi]

= {JkeziUfeg f1 ? hi ? /) I11 eacn câss M nx one representative

that is primitive, say pi. Let Xi,yi denote, respectively, the repelling and the

attracting fixed points of ft?. Let Stabi be the subgroup of G whose elements fix the set {xi,yi}. This is an elementary group and therefore there exist an elementary Fuchsian group Staty and a if-quasisymmetric map i?i : T ?

T, which conjugates

Stab'i and Stabi. We can choose i?i so that i?i(xi)

? Xi, i?i(yi) = y?. Let i?i be some if

quasiconformal extension of i?i. Let Ei(l) be the symmetric crescent of hyperbolic width 1 around the geodesic that connects x? and yi, and set Ui(l)

= i?i(Ei(l)).

Now fix i E I. For every h E Stabi put h = i?i o Ui o i?^1, where i?? G Staty such that h = i?i o m o i?i~ . For /, g G ?, we say that f ~ g, with respect to the pair (xi,yi), if f(xi)

= g(xi) and f(yi)

= g(yi). Denote the corresponding

equivalence class by [/]?. Note that for / ~ h, where h E Stabi, we have already defined the extension / because each such / must be in the corresponding stabilizer

Stabi. In every other equivalence class choose one representative, say /, and let

/ be a fixed if-quasiconformal extension. For every other g E [f]i, there exists




h G Stabi such that f'1 o g = h. Set g = f o h. Therefore, for / G G and for

fixed i G I, we have defined the extension /, which is if3-quasiconformal. Set

[[/"?(I)] =

Ufeg f(Ui(l)). Now repeat this process for each i. Note that formally we should denote the extension / by fi, because a separate extension is defined for

every i G I. We avoid doing this to simplify the notation, and no confusion should arise.

By Lemma 4.1 and Lemma 4.2 there exists ei(if) > 0 small enough that if we set e = ei(if), then for every / G G we have

d(f{Uj{\)),g{Uk{l))) > 1, whenever f(Uj(l)) and #(i7fc(l)) do not touch the unit circle T at the same points

(otherwise, for every e > 0, there would be z E D such that Gz is not cyclic). Note that this choice of e yields that f(xi)

= g(xi) implies f(yi)

= g(yi), which we

already know to be true. Let R =

R(K) be the constant from Lemma 3.1. By the same argument as above, we can choose e2(K) > 0 and the symmetric crescent Ei (Ei contains the crescent

Ei(l)) with the following properties. Set ?/? = i?i(Ei) and [Ui] = [jfeg f(Ui). If / G Stabi, then

(4.1) e2(K)<d(z,f(z)), for any z E dUi. If f,g E G are such that f(xj) ^ g(xk) (or equivalently f(yj) ^ g(yk)), where j, k E I are any two numbers (j, k may be equal), then

(4.2) d(f(Uj),g(Uk))>R. Set ?? = D ?

Utî]- We have that Q is a disconnected set, but any connected

component of Q is simply connected. This follows from (4.2). Since G is not an

elementary group, there are infinitely many connected components of ??. If f?o is a

connected component of Cl, then Qo is uniquely determined by its boundary points that lie in T. Therefore, if / G ?, we can properly define the connected component /*i?o of f?. We denote by ?j, j E V, mutually non-conjugated components such that every other component is in one of the classes [ilj]

= U/ee f*f?j)- Here I' is

the range for j, and it may be either a finite set or I' = N (see Figure 2). For / G G, we first define E(f) on each [Ui]. Let U[ be a component of [Ui]

(U? touches T at x'^y?. Set x'? =

f(x'i), y'? =

f(y[). Let g',g" E G be such that

g'(xi) =

x\, g'(yi) =

y[, and gn(xi) =

x'?, gn(yi) =

y'[ (such g',g" E G are not

uniquely determined). Then, there exists h G Stabi such that g"ohog'~ = f. Set

for z EU[. This defines E(f) on \J[Ui] = D - ?X It follows from the definition of

the corresponding extensions of g',g" that E(f) is well defined.

Lemma 4.4. E(f) is a homeomorphism of[J[Ui]. Also,

(1) E(f) o E{g) = E(f o g); E(id) - id; (2) for every f EG, there exist ifi = ifi (if) and a ifi-quasiconformal map f

which extends f such that f = E(f) on U[?/?] = D ? f?.

Proof. Item (1) and the fact that E(f) is a homeomorphism follow from the defini tion of E and the discussion above. On each component of [Ui], the map E(f) is a




x'j Vk

Figure 2

restriction of a if8-quasiconformal map g" o h o g' . Combining this with (4.2), it

follows from Lemma 3.1 that there exists the map f with the stated properties. D

Next, we define E on the rest of D. In each class [?j], j E V, fix one repre

sentative, say ?j. For f E G, f*?j =

?j, let e(f) be the restriction of the map

E(f) on d?j. Let Stj be the group of all maps e(f). The group Stj is isomorphic to the subgroup of G that contains all / so that f*?j

= ?j. From Lemma 4.4 it

follows that Stj is a ifi-quasisymmetric group on ?j. By conjugating the group

Stj via the boundary values of the Riemann map that maps ?j to the unit disc, we

obtain the ifi-quasisymmetric group Gj- We get from (4.1), from Lemma 4.1, and from the definition of ? that there is an 63 (if) > 0 such that Gj does not contain

e3(if)-small hyperbolic elements, for some 63 (if) > 0. Note that for a hyperbolic f E G such that e(f) E Stj, we have that e(f) is hyperbolic. It follows from the definition that the length of / is greater than or equal to the corresponding length of e(f). This follows from the fact that a conformai map of D to a subset of D is a contraction with respect to the hyperbolic metric on D. By repeating this for

every j, we obtain the collection of groups from the statement of this theorem. It follows from the construction that none of them has elliptic elements of order three or more (because G does not have any such elliptic elements).

If we assume that Gj is if'-quasiconformally conjugated to a Fuchsian group, then there exist if2 = if2(if ), a Fuchsian group St^ (Stj

acts on D of course), and

a if2-quasiconformal map i?j : D ?> ?j such that the induced map i?j : T ?

d?j conjugates the groups St[ and SU.



QUASISYMMETRIC groups 697

Now fix i G I. For every e(f) E Stj, we put / =

i?j o Ui o i?i x, where u E

St'j, such that e(f)

= i? j o u o i?j'1. For f,g E G, we say that /

~ g with respect to ?j, if /*(?2?)

= <7*(fi?). Denote the corresponding equivalence class by [/]?.

Note that for f ~ h, where e(/i) G St?, we have already defined the extension /, because for each such /, the corresponding map e(f) must be in the stabilizer SU.

In every other equivalence class, choose one representative, say /, and let / be the

ifi-quasiconformal map from Lemma 4.4 (note that / agrees with E(f) on U[^D For every other g E [f]i there exists h E SU such that f~l o g = h. Set g = f o h.

Therefore, for / G (?, we have defined the extension / which is if3-quasiconformal,

if3 = KS(K). Let

?'j be a component of [?j]. Set ?" =

f*?'j. Let g',g" E G be such that

g'+?j =

?'j and g"?j =

?'j (such g',g" G G are not uniquely determined). Then,

there is an h E G-, h*?j =

?j, such that g" ohog'~ = /. Set

(?o?o5'-1) (*)=?(/)(*), for z E ?'j.

