Roger W. Barnard and G. Brock Williams- Combinatorial Excursions in Moduli Space

8/3/2019 Roger W. Barnard and G. Brock Williams- Combinatorial Excursions in Moduli Space

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PacificJournal of

Mathematics

COMBINATORIAL EXCURSIONS IN MODULI SPACE

Roger W. Barnard and G. Brock Williams

Volume 205 No. 1 July 2002


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PACIFIC JOURNAL OF MATHEMATICSVol. 205, No. 1, 2002

COMBINATORIAL EXCURSIONS IN MODULI SPACE

Roger W. Barnard and G. Brock Williams

Given an abstract triangulation of a torus, there is a unique

point in moduli space which supports a circle packing for this

triangulation. We will describe combinatorial deformations

analogous to the process of conformal welding. These combi-

natorial deformations allow us to travel in moduli space from

any packable torus to a point arbitrarily close to any other

torus we choose. We also provide two proofs of Tokis result

that any torus can be transformed into any other by a confor-

mal welding and compute the maps necessary to accomplish

the welding.

1. Introduction.

A circle packing is a configuration of circles with a prescribed pattern oftangencies. The interplay between the combinatorial pattern and the ge-ometry provided by the circles has generated intense interest in recent years.As a result, a discrete version of complex analysis based on maps betweencircle packings has now emerged. The seeds of a discrete Teichmuller theoryhave been sown, but it is still in the very early stages of development.

It has been understood that the underlying combinatorial structure of acircle packing determines its geometry. For packings on surfaces, the com-

binatorics determine a unique point in moduli space on which the packinglives [4]. As we will see in Section 5, changing the combinatorics generallychanges the point in moduli space, but the precise mechanism remains verymysterious. Our goal then is to develop some controlled combinatorial pro-cedure for which we not only understand the resulting movement in modulispace, but which we can actually use to drive us to a specified target.

Toki [26] proved that for the torus, any point in moduli space can bedeformed into any other by a conformal welding. That is, any torus can

be cut open along an appropriate geodesic and then glued (unevenly) backtogether to form any other torus. After introducing some background ma-terial in Section 2, we present another proof of Tokis result in Section 3.Our approach allows us to explicitly compute the welding maps involved.

Theorem 1.1. Consider a point H. If {z| 0 < Im z Im }, thenthere is a conformal welding along any vertical geodesic which will transform

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4 ROGER W. BARNARD AND G. BROCK WILLIAMS

R to R. Conversely, if {z| Im z Im }, then there is a conformal

welding along any horizontal geodesic which will transform R to R.

In Section 6, we describe a combinatorial deformation analogous to weld-ing. Packings are cut open along a geodesic, then reconnected in a differentpattern than before. In Section 7 we prove that these discrete weldingsapproximate their classical counterparts.

Theorem 1.2. Fix any abstract triangulation K of a torus and any point H. Then combinatorially deforming K by hex refinement and combi-natorial welding along an appropriate geodesic produces packable surfaces

which converge in the Teichmuller metric to R.

Thus we can prescribe the combinatorial changes necessary to move inmoduli space from any packable surface to a point arbitrarily close to anyother surface we choose. This also gives a second proof of Tokis theoremusing only circle packing techniques. Moreover, our proof holds for anybilipschitz welding map, not only the ones needed for our excursions inmoduli space. Thus we have a discrete method for approximating the surfacecreated by any bilipschitz welding.

Our procedure allows for several arbitrary choices in the small scalechanges made to the complexes. However, the welding process is remarkablystable, forcing these variations in the construction to disappear in the limit.

Theorem 1.2 also yields a genus one version of Brookss density resultfor compact surfaces [8]. Not only are the packable tori dense, but anyone packable torus generates a dense packable family by welding. A similarresult for hyperbolic surfaces was given by the second author in [28] usingearthquake deformations.

Finally, we would like to thank the referee for several very helpful sugges-tions.

2. Teichmuller theory.

2.1. Riemann surfaces and their deformation spaces. We begin witha brief review of Riemann surfaces and Teichmuller spaces. Some of themany excellent references include [10, 11, 15, 18, 13].

Definition 2.1. A Riemann surface is a one complex-dimensional man-ifold with charts whose overlap maps are conformal. A maximal collection

of such charts define a conformal structure.