This defines E(f) on fi. It is clear from the above discussion that E

satisfies all the required properties. In particular there exists if[ =

K[ (if) such that

E(G) is a if (-quasiconformal group, which implies that G is if "-quasisymmetrically

conjugated to a Fuchsian group, for some if" = if"(if). This completes the proof of Theorem 4.1. D

5. Hyperbolic quasisymmetric groups

Throughout this section G denotes a if-quasisymmetric group all of whose el

ements are hyperbolic. We also assume that for some 0 < e(K) < e(K), G does

not contain any e(K)-small elements. The constant e(K) is from Theorem 4.1. We

show that such a group is a quasisymmetric conjugate of a Fuchsian group.

5.1. The pants-annuli decomposition of a quasisymmetric group. Recall

that if T is a Fuchsian group, then T acts on T, and the corresponding group that

acts on D is denoted by T. We first assume that G is finitely generated. Let T be a Fuchsian group and T a homeomorphism such that G = ^pT^p~x. Then

T is finitely generated, and therefore D/J7 = S is a topologically finite Riemann

surface. This means that S is biholomorphic to a closed Riemann surface of finite

genus with at most finitely many discs removed.

Remark. The only reason why we temporarily assumed that G is finitely generated is that we can more clearly describe the pants-annuli decomposition of the surface S

(see [24]). A similar decomposition is valid for all Riemann surfaces, but we choose

to work with the finitely generated case and when needed, we apply Proposition 1.3.

Consider the induced hyperbolic metric on S (this metric agrees with the hyper bolic metric that S inherits from D/.F

= S). We recall the standard decomposition

of a topologically finite Riemann surface and the corresponding lift into the univer

sal cover. Let So be the convex core of S (if S is closed, then So = S). S is a union

of So, the simple closed geodesies (in the future just geodesies) that represent the

boundary of So, and the annuli (each annulus is bounded by a geodesic which is

also a boundary component of Sq and a boundary component of S). Now, cut up




the surface So into pairs of pants (recall that a pair of pants is an open Riemann surface which is biholomorphic to the Riemann sphere minus three discs). We have that S is a disjoint union of the pairs of pants, the annuli, and the geodesies that border the pairs of pants (some of them also border the annuli if they exist). De

note by Pi, 1 < i < ni, the pairs of pants; by Ai, 1 < i < n2, the annuli; and by Z?, 1 < i < n$, the geodesies that border the pairs of pants in this decomposition of S. Here, n\, n2,n% G N depend on the Euler number of S and the number of

free boundary components of S. We denote by [Pi], [Ai], and [Z?] the totality of the

corresponding lifts (the ?'s run throughout the corresponding ranges) in D.

Each connected component in D of a fixed [Pi] is a convex subset of D, whose relative boundary (in D) is contained in the collection of geodesies which are the

lifts of geodesies on S that border Pi (this is a subset of the union of [li], 1 < i < ns, which we denote by U&D- We also denote by P? the closure of a fixed single lift

of a given pair of pants Pi in S. Here Pi is the closure of a chosen fundamental

region for the corresponding pair of pants. We choose it so that Pi is a right-angled

octagon. Four sides of this octagon are contained in four geodesies from \J[h], and the other four sides are geodesic arcs that are orthogonal to the first four sides, so

that all together they complete the boundary of the right-angled octagon Pi. Each

[Pi] is equal to the union \Ju(Pi), where u E T (this union is disjoint except that

the points which belong to certain sides of P? may occur more than once in this

union). Note that a fixed (connected) component of [Pi] contains many copies of

Pi (see Figure 3). A connected component of a fixed [Ai] is a hyperbolic halfspace bounded by a

lift of the geodesic in S that borders ??. We also denote by Ai the closure of a

fixed single lift of a given annulus Ai C S. Again, Ai is the closure of a chosen

fundamental domain of the corresponding annulus. We choose it so that Ai is a

right-angled parallelogram. One of its sides is contained in a geodesic from |J[Z?], its opposite side is an arc contained in T, and the remaining two sides are geodesic rays that connect the first two sides to complete the boundary of the right-angled

parallelogram Ai. Note that each [Ai] is the union of u(Ai), where u E T (this union is disjoint except that points in certain sides of Ai appear twice). Note that

a fixed (connected) component of [Ai] contains many copies of Ai. A connected component of a fixed [li] is a geodesic in D. We also denote by Z?

a fixed connected component (or just a component) of [ZJ (note that this is much more than a fundamental domain of the corresponding closed geodesic on S). In

this case we also have that [li] is equal to the union \Jêû(li) =

UuejrU*(li), but this union is not disjoint. Note that to each k corresponds an infinite cyclic subgroup of the group T, which consists of the elements of T that fix its endpoints.

We denote the generator of this subgroup by hr. Also, if u is an element of T

which fixes the endpoints of Z?, then u belongs to this cyclic subgroup. T has only

hyperbolic elements; therefore, no element of T can permute the endpoints of Z?. Note that no two geodesies from \J[k] intersect.

In the remainder of this section P?, Ai will always refer to a choice of single lifts

(described above) of the corresponding pair of pants and annulus, while k will stand

for a fixed component in [li]. Now, fix Z?. Since *(??). We set [k] =

\Jfe? /*(/*) =

Uue.7^*(u*(^)) Also, to each Z? corresponds an infinite cyclic subgroup of G whose elements fix its

endpoints. We denote the generator of this subgroup by h\i. We have

h\i =

(f o hj o

(p^1.

Note that if f(li) = li, for some f E G, then because G contains only hyperbolic

elements, we have that / belongs to the cyclic subgroup generated by h?^ Similarly as above, no element of G can permute the endpoints of Z?. Again, no two geodesies in U[k] intersect.

5.2. Extending the action of G to the unit disc. Our aim is to extend the action of G to D. We will define a homomorphism E (which is an isomorphism onto its image) of the group G into the group of quasiisometries of D, so that for each / G G, E(f) extends / to D. The image of the group G under E is denoted

simply by E(G) . We do this in steps. First we define E(f) for every / G G, on

every [li], 1 < i < 77,3, and then on every [P?] and [Ai] (see below for the definition of [Pi] and [Ai]).

Fix i E [1,723], and let Z? be a fixed component of the set [li]. Recall that hi. is the generator of the cyclic group that fixes the endpoints of Z?. Let i? be a

if-quasisymmetric map and u E M such that i? o u o i?-1 = hi.. Let i? be a if

quasiconformal and bilipschitz extension of i?. Set i? o u o i?-1 =

h^. Since i? is

bilipschitz, we conclude that hii is bilipschitz, where the corresponding constant is a function of if. For f,g EG, we say / ~ g if /*(Z?)

? g*(h). Denote by [/]? the

corresponding equivalence class. Fix one / G [/]?. Let / be a if[-quasiconformal map, if{

= if ((if), which extends / and which maps li onto the geodesic /*Z? (such exists by Lemma 3.5). We also assume that / is bilipschitz, where the corresponding




constant is a function of if. For an arbitrary g E [f]i, there is k E Z, such that ^ _/~

f-1 o g = hf.. Set g = f o /??. . Therefore, for each / G ?, we have defined a

ifi-quasiconformal extension, ifi = ifi (if). Now, repeat this for every i. Let / be an arbitrary element of G, and let l\ be a component from [li] (l[ is not

necessarily equal to Z? which is fixed). Let Z" = /*(Z?)- Choose g',g" E G such that

9*(h) ?

li and g"(k) =

l'? (such g',g" are not unique). Since (g"~ ofog')*(li) = li,

there exists k E Z such that g"~ o f o g' =

/??fc. We set

(5.1) E1(f)(z) = (?pohlogr1")(z) for z E l[. Repeat this for every i. In this way we have defined the mapping Ei(f) for every point in every geodesic from any of the [li], 1 < i < 723. Since no two

geodesies from \J[h] intersect, Ei(f) is well defined. Also, this definition does not

depend on the choice of g', g" because of the way we have defined the corresponding extensions.

We need to slightly modify Ex(f) (to obtain E(f)), f E G, so that E(f) is ?(e) continuous on \J[h], where 8 : R+ ?> R+ is a function that is itself a function of if. We have already shown that the restriction of Ei (/) on every geodesic from

\J[h] is bilipschitz. Let eo,ei > 0 such that 0 < ei < e0 < \. Let 1,1' be two

geodesies from \J[k] such that d(Z,Z') < e0. Let a C Z be a subarc of Z such that for z E (I

? a) we have that the distance between z and I' is greater than ei. We

similarly define a' C I'. Denote by h,h' E G the corresponding elements that fix the

endpoints of Z and I', respectively (h, h' generate the corresponding cyclic groups). Then there exists Lo =

Lo(eo, if) such that the lengths of h and h' are greater than

Lq. Moreover, we have that L0 ?> 00, for eo ?? 0 (if not, we would have that the

geodesic hj' E [j[h] intersects the geodesic V). This implies that we can choose

0 < ei < e0, both of them being functions of if, such that Ei (h)(a) is disjoint from a. Similarly, Ei(h')(a') is disjoint from a'. For ei small enough, the hyperbolic length of both a and a' is greater than 1. Now fix eo, ei with the above properties.

For f E G, let ctf =

Ei(f)(a), a'f =

Ei(f)(a'). Since / is if-quasisymmetric and since the value of Ei(f) on Z is a restriction of a bilipschitz map (see (5.1)) of D onto itself, it follows that there exists e2 = e2(K) such that for any z E (fj

? af)

we have that d(z, fj') > e2. We have the analogous conclusion for a'*.

We now define a new map E(f) that agrees with Ei(f) on \J[k] ?

U/e? a/- Let

If : ctf ?> a'* be the affine map that maps each endpoint of a/ to the closer of the

two endpoints of a'f.

If a/ is very long, then so is a'f,

and i/ is nearly an isometry. In any case i/ is bilipschitz (with the constant that is a function of if) regardless of the length of a/ (since the hyperbolic length of a is at least one, we conclude

that the length of oj/ is not too small). For f E G and z E a' = a'id set

E(f)(z) = (IfoE1(f)olrdi)(z).

Let g gG, and let z 6 a'f,

for some / EG- Set

E(g)(z) = E(gof)oE(f)-1. Since Ei(h)(a) fl a is an empty set, it follows that E(f) is well defined. Let

h,l'i E G be a pair of geodesies that is not in the orbit of the pair 1,1' and such

that d(Zi,Zi) < eo- Repeat the same process. The operator E(f) is conjugated to

Ei(f), and therefore it respects the group structure.




By this, we have defined the extension E(f), f EG, which is a modification of

Ei(f). It follows from the construction that E(f) is ?(e)-continuous, where 8(e) is a function of if.

Lemma 5.1. Let f,g E G and let id denote the identity mapping. There exist

L = L(K), 8 : R+ ?> R+7 p = p(K), and an L-quasiisometry f which extends f

such that

(1) E(f) is a homeomorphism of every geodesic from \J[h]; (2) E(f) is 8(e)-continuous on \J[k], where 8(e) is a function of if; (3) for every z E \J[k], f EG, we have d(E(f)(z),z) > p; (4) E(f o g) = E(f) o E(g), and E(id) = id; (5) / agrees with E(f) on \J[k]; (6) / is 8(e)-continuous on D.

Proof. It follows from (5.1) that Ex(f o g) = Ex(f) o Ex(g) and Ex(id) = id. The modification preserves this property. We have already proved that E(f) is a

uniformly continuous homeomorphisms of \J[h]. The existence of the quasiisometry

/ follows from Lemma 2.2. Let Z G \J[h], and let h EG be the corresponding hyperbolic element. Let z El.

If E(f)(z) is very close to z, then z cannot be in Z since h cannot be e(if)-small. This implies that / does not preserve Z. Let I' = fj be the geodesic from \J[h] that contains E(z). Since Z and I' are disjoint, we have that the corresponding (repelling and attracting, respectively) endpoints of Z and I' are very close to each other (because z and E(z) are very close). But this implies again that the length of / is very small, which is a contradiction (recall that we assume that G does not contain any e(K)-small hyperbolic elements). This shows the existence of p. D

Next, we define E(f) on the rest of the unit disc. To do this, we first need to define an appropriate extension (p : D ?> D of the map (p. Since no two geodesies from \J[h] intersect and since (f*(k)

= h, we can choose a homeomorphism (p so

that (p(li) = li and so that the equality

(5.2) ?p o hj = /i/, o (p

holds on li, for every Z?. Because of (5.2) one can arrange that for every geodesic arc

a that is a side of some octagon P\ E [Pi], we have that (p(cx) is also a geodesic arc

(if a is contained in one of the geodesies from \J[k], then this is already the case). This directly follows from the fact that the endpoints of a belong to two geodesies from \J[li], and no two such sides can intersect (except that they can meet at the

endpoints which are always in (JM)- Similarly, if a is a side in the parallelogram

A'i E [Ai], we can arrange that (p(a) is a geodesic arc, geodesic ray, or a circular arc in T, depending on which one of these is a.

We set (p(Pi) = Pi and (p(Ai) = Ai, for every P? (i G [l,î]) and Ai (i G [l,n2]). Note that each P? is an octagon and each Ai is a parallelogram (P? and Ai are not

necessarily right-angled anymore). Also, (p([Pi]) =

[Pi] and ^([A?)] =

[Ai], Since D is a union of \J[Pi] and (J[^]> we nave tnat tne same is true for [P?] and [Ai],

Lemma 5.2. Let f E G and fix Pi, for some i E [l,ni]. Then we can choose

Li = Li(K), pi = pi(K), and an Li-quasiisometry f, which extends f such that

the following hold.




(1) / agrees with E(f) on (J[Z?]. (2) f(Pi)

= ?p(u(Pi)), where u = (p'1 o f o cp.

(3) Let f EG be such that for the above-defined extension f we have f(Pi)C\Pi =

?, where ? is a side of Pi. Then, ?/_1 o (/_1) J (z) = z, for z E ?.

(4) / is 81(e)-continuous, where 8i : R+ ?> R+ is a function of if.

(5) For every z G D, we have d(f(z), z) > pi.

Also, for a fixed Aif i E [l,n2], we can choose Li = Li(K), pi = pi(K), and a

Li-quasiisometry f, which extends f, such that the following hold.

(1) f agrees with E(f) on \J[k]. (2) f(Aj)

= (p(u(Ai)), where u = (p'1 ofo(p.

(3) Let f E G be such that for the above-defined extension f we have f(Ai) C\

Ai = ?, where ? is a side of Ai. Then, ? f~l o (/-1)j (z)

= z, for z E ?.

(4) / is 8i(e) -continuous, where 8i : R+ ?> R+ is a function of if.

(5) For every z E D, we have d(f(z), z) > pi.