Surfaces R1 and R2 are said to be conformally equivalent if there is aconformal homeomorphism f : R1 R2. The equivalence classes of surfacesof the same topological type as R1 constitutes the moduli space ofR1. Formany purposes, however, we need an equivalence relation that keeps trackof more information. Equivalent surfaces should not only share the same


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COMBINATORIAL EXCURSIONS 5

conformal structure but generators of their fundamental groups should alsocorrespond.

Definition 2.2. Let R be a Riemann surface and a collection of canonicalgenerators of 1(R). is then called a marking for R. Two markings areequivalent if they differ only by the choice of basepoint.

Definition 2.3. Two marked Riemann surfaces (R, ) and (R, ) areequivalent if there is a conformal map f : R R for which the mark-ing f() is equivalent to

. The Teichmuller space of R1 is the set ofthese equivalence classes.

An alternative definition can be given using maps from a reference surface.

Definition 2.4. Quasiconformal maps f1

and f2

of a Riemann surface Rare Teichmuller equivalent if f2 (f1)

1 is homotopic to a conformal map.

Proposition 2.5. Fix a reference Riemann surface R and suppose f1 andf2 are quasiconformal maps from R to R1 and R2, respectively. Then R1and R2 are equivalent in Teichmuller space if and only if f1 and f2 areequivalent mappings.

There is a natural metric on Teichmuller space determined by how nearlyconformal a homeomorphism which respects the markings may be. More

specifically, the distance between surfaces R1 = f1(R) and R2 = f2(R) isgiven by

1

2log K,

where K is the infimum of the dilatation of g2 (g1)1 with gi equivalent

to fi, i = 1, 2. This infimum is attained by a unique Teichuller mapping.The collection of homotopy classes of diffeomorphisms of R is called the

mapping class group of R. It acts on the Teichmuller space of R trans-forming one marking into another. Teichmuller space modulo the mappingclass group is precisely the moduli space. Thus the Teichmuller metric in-duces a metric on moduli space.

2.2. Tori. Consider a point H and the group of translations ofCgenerated by z z + and z z + 1. The quotient surface R = C/ isa torus. A fundamental region for R is the parallelogram with verticesat 0, 1, , and 1 + .

Such a torus has two families of geodesics corresponding to its generatorsz z + 1 and z z + . The horizontal geodesics are the projections ofhorizontal lines, that is, lines parallel to the translation z z +1. Similarly,the vertical geodesics are the projections of lines parallel to the translationz z + .

Any of the tori R are topologically identical, but their conformal struc-tures change with . In fact the Teichmuller space for the torus is isometric


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to H with the hyperbolic metric. The mapping class group is isomorphic toPSL2Z, and the moduli space is isometric to H/PSL2Z.

3. A moduli space trip in a classical vehicle.

3.1. Conformal welding. Suppose R is a Riemann surface and a simpleclosed geodesic. If we cut R open along , we obtain a bordered surface Rwith geodesic boundary arcs + and . Notice that each point z splitsinto points z+ + and z .

Definition 3.1. If R is a bordered Riemann surface with geodesic bound-aries + and as above and : + , then a conformal welding for is a map f from R to some surface R which is conformal off + and

satisfies the welding propertyf (z+) = f(z),

where z+ + and z are copies of the same point z .

From a topological viewpoint, gluing + and together is relativelyuninteresting. The new surface R is topologically equivalent to the originalsurface R, and the welding curve = f(+) = f() is homotopic to .From a conformal point of view, however, R and R may be dramatically

different surfaces. The welding map stretches and compresses + as itattaches it to . The new seam must absorb this distortion, bendingand warping as the competing metric forces do battle. See Figure 1.

The existence of a conformal welding allows us to define coordinate chartson R \ using the charts on R. That is, if {} is the conformal structureon the bordered surface R, then {f

1} is a conformal atlas on R \ . Asa result, the metric distortion between R and R is completely captured inthe geometry of .

Notice that if g : R

R

is conformal, then gf is another conformalwelding for . If every welding for is of this form, then determinesa unique point in moduli space. Oikawa [20] proved that unique weldingsexist for locally quasisymmetric maps (that is, maps which can locally beextended to quasiconformal maps) gluing the sides of infinite strips to forma cylinder. Since a torus is covered by a cylinder, locally quasisymmetricwelding maps on a torus can be lifted to periodic locally quasisymmetricwelding maps on a cylinder. Oikawas result can then be restated as follows.

Theorem 3.2. If is a locally quasisymmetric welding map on a torus,then welding by determines a unique point in moduli space.