Proof. Let / be the L-quasiisometry from Lemma 5.1. Here / already agrees with

E(f) on \J[k]. This implies that the octagons /(P?) and (p(u(Pi)) have the cor responding four sides in common. Now, by Lemma 2.2 we can choose an Li

quasiisometry /, which agrees with / on \J[k], such that for a fixed P? we have

f(Pi) =

(p(u(Pi)) and such that / is ?i(e)-continuous. Let / G G be such that for the above-defined extension / we have /(P?) H Pi = ?,

where ? is a side of P?. First we define the extension ui of /_1 which satisfies (1),

(2), and (4) of this lemma. Then we post-compose Hi with a homeomorphism H2 of D which pointwise fixes the set \J[k] and all the sides of P? except ?. Since the

hyperbolic distance between i?i and (/)_1 is bounded above (by a bound which is a

function of if), we can choose H2 to be a quasiisometry and uniformly continuous, so that H2o Hi = f~l satisfies (3).

Item (5) follows from Lemma 5.1. We proceed similarly to define the / that

corresponds to a fixed Ai. D

Let f E G and let z E D. If z E [k], then we have already defined E(f)(z). Suppose z E [Pi]. Then there exists P? C [P?] that contains z. If z belongs to a

side of the octagon P[, then there will exist at least two and at most three (this can happen only in the case when z is in some [Z?]) octagons from this partition that contain z. Recall that four sides of the octagon P[ lie in (JM Since E(f) is

defined there, E(f) uniquely determines the octagon P" = E(f)*(P?) whose four

sides are the images of the four sides of P[ under E(f). In the same way we can

find the elements g',g" E G such that g'^(Pi) =

P[ and g'i(Pi) = P" (such g',g" are

unique). We define E(f)(z) by

(5-3) E(f)(z) = (ôg~1)(z), where g' and g" are the extensions from Lemma 5.2 of g' and g", respectively (these extensions correspond to the fixed Pi). It follows from Lemma 5.2 that if z belongs to a side of P[, then the definition of E(f)(z) does not depend on the choice of

the octagon which contains z. This, together with Lemma 5.1, shows that E(f)




is continuous and therefore a homeomorphism on every [Pi]. We similarly define

E(f) on every [Ai].

Lemma 5.3. Let f E G- Then E(f) is an L2-quasiisometry ofD, L2 = L2(K),

and E : G -* E(G) is an isomorphism. E(f) is 82(e)-continuous, where 82(e) is a

function of if. Also, there exist pi = pi(K) such that d(E(f)(z),z) > pi.

Proof. The fact that E(f) is an isomorphism and #2(e)-continuous follows from the definition of E. It also follows from the definition of E that there exists L2

= L'2(K)

with the following properties. For every z E D, there is an L^-quasiisometry / that extends / such that E(f)(z)

= f(z) (at every step of the way we have always

defined E to be a restriction of some quasiisometric map). The rest follows from

Proposition 2.1, Lemma 5.1, and Lemma 5.2. D

Theorem 5.1. Let G be a if-quasisymmetric group that does not contain anye(K) small elements for some 0 < e(K) < e(if) (e(K) is the constant from Theorem J^.l). Then there exist a Fuchsian group T, if3 = if3 (if), and a if3-quasisymmetric mapping (p : T ?> T such that G = ^pT^p"1.

Proof. If G is finitely generated, we have established above that E(G) is an L2

quasiisometric group (this means that every element of E(G) is an L2-quasiisometry) so that d(E(f)(z), z) > pi and E(f) is ?2(e)-continuous. By Lemma 2.3 there exists a if3-quasisymmetric mapping <p : T ?? T such that G = <p)T^p~x, for some Fuchsian

group T. The case of infinitely generated G follows from Proposition 1.3. D

6. Torsion-free quasisymmetric groups

Let G be a torsion-free if-quasisymmetric group and assume that G does not con

tain any e(if )-small elements, for some 0 < e(K) < e(K). Here e(if ) is the constant from Theorem 4.1. We show in this section that such a group is a quasisymmetric conjugate of a Fuchsian group. Our aim is to construct a homomorphism E (which is an isomorphism onto its image) of G into the group of quasiconformal selfrnap pings of D. We first assume that G is finitely generated. Since G is topologically conjugate to a Fuchsian group, we conclude that there are pi, ..,pn E G, n E N,

mutually non-conjugate parabolic elements such that every other parabolic element of G is contained in some conjugacy class [p?]

= Ufcez(U/e? f1 ? Pi ? /) Note

that this implies that each pi is primitive; that is, it is not a power of another element from G- Also, if f E G fixes the fixed point of some pi, then / belongs to the cyclic group generated by pi. In each class [p?] fix one representative that is

primitive, say pi, and let Xi E T be the point fixed by pi. There exists a parabolic M?bius transformation Ui and a if-quasisymmetric map i?i : T ? T such that

i?i o m = Pi o i?i (also i?i(xi) =

Xi). Let i?i be some if-quasiconformal extension of

i?i. Let Hi be a horoball that touches T at Xi and set Ui = i?i(Hi). Now, fix i E [l,n]. Let /

= p\, for some k E Z. Let / be a if2-quasiconformal

extension of / defined by / = p% =

i?i o u* o i?r1. For f,g E G, we say that f ~ g

with respect to Xi, if f(xi) =

g(xi). Denote the corresponding equivalence class by

[f]i. Note that for / ~ pi we have already defined the extension / because each such

/ must be in the corresponding cyclic group. In every other equivalence class [/]? choose one representative, say /, and let / be a fixed if-quasiconformal extension. For every other g E [f]i there exist k E Z such that f"1 o g = Pik. Set g = f o j5??\




Therefore, for / G G we have defined the extension / which is if 3-quasiconformal. Set [Ui]

= U/e? f(Ui)- Now, repeat this process for each i (see Figure 4).

Let R ? R(K) be the constant from Lemma 3.1. By Lemma 4.1 and Lemma

4.2, we can choose the horoballs Hi, i E [l,n], so that if f,g E G are such that

f(xj) ^ g(xk), where j, k E [1, n] are any two numbers (j, k may be equal), then

(6.1) d(f(Uj),g(Uk))>R. Let U' be a component of U[t/?], and let f E G be the parabolic element that fixes U' (such exists by construction). Since R is fixed and by choosing the horoballs

properly, we can also arrange (see Lemma 4.1, item (5)) that

(6.2) d(z,f(z))>e1(K)>0, for z from the quasihorocircle dU' and some constant ei(if ) > 0. Set ? =

D?|J[[/?]. It follows from (6.1) that ? is a simply connected region whose boundary consists

ofTand|J0p7?]. For / G G we first define E(f) on each [Ui]. Let U[ be a component of [Ui] (U[

touches T at x^). Set x'( =

f(x'i). Let g',g" E G be such that g'(xi) =

x\ and

g"(xi) =

x'l (such g',g" E G are not uniquely determined). Then, there is k E Z

such that g" o p\

o g'~x = f. Set

ôfiOgrlyz) = E(f)(z),

for z eU[. Repeat this construction for every i E [1, n]. This defines E(f) on \J[Ui]

= D ? ?. From the definition of the extensions g' and g", it follows that E(f)(z) is well defined; that is, it does not depend on the choice of g',g".

Lemma 6.1. E(f) is a homeomorphism of\J[Ui] such that the following hold.

(1) E(f) o E(g) = E(f o g); E(id) = id. (2) For every f E G there exist if' =

if'(if) and a if'-quasiconformal map

f : D ?? D, which extends f, such that f = E(f) on \J[Ui] = D ? ?.

Proof. Item (1) and the fact that E(f) is a homeomorphism follow from the defi nition of E and the discussion above. On each component of a given [Ui], the map

E(f) is a restriction of a if8-quasiconformal map g" o p% o g' . Combining this

with (6.1), it follows from Lemma 3.1 that there exists a map / with the stated

properties. D

Let Gi be the group (isomorphic to the group G) whose elements are the map

pings e(f) : d? ?> d?, where e(f) is the restriction of E(f). Let </> : ? ?? D be the Riemann map. Then <?Gi<?~x is a if'-quasisymmetric group, where if' is from Lemma 6.1. Note that the group <?Gi<?~x contains only hyperbolic elements. More

over, we have assumed that G does not contain any e(if )-small elements. Then from (6.2) it follows that there is e(K) > 0 such that (?Gi^"1 does not contain

any e(K)-small elements. From Theorem 5.1 we conclude that the group (?Gi<?~x is

ifi-quasisymmetrically conjugated to a Fuchsian group, ifi = ifi (if). Conjugating the action of this Fuchsian group by the Riemann map, we get that for each / G G

there is a ifi-quasiconformal map e(f) : ? ?> ?, which extends e(f) to ? and such

that the map e(f) ?

e(f) is an isomorphism of the group Gi onto a subgroup of




Figure 4

the group of quasiconformal selfmaps of ?. Set E(f) =

e(f) on ?, for each f EG Therefore, we have constructed an isomorphism of G onto a subgroup of the group of quasiconformal selfmaps of D. Moreover, each E(f) is if {-quasiconformal, where

if{ =

if{ (if ) is the maximum of the set {ifi, K8}.

Theorem 6.1. Let G be a torsion-free K-quasisymmetric group that does not con tain any e(K)-small elements, for some 0 < e(if) < e(if). Then there exist

K2 = if2(if) and a K2-quasisymmetric map (p : T ?? T such that G ? tpTtp~x

for some Fuchsian group T.

Proof. If G is finitely generated, then the existence of a if2-quasisymmetric map (p and a Fuchsian group T follows from the fact that E(G) is a if {-quasiconformal group. The infinitely generated case follows from Proposition 1.3. D

7. Elliptic elements of order greater than 2 and the barycentric extension

7.1. Quasisymmetric groups with elliptic elements of order at least three. For integer m > 3 let u(z)

= exp(^)z be the rotation, z E T (exp will sometime

stand for the exponential function). Denote by Ou the orbit of the point z ? 1.

Then, irrespective of the order m, we can choose three points Xi,x2,x^ E Ou such

that J2 < a(xi,Xj) < n ? f^, for every i ̂ j (j^ is not the best bound). Here

a denotes the standard spherical metric on T. By la we will denote the spherical length.

Let ip : T ?> T be a homeomorphism, normalized by (p(xi) = Xi, i = 1, 2, 3, such

that the cyclic group generated by (pouo<??_1 is a if-quasisymmetric group (in this section the symbol ip always refers to a homeomorphism with these properties).




From Proposition 1.2 we have that ip = i? o </>, where <?> fixes pointwise the set Ou and (? commutes with u (or, equivalently, <? conjugates u to itself) and i? is a

if-quasisymmetric map. Clearly i?(xi) = x?, i = 1,2,3.

Set Zi = Zi(if) = sup la(i?(a)), where the supremum is taken with respect to

all arcs a C T of length at most 2tt ? f^ and all if-quasisymmetric maps i? : T ?> T that fix three points, say Xi,x2,x$, such that j^ < o~(x?,Xj) < n ?

j^, i t? j. Since this family of if-quasisymmetric maps is a normal family, we have 0 < Zi < 2-K. On the other hand, let ? C T be an arc of length at least 2tt ? ^. Let l2 ? infm>3 lcr((?(?)), where the infimum is taken with respect to all such ? and all

homeomorphisms (? which commute with the rotation u(z) =

exp(^)z. Clearly l2 > tt. This proves the next lemma.

Lemma 7.1. With the notation as above, we have the following. Let a be an arc

of spherical length at least l\. Then the spherical length lcr(ip~1(a)) is at least l2. Here l2 is a fixed constant (does not depend on K).

Set 1'2 = l2

? (1-2y!L)> Recall that for z E D, az is the hyperbolic M?bius trans

formation such that az(Q) = z and which preserves the geodesic that contains both

0 and z. Let do > 0 be small enough such that for every z E A(0, do) we have

l<j(az(a)) > 1'2, for every arc a of length at least Z2. Denote by (p the barycentric extension of the homeomorphism (p defined above.

Lemma 7.2. With the notation as above, the following hold.

(1) There exists ro = ro(K) > 0 so that for every z E A(0, do) we have

d((p(z),0)<ro. (2) (p is Ki-quasiconformal in A(0, di). Here, di = di(K), ifi = ifi (if).

Proof. Let r0 > 0 be such that for every z E D, d(z, 0) > r0, there exists an arc a C T of spherical length at least Zi such that 0 < la(az(a)) < 1 ? y. Clearly ro < 00. Let w = (p(z) forz E A(0, do). Then the barycentric extension of the map aw~x oipoaz is equal to a"1 o (poaz. Moreover (a"1

o (p o az) (0) = 0. This implies

(see [4] and [1])

(7.1) J

(aw-1oiPoaz)(Q\dC>\=0.

Suppose that there exists z E A(0, do) such that d(ip(z),0) =

d(w, 0) > r0. There exists an arc a C T, la(a) > h, with the property that la(a~1(a)) < 1 ? y. Let y be any point in a"1 (a). Then for ?

? (tpoaz)~1(a) we have

(7.2) ijm-jiaôipoaz)

(OKI < lA?) (l

- Y2)

< 2(*2 -* O

Here we used that la(?) > l2> This follows from Lemma 7.1 and the definition of

1'2. But then (7.2) contradicts (7.1). This proves (1). Let v : R ? R be an increasing function such that ip(z)

= <p(elt)

= elv(?\ Since 4> commutes with a rotation u of order at least three and since i? is a normalized

if-quasiconformal map (with fixed points xi,x2,x$), we conclude that there exists C =

C(K) such that v(t + s) -

v(t) > C, for ̂ < s < n, and every t E K. Now

(2) follows from Lemma 3.5. D




Let /i be an element of the unit ball of the Banach space L??(D) of essentially bounded measurable functions on D. Set fco = T^pp

For t E [0, fco], let rjt : D ?> D

be the quasiconformal map with the complex dilatation t/i and which fixes the

points xi,x2, #3 as above. Let rjt : T ?> T be the map such that rjt extends r?t. Set

<Pt = Vt0 <P- By fit we denote the barycentric extension of Kp?.

Lemma 7.3. With the notation as above, we have the following. Let h : [0, fco] ?* D be the curve h(t)

= (rj^1

o (pt)(0), t E [0, fco]. Then for every e > 0 there exists

?(if,e) > 0 such that d(h(t),h(s)) < e, for \t ?

s\ < ?(if,e). In other words, h(t) is 8{if, e)-continuous.

Proof. It is enough to prove the statement of the lemma for the curve e(t) =

f>t(0) This follows from the fact that the family rjt is uniformly Holder continuous with

respect to both t E [0, fco] and the variable in any fixed compact set in D and that bound depends only on fco and that fixed compact set in D. It follows from the

proof below that (pt(0) does not leave the disc A(0,ro). Proof by contradiction. Suppose that for some fixed e > 0 there exist sequences

{(pn} and {/jLn}, n E N, with the above properties and such that for each n there are tn, sn E [0, fc0], \tn

- sn\ < ?,

so that

(7.3) d(en(tn),en(sn))>e.