Beyond uniqueness, we are interested in the geometry of tori deformedby welding. The following result appears in a paper of Toki [26]. Tokisproof, however, filters the welding through the exponential map and doesnot provide a simple description of the welding f. Our proof is similar in


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R

_

R

R

Figure 1. A torus before, during, and after welding. Noticethat before welding (top) the torus is much fatter thanafter the welding (bottom). The global geometric distortionis also reflected in the distortion of the image of the geodesic.

spirit, but yields an explicit description of the deformation needed to convertone torus to another. In Section 7, we present a second proof using discretemaps induced by circle packings.

Theorem 1.1. Consider a point H. If {z| 0 < Im z Im }, thenthere is a conformal welding along any vertical geodesic which will transform


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R to R. Conversely, if {z| Im z Im }, then there is a conformal

welding along any horizontal geodesic which will transform R to R.

Proof. First suppose 0 < Im Im and choose a geodesic of R corre-

sponding to the translation z z + . Recall that a fundamental region forR and R is given by the parallelogram having vertices 0, 1, , 1 + .By an appropriate choice of basepoint, we can assume lifts to the left andright sides of . We will label these sides + and , respectively.

Similarly, R lifts to a fundamental region given by the parallelogram with vertices 0, 1,

, 1 + . In general, and will have differentconformal moduli (as quadrilaterals) and hence no conformal map betweenthem. However, if we distort , we can obtain a fundamental region for

R

whose conformal modulus equals that of .By decreasing the angle at and 0 and simultaneously increasing theangle at + 1 and 1, we bend the middle of to the right. See Figure 2.

1+

0 1

1+ 1+

0 1 0 1

0

Figure 2. The quadrilateral (left) has a smaller modulusthan (middle). By sliding the middle of to theright, we can lower the modulus of

() (right).

If 0 = Arg() is the angle in at , then for 0 < 0, let

() be the 6-gon with vertices at 0, , 1 + , and 1, angle at 0 and

, and angle (1 ) at 1 and 1 +

Notice that mod((0)) = mod() mod() and mod(())

decreases continuously to 0 as 0. Consequently, there is a unique valueof > 0 for which mod((

)) = mod().

Since the moduli are equal, there is a conformal map f : ().By Schwarz reflection, f sends points on the edges of which correspondunder to points on the edges of (

) which correspond under . Thusf projects to a map f : R R which is conformal off .Now let , = f1( f(z) + 1) on +. Then projects to a welding map

on . Notice that f(z+) = f (z) on .


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The case Im Im is similar. By bending a horizontal geodesic, wecan increase the modulus of until it equals that of . The existence ofa welding then follows precisely as before.

It follows from the well-known behavior of conformal maps at corners thatin shrinking the angle at and 0, must act locally at and 0 like a powermap z z, with < 1. As a result, 0 and are not only fixed points of, but attracting fixed points.

Conversely, at the midpoint of the side between 0 and , the straight lineis bent to form an angle measuring more than . Thus acts locally atthe midpoint like a power map z z, > 1. Hence, the midpoint is arepelling fixed point. See Figure 3.

Figure 3. The welding map constructed in the proof ofTheorem 1.1 glues two points of + back to their originalpositions and then pulls the remainder of + away fromone fixed point and toward the other.

Notice also that there was no great need to take 0 in the aboveproof. There is also a unique 0 so that mod((

)) = mod().

This will bend to the left; thus we obtain welding maps which stretch inthe other direction.

3.2. Computing the welding map. The welding map , used in theproof of Theorem 1.1 can be expressed explicitly using Schwarz-Christoffelmaps.

First observe that the unit disk D is taken onto the parallelogram bythe Schwarz-Christoffel map

f1(z) = c1 z1

(1 )1(1 ei0)(1 + ei0)1(1 ) d,(3.1)

where = Arg() is the exterior angle at + 1 and c1 and 0 aredefined by the requirements that

f(1) = c1

11

(1 )1(1 ei0)(1 + ei0)1(1 ) d = 1


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f f3

3,

-1

1

f

f2

Figure 4. Our conformal welding is given by the compo-sition f3f2f

11 . The images of the four corners of are

indicated by dots.

and f1(ei0) = . See Figure 4.

The map f1 takes 1, ei0 , ei0 and 1 onto the corners 0, , 1 + and

1, respectively. To take advantage of the symmetry of the region (),

we use the map

f2(z) =z ir

1 + irz,(3.2)

with 1

ir1+ir = e

i

0

2 , to take 1, e

i0 , ei0 and 1 to the symmetric pointsei

02 .