Here, en : [0, fc0] ? D denotes the curve en(t)

= ?n,t(0), where (pn?

= rjn?o(pn, and

rjnj is the normalized (fixing xi,x2,xs) quasisymmetric map that can be extended to the normalized quasiconformal map with the complex dilatation tfin. Let n ? oo.

Then we can assume that tn, sn tend to t0 E[0, ko]. After passing to a subsequence if necessary, the mappings rjn,tn, Vn,sn converge to a if-quasisymmetric map 7700

which fixes x\, x2,x$. Write <pn = i?n o (?n. Here (?n fixes pointwise the set 0Un and it commutes with un, and i?n is a if-quasisymmetric map which fixes the three

points xi, x2, xs E Ou satisfying ̂ < a(xi, Xj) < it ? j~, for every i ̂ j. Here, for

every n E N, un is the rotation of order mn > 3. After passing to a subsequence if

necessary, i?n converges to a normalized if-quasisymmetric map ^00

Consider the sequence (?n*o~o of the corresponding probability measures on T. Here ao denotes the normalized Lebesgue measure on T (we have already used a to denote the ordinary (non-normalized) Lebesgue measure on T). We have the induced sequences <pn,tn*^0 and ipn,sn*vo of probability measures. After passing to a subsequence if necessary, </>n*cro converges to a probability measure 0oo on T. Since (?n commutes with a standard rotation of order at least three, we conclude that if 0oo has atoms, then </>oo has at least three atoms of the same mass, and therefore the mass of every atom is at most |.

Remark. The above observation is the key observation of this section. It is not valid if the order of the rotation is allowed to be two, because then ̂ can contain

strong atoms.

This implies that both (pn,tn*o-o and ^n,snj|ecro converge to the probability mea sure (?>oo =

(r/oo o V;oo)*0oo- We have seen that every atom (if any) of ip^ has mass at most

|. Therefore, the sequences i>n,tn(0) =

Bar(ipn^nô) and i>n,an(0) =

Bar((pniSnâo) both converge to the point ?>oo(0). Here Bar stands for the barycen ter of the corresponding measure. Note that since the measure (p^ has no strong atoms, we have that ̂ 00 (0) G D. On the other hand, from Lemma 7.2 we know that the points <pn,tn(0) and (pn,Sn(0) are contained in A(0,r*i), ri = r*i(if). Therefore




the sequences {?>n,tn(0)}, {?Vsn(0)}, after passing to a subsequence, converge to the same point in the closure A(0, r?). But this contradicts (7.3). D

7.2. Removing elliptic elements of order at least three. We are in a position to prove the following theorem.

Theorem 7.1. For an arbitrary K-quasisymmetric group G there exists a Ki

quasisymmetric group Gi, ifi = ifi (if), with the following properties.

(1) Gi does not contain elliptic elements of order three or more.

(2) IfGi is K'-quasisymmetrically conjugated to a Fuchsian group, K' ? K'(K),

then there exists K" = if" (if) such that G is K" -quasisymmetrically con

jugated to a Fuchsian group.

Proof. Let G be an arbitrary if-quasisymmetric group. Let T be a Fuchsian group and let (p : T ?> T be a homeomorphism, so that (pT(p~x

= G- Denote by E' the

set of all points in D such that z E E' if u(z) = z, for some elliptic element uET

of order at least three. We have by Lemma 4.3 that E' is a po-discrete set for some

universal constant po > 0. Let (p be the barycentric extension of (p. Set E = (p(Er)

and S = D ? E. Clearly, E is a discrete subset of D, and S is a Riemann surface.

The group of homeomorphisms G = (pT(p~x naturally acts on S. Denote by G' the subgroup of the mapping class group of S induced by ?/. We have that Q' is

isomorphic to G (because G is). We show that G' is a ifi-quasisymmetric group. Fix f E G and let /

= (p o v o (p"1 where v E T such that ip o v o ^p~x

= f.

We need to show that / is isotopic (as a selfmap of S) to a ifi-quasiconformal

map. Let n : T ?? T be a if-quasisymmetric map such that rj o f o n~l = u for some eu E Ai. Because of the conformai naturality of the barycentric extension, we

conclude that / is isotopic (rel dS) to the map A"1 o Q o A. Here A = B o (p~l, where B is the barycentric extension of no (p. So, it is enough to show that A is a

ifi-quasisymmetric map of S, K[ =

K[(K) (note that A does not have to map S

onto itself). Let pi E L??(D) be the complex dilatation of 77, and set ?i = -An. For

t E [0, fco], let fft : D ?? D be the quasiconformal map with the complex dilatation

tfji and which fixes the points 1,2,-1. Let rjt : T ?> T be the map such that

fft extends nt. Here t E [0, fco], fco =

î- Set <pt = rjt o (p, and let (pt be the

barycentric extension of <pt. Finally, let At = <pt ? <^~1 and Et = At(E). Note

that A ? Ak0. From Lemma 7.2 (item (2)) and Lemma 4.3 we have that Et is a

p(if)-discrete set, t E [0, fco]. Consider the map rj^1 o At. Note that the restriction

of rj^1 o At on T is the identity. After properly pre-composing and post-composing the map rj^1 o At by M?bius transformations, it follows from Lemma 7.3 that this

map satisfies the assumptions of Lemma 3.3. This proves that / is isotopic (rel dS) to a ifi-quasiconformal map.

Now, we cover the surface S by the unit disc and lift the action of G1 to the unit

disc. This lifted group G" is a if2-quasisymmetric group of T. Moreover, G" does not have any elliptic elements of order three or more. Note that there is a natural

homomorphism N : G" ?> G'', and the kernel Ker^ of this homomorphism is the

group of covering transformations from the covering of S. Define Gi = G" (Gi is the group from the statement of this theorem). We have that Gi satisfies (1).

Assume that Gi = G" is if'-quasisymmetrically conjugated to a Fuchsian group T". Let M be the normal subgroup of T" which corresponds to if er^ under this

conjugation. Then D/M is a Riemann surface which is biholomorphic to the disc D




minus a discrete set of points. Let S' = D/M. Note that there is a homomorphism

of T" onto a conformai group T' and the kernel of this homomorphism is M. Here

T' acts on S', and if we see S' as a subset of D (which we can because of the

biholomorphic type of S'), then T' acts on D as well. Therefore, T' is a Fuchsian

group. It follows that the above conjugation induces a homeomorphism between S and S' such that the restriction of this homeomorphism on T, (p : T ?? T, is a if "-quasisymmetric map, if" = if" (if), which conjugates the group G to the Fuchsian group T'. This proves (2). D

8. Elliptic elements of order 2 and the proof of Theorem 1.1

Throughout the next two subsections we assume that G is a if-quasisymmetric group that has no e(if )-small hyperbolic elements and that G does not contain any

elliptic elements of order three or more. Here e(if ) is the constant from Theorem 4.1.