Now these points are mapped to the corners of () by the Schwarz-

Christoffel map

f3(z) = c3

z1

(1 )12

(1 2cos(0/2) + 2)

(3.3)

(1 + 2 cos (0/2) + 2

)

1

(1 + )

21

d + 1

2 ,

where c3 is chosen so that f3(ei0) = 1 + . Our desired welding map is

then given by , = f3f2f11 restricted to the left side of .

Notice, however, that the sharp corners of n destroy the smoothness off3 there. We can obtain a bilipschitz approximation to f3f2f

11 if we re-

place () by a region () in which these corners have been smoothly


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rounded. This technique was used in [3]. We replace f3 by

f3,(z) = c4

z1

[(1 ei)2

+ (1 ei)2

](3.4)

[(1 ei(0/2)

)1

+ (1 ei(0/2+)

)1

][(1 + ei(0/2))1

+ (1 + ei(0/2+))1

]

[(1 + ei(0/2))

+ (1 + ei(0/2+))

]

[(1 + ei(0/2))

+ (1 + ei(0/2+))

]

[(1 ei)12

+ (1 ei)12

] d + 1

2

where is small and c4 is chosen so that f3,(ei0/2

) = 1 +

.Thus we obtain a C1 welding map ,, = f3,f2f11 . The smoothness of

, will prove useful in constructing a circle packing version of Theorem 1.1.

Remark 3.3. The reader may object to the description of, as explicitsince an inverse for f1 was not given in closed form. In practice, however,the inverse is obtainable by recently developed Schwarz-Christoffel packagesto the desired accuracy [9].

Remark 3.4. The nature of conformal weldings of tori is in marked con-trast to the well-studied problem of welding discs [11, 14, 16, 19, 21, 27,29]. Since there is only one possible conformal structure on the sphereformed by welding discs, welding discs does not alter position in modulispace. Moreover, except for a few special cases, it has not been possible tofind either a closed form of the conformal welding or an explicit descriptionof the seam resulting from a given welding. The best known result is thediscrete approximation scheme developed in [27].

4. Circle packing.

4.1. Abstract triangulations. We will use the term abstract triangu-lation to mean a simplicial 2-complex. In particular, these are purely com-binatorial objects. They possess no geometric structure until they are em-bedded in a surface.

We also require our abstract triangulations to be CP-complexes; how-ever, notice that the restrictions imposed by Conditions (2) through (4) are

extremely mild and are met by any reasonable complex.

Definition 4.1. A CP-complex K is an abstract simplicial 2-complex suchthat:

(1) K is simplicially equivalent to a triangulation of an (orientable) surface.(2) Every boundary vertex of K has an interior neighbor.(3) The collection of interior vertices is nonempty and edge-connected.


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(4) There is an upper bound on the degree of vertices in K, that is, on thenumber of edges incident at each vertex.

4.2. Circle packings.

Definition 4.2. A circle packing is a configuration of circles with a spec-ified pattern of tangencies. In particular, ifK is a CP-complex, then a circlepacking P for K is a configuration of circles such that:

(1) P contains a circle Cv for each vertex v in K,(2) Cv is externally tangent to Cu if [v, u] is an edge of K,(3) Cv, Cu, Cw forms a positively oriented mutually tangent triple of cir-

cles if v,u,w is a positively oriented face of K.

A packing is called univalent if none of its circles overlap, that is, if no pairof circles intersect in more than one point.

A univalent circle packing produces a geometric realization of its under-lying complex. Vertices can be embedded as centers of their correspondingcircles, and edges can be embedded as geodesic segments joining centersof circles. This embedding for K is called the carrier of the packing, anddenoted by |K|.

If K is embedded in C in two different ways, there is a natural piecewise

affine map sending triangles in one embedding to triangles in the other. Ifthe embeddings for K are created using circle packings, we will refer to theinduced map as a discrete conformal map. As the name suggests, thereare deep connections between discrete conformal maps and their classicalcounterparts.

4.3. Packable surfaces. Various existence results for circle packings havebeen given by Thurston [24], Minda and Rodin [17], and Beardon and

Stephenson [4]. Moreover, Brooks [8] showed that compact packable surfacesare dense in moduli space. His result was extended to surfaces of finiteanalytic type by Bowers and Stephenson [6, 7]. In summary:

Theorem 4.3. Let K be an abstract triangulation of a surface R of finiteanalytic type. Then there is a unique point in the moduli space for R whichsupports a packing for K. Similarly, K and a marking for R determine aunique packable point in the Teichmuller space ofR. Moreover, the collectionof all packable surfaces is dense in both the moduli space and Teichmuller

space of R.