8.1. Quasicenter of an elliptic element of order two. Denote by ? the set of

elliptic elements of G of order two. We adopt a few notions introduced by Gabai

(see [8]). Let Z be an oriented geodesic in D with the endpoints a and b (positive orientation is from a to b). Hr and Hl denote, respectively, the right and the left

halfspaces determined by Z. This is chosen so that the arc [a, b] C T borders Hr

(we use the standard counterclockwise orientation on T). Let e G ?. A pair of

points x, y E T is said to be an orbit of e if e(x) = y. We say that e is to the left

of Z if there is a pair of points that is an orbit of e and that belong to the open arc

(b, a) C T. Similarly we define the notion to the right. We say that e is on Z if the

pair a, b is an orbit of e (it is clear that every e G ? has to be in one and only one

category). If e were a M?bius transformation, then the center of the map e would have been in Hr or in Hl or on Z, depending on whether e is to the right of Z or to the left of Z or on Z, respectively. Let h E G be a hyperbolic element, and let ah, bh denote its repelling and attracting fixed points, respectively. Let lh be the oriented

geodesic with endpoints ah,bh (the positive orientation is from a^ to 6^). il? and

Hlh denote, respectively, the right and the left halfspaces determined by Z^. We say that e is to the left of h or to the right of h or on h if e is to the left of lh or to the

right of lh or on lh, respectively. Let L > K and e G ?. We say that a point z E D is an L quasicenter of e if

there exists an L-quasiconformal map e : D ?> D which extends e and such that eoe is the identity map. Since G is if-quasisymmetric, there exists at least one if2

quasicenter for every e G ?.

Lemma 8.1. With the notation as above, there exists D = D(K) > 0 such that

the following hold. Let I be an oriented geodesic and e E ?. If z E D is an L

quasicenter of e, the following hold.

(1) Ife is to the right (left) of I, then z either belongs to Hr (Hl) ord(l, z) < D. (2) If e is on I, then d(l,z) < D. (3) For any r > 0 there exists L' =

L'(L,r) > L, so that if z E D is an L

quasicenter for e, then every point in A(z,r) is an L' quasicenter for e.

(4) There exists r(L) > 0 such that every L quasicenter of e E ? is contained in a fixed hyperbolic disc of radius r(L).




Proof. Let e : D ?> D be as above; that is, e extends e and eoe = id. Then we can

choose a if-quasiconformal map / such that e = f~l o ?o o /, where ?o(z) = ?z,

for z G D, is the standard order two rotation. Note that f(zo) = 0, where zo is the

L quasicenter that corresponds to e. Let 7 = f(l), and let I' be the geodesic with the same endpoints as 7. Then 0 is to the left of I' or to the right of I' or on I' if and only if e is to the left of Z or to the right of Z or on Z, respectively. Therefore

Zo is to the left of f~l(l') or to the right of /-1(Z') or on /~1(Z/), respectively, if

and only if e is to the left of Z or to the right of Z or on Z, respectively. Since / is

if-quasiconformal, (1) and (2) follow. If z E A(zo,r), let g : D ?> D be a if'-quasiconformal map, if' =

K'(L,r), which maps z0 to z and which is the identity on T. Then, z is the fixed point of the map e* = g~x o e o g. This proves (3).

Let z, z' be two L quasicenters of e, and let / and /' be the corresponding if

quasiconformal maps that conjugate e to eo, where z, z' axe, respectively, the fixed

points of f~l o ?o o / and /' o ?o o /'. Therefore, there is a if2-quasiconformal map which maps z to z'. This proves the last part. D

Recall that we assume that G has no e(if )-small hyperbolic elements nor elliptic elements of order three or more.

Lemma 8.2. Let G be a K-quasisymmetric group. Let c : ? ? D be the map that associates to each e E ? an L quasicenter c(e), L =

L(K). There exists N =

N(K) E N such that in each geodesic ball of radius 1 there are at most N

points from c(?). In addition, if we assume that E is a p(K)-discrete set, p(K) > 0, then for each f E G, there exists a K = if (if) -quasiconformal map f which extends

f and such that f(c(e)) =

c(f o e o /_1), for every e E ?.

Proof. We prove the first part by contradiction. Assume that Gn is a sequence of if-quasisymmetric groups such that for each n E N, we have at least n mutu

ally different en,...,e E ? and the corresponding L quasicenters cn(ei), ...,cn(en) are in the hyperbolic disc A(0,1). For each 1 < i < n, by f we denote a if

quasiconformal map that conjugates en to the standard order two rotation eo and

such that there is a if-quasiconformal extension /f, with fi (0) =

c(en). The

family of if-quasiconformal maps which map 0 into A(0,1) is a normal family. We conclude that for every e > 0, there is an n large enough and a choice of two maps

fp and fj,

i t? j, such that f o fv- is e close to the identity map in the C?

topology. In particular, we have that e^ o (e^)_1

is e close to the identity in C?

topology. The composition of two non-identical elements of order two is always a non-identity hyperbolic element; that is, eôej"1 ^ id. By choosing e small

enough, we obtain a contradiction, because none of the groups Gn has e(if )-small hyperbolic elements.

If / G G and / a if-quasiconformal extension of /, then it follows from the

proof of the previous lemma that d(f(c(e)),c(f o e o /-1)) < r(if). Set E =

Ufe? f~x(c(f o e o /-1)). Since E is p(if)-discrete and / is if-quasiconformal, it

follows that E is pi (if )-discrete, for some pi(K) > 0. The proof of this lemma follows by applying Lemma 3.2. D




8.2. Removing the elliptic elements of order two. Let e G ?. We can find a

if-quasisymmetric map that conjugates e to the map e0(z) = ?z. By conjugating

the whole group G by this quasisymmetric map, we may assume that G, if ? is not

empty, always contains the transformation eo.

Now, fix x E T. Let Gx be the subgroup of G which contains all elements from G

fixing the set {x, ?x}. Each Gx contains at least eo, and it is an elementary group. There is a if-quasisymmetric map i?x : T ? T with the following properties. i?x conjugates Gx to a Fuchsian group Fx which is an elementary group that fixes the same set {x, ?x}. Denote by i?x its if-quasiconformal extension. This Fuchsian

group may be assumed to contain eo and we may assume that the conjugation map

conjugates eo to itself. Also, by Lemma 3.5 we may assume that the conjugation map preserves the origin and the geodesic sx that connects x and ? x. An element of Fx is either a hyperbolic transformation which preserves the points x,?x, or it is an elliptic element of order two that permutes x and ? x. Denote by E'x the set

of fixed points (in D) of all elliptic transformations from Fx (note that E'x C sx). Set E'x

= i?~1(E'x[). Let c : E'x ?> sx be the induced map that associates to each

e G Gx the corresponding if2 quasicenter c(e). Repeat this process for every x E T (by choosing the appropriate quasisymmetric

map i?x for every fixed x E T). Set E' = \JxeT E'x. Note that the origin 0 belongs to

all E'x C sx (and all E'x C sx). Let c : E' ?> D be the induced map that associates

to each e G ?, and in particular e G Gx, the corresponding if2 quasicenter c(e) E E'.

Remark. Here we use the fact that every e E ? must be contained in E'x, for some x E T. That is, for every e G ?, there exists x E T such that x,

? x is an orbit of e. In particular, if h = e o eo is the corresponding hyperbolic transformation, then

x, ? x are the fixed points of h. This is obviously true for Fuchsian groups, and

Proposition 1.1 implies it for quasisymmetric groups.

If z,w E E' both belong to a fixed E'x, for some x E T, then there exists

p'(if ) > 0 such that d(z, w) > p'(K). This follows from the fact that G contains no

e(if )-small hyperbolic elements and that i?x is if-quasiconformal. For each x E T, let Tx be a diffeomorphism of sx onto itself which preserves each x,

? x and 0 and such that

(8.1) d(?!,?2) - ^p < d(Tx(fi),Tx(t2)) < d(ti,t2) + ^p-.

It follows, with the aid of Lemma 8.2 (since E' is the set of if2 quasicenters of ?, it follows from Lemma 8.2 that E' is not too dense) that for each x we can choose

Tx which satisfies (8.1) and such that the set

(8-2) E = (J TX{E'X)

x?T

is p(K)-discrete, for some p(K) > 0. We use the same notation for the induced

map c : E ?> D which associates to each e E ? the corresponding L" quasicenter

(since we moved points from E' only a finite distance, we have by Lemma 8.2 that L" =

L"(K)). By Ex we denote the subset of E that is contained in sx. For a given group (?, we have constructed the set E of L" quasicenters. This

set is fixed from now on. Let T be a Fuchsian group, and let p : T ?? T be a homeomorphism such that G ? pT^p~x. We may assume that T contains the transformation eo and that ip conjugates eo to itself. For each y E T, let Fy be the




subgroup of T containing elements that fix the set {y, ?y}. The set of centers of

elliptic elements (which are all of order two) that lie on sy is denoted by Ey. Set

E = \JyeT Ey. Let c:?-^Dbe the map that associates the center to each elliptic

element from T. Let (p(y) = x. Note that (p induces the map from Ey onto Ex by

cooôc"1. It follows from (8.2) (the definition of E) that this map is orientation

preserving, where the orientation for sets Ey and Ex comes from the way they lie in sy and sx, respectively.

We define an appropriate extension (p of the above homeomorphism (p. For each

y E T, the restriction of (p to sy is a homeomorphism which maps sy onto sx, where x = (p(y) ((p maps a radius onto a radius). Also <?>(0)

= 0, (p(y) = x, (p(?y)

= ?x.

Furthermore, we can choose (p with the following property. If w = c(u), for some

elliptic u E T of order two, then (p(w) = z, where z = c((p ouo (p)_1).

Set S = D?E, and let Gi be the subgroup of the mapping class group of S defined as follows. To / G G we assign the isotopy class of the map (p ouo (p~x : S ?> S, where u = tp~l o f o (p. The group of the corresponding isotopy classes is Q\. The

map from G onto G\ is an isomorphism.

Lemma 8.3. There exists K2 = if2(if) such that the group Gi is K2-quasisym

metric.

Proof. Fix f E G- Let / =

(po u o (p~x, where u = (p~l o f oip. Our aim is to show

that / is isotopic in S to a if2(if)-quasiconformal map. E is p(if)-discrete and E is the set of L" quasicenters. Let / be a if-quasi

conformal map from Lemma 8.2; that is, for every z E E, z = c(e), we have

/0)=c(/oeo/-i). Fix x E T, and set jx =

f(sx). Note that i?-1^) =

7^ is a geodesic in D and

u(sy) =

jy, where (p(y) = x and (p conjugates u to /. Let v E T be an elliptic

element of order two. Since the position of c(v) (the center of v) with respect to

j'y respects whether v is to the right of, to the left of, or on j'y,

we have that the same is true for the curve 7X and the corresponding quasicenter from E. Also, if the endpoints of jx are not an orbit of eo, then jx intersects each geodesic sy at

most once (this is because the corresponding statement is true for 7^) (see Figure

5). Denote by [7^] the homotopy class of jx in S. For y E T let ay

= jx D sy if this intersection is not empty. For sy, where this intersection is non-empty, denote by

Iy C Sy the maximal open geodesic arc which contains jx fl sy =

ay and such that

Iy does not contain any points from E (except for ay, if ay E E). If sy contains no

points from E other than 0, then Iy is one of the two geodesic rays that end at 0 and which constitute sy .

Let a : sx ? D be a curve in D that intersects each sx, x E T, at most once,

and set by = a fl sy. We have that a E [jx] if and only if the map a is isotopic to

the restriction of / on sx, which is equivalent to the following two conditions being satisfied.

(1) If ay E E, then ay =

by. (2) If ay does not belong to E, then by E Iy.

Now we show that for each x E T we can choose an Li bilipschitz quasigeodesic a : sx ?> D, a E [yx], Li

= Li(if). If jx is one of the geodesies that contain the

origin, then a = jx. If not, then jx intersects each sy at most once. Let Z be the




Figure 5

geodesic with the same endpoints as the curve jx. Then, by Lemma 8.1 we have that

d(Z,/w)<r(?-) and

(8.3) d(Z, ay) < r(K), ay E E,

for each Iy and for each ay E E. Since the set E is p(K)-discrete, it follows from

(8.3) that we can choose an L[ bilipschitz quasigeodesic ? : I ? D, L[ =

Li (if), such that the curve ? is in [7J.

Let a' : sx ?> ? be defined as follows (here ? =

?(l)). For z E sx, let a'(z) be the point ?f]sy, where y E T is the unique point such that sy contains the point

f(z). Either f(z) =

a'(z) or f(z) and a'(z) belong to the same Iy. The map o?

does not have to be a quasiisometry, but we have a'(z) =

f for z E Ex. Since / is if-quasiconformal (recall that / is the map from Lemma 8.2 that corresponds to the set E) and since ? =

a'(sx) is a bilipschitz quasigeodesic, we can construct a : sx

? ? = a'(sx), so that a is Li bilipschitz quasigeodesic.

Consider the map f~x o f : S ?*5. This map is the identity on T, and it fixes

every point in E. Also, for each sx, the restriction of the map /_1 o / on sx is

isotopic (rel dS) to an Li bilipschitz quasigeodesic. It follows from Lemma 3.4 that

/ is isotopic (rel dS) to a if2-quasiconformal map, for every f E G, and the group Gi is if2-quasisymmetric. D




Note that Gi, as a group that acts on the Riemann surface S, does not have any elliptic elements; that is, Gi is a torsion-free group.

8.3. Proof of Theorem 1.1. Let G be an arbitrary if-quasisymmetric discrete

group. We want to prove that such a group is a quasisymmetric conjugate of a Fuchsian group.

By Theorem 7.1 we can assume that G has no elliptic elements of order three or more. Then, we can apply Theorem 4.1 to the group G and it follows from Theorem 4.1 that we can further assume that G does not contain any e(if )-small hyperbolic elements, where e(if) > 0 is the constant from Theorem 4.1.

Since G has no e(if )-small hyperbolic elements and no elliptic elements of or der three, it follows from Lemma 8.3 that the corresponding group Gi is a K2 quasisymmetric, if2 = if2(if), torsion-free group. But Gi acts on the Riemann surface S = D ? E. We now cover the surface S by the unit disc, and in the same

way as in the proof of Theorem 7.1 we produce a new if2-quasisymmetric, torsion free group G2 (that acts on D) such that G2 naturally projects to Gi- Moreover, since G does not have any e(if )-small hyperbolic elements and since the set E is

pi (if) discrete (this implies that there are no very short closed geodesies on S), we can find e(if ) > 0 such that G2 does not contain any e(if )-small hyperbolic elements. As in the proof of Theorem 7.1, we have that G2 is quasisymmetrically conjugated to a Fuchsian group if and only if G is.

So, G2 is a if2-quasisymmetric, if2 = if2(if), torsion-free group, that does not contain any esmall hyperbolic elements. We can now apply Theorem 6.1. This concludes the proof of Theorem 1.1.

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