Unfortunately, a thorough description of the effect of K on the globalgeometry of R has been extremely elusive. For hyperbolic surfaces, theeffect of combinatorial manipulation of K by earthquakes has been studiedin [28]. In Section 6, we will consider combinatorial deformations of torianalogous to the conformal weldings of Section 3.1.


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4.4. Combinatorial influence on geometry. It has been known for sometime that the local combinatorics of a packing exert an important influenceon the local geometry. Rodin and Sullivans Ring Lemma [22] connects thecombinatorial properties of packings and the function theoretic properties

of associated discrete conformal maps.

Ring Lemma. Suppose K is a complex of degree D. If v an interior vertexof K and w is a neighbor of v, then there is a lower bound CD on the ratioof the radius of Cw to Cv in any univalent packing for K. In particular, thelengths of any two sides of a triangle in |K| are CD-comparable.

The constant CD of the Ring Lemma has been computed by Aharonov [1,2] to be

CD =1

a2D2 + a2D1 1

,

where ai is the ith Fibonacci number. This bound is sharp.

It is easy to show that affine maps between triangles are quasiconformalwith dilatation depending only on the difference in corresponding angles ofthe triangles. The Ring Lemma implies that angles at interior vertices in thecarrier of a univalent packing are bounded away from 0 and . This gives anupper bound on the difference between corresponding angles in the carriers oftwo different packings for the same complex. Consequently, discrete analyticfunctions are k-quasiconformal on faces which do not contain a boundaryvertex, where k depends only on the degree of the packing.

Even greater control on the dilatation is possible by considering moregenerations of circles. The resulting Packing Lemma was first shown forpackings in which every vertex has degree 6 [22, 25] and later extended byStephenson [23] and He and Rodin [12].

Definition 4.4. A chain of circles in packing P for K is a collection ofcircles Cv1 , Cv2 , . . . , Cvn of P so that vi and vi+1 share an edge in K, i =1, 2, . . . , n 1, and vi = vj , if i = j. Thus a chain describes a non-self-intersecting edge path in K. A chain is closed if v1 = vn.

Packing Lemma. If a circle Cv of a circle packing P is surrounded by nclosed chains of circles, each circle having degree at mostm, then the dilationof a discrete conformal map defined on faces containing v decreases to 0 asn .

Finally the Length-Area Lemma [22] provides combinatorial control overthe size of circles. It is a type of combinatorial isoperimetric inequality.

Length-Area Lemma. Let P be a univalent circle packing inD and Cv acircle in P with euclidean radius r. Assume there are m disjoint chains of


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circles in P having combinatorial lengths n1, n2, . . . , nm, and each separatingCv from the origin and a point on the boundary. Then

r 1

mi=1 1ni.

5. Simple combinatorial deformations.

5.1. Changing the triangulation vs. changing the embedding. Re-call that our abstract triangulations are purely combinatorial objects untilthey are embedded in some surface, usually by means of a circle packing.If a triangulation K is realized by a packing of a surface , then changingK will lead not only to a new complex K, but K will also induce a new

packing on a (most likely different) surface .If we wish to directly compare the circle packing-induced geometries of

and we must limit the type of changes we make to K. In particular, anycombinatorial change we make to K should be realizable in by adding ormoving edges of|K|. These in situ changes to |K| will produce an embeddingof the new complex K in . Now we construct a discrete conformal map fsending triangles in the embedding ofK in to the corresponding trianglesin the embedding of K in . By thus comparing the embeddings of K in

and in , we can compare the geometry of and .If the in situ modification to |K| can be made in such a way that the angles

in the newly embedded triangles are bounded away from 0 and and if thedegree of K is controlled, then using the Ring Lemma, the dilatation of fcan be estimated and an upper bound on the Teichmuller distance between and computed.

For the remainder of the this section, we will consider examples of thesecombinatorial modifications which

1) permit bounds on the degree of the new complex2) can be realized by an in situ deformation of the original embedding

while maintaining control over angles.

5.2. Hex refinement. A particularly useful example of this type of com-binatorial deformation is the hex refinement scheme of Bowers and Stephen-son [5].

Definition 5.1. The hex refinement of a complex K is the complex

formed by adding a vertex to each edge and adding an edge between newvertices lying on the same face. See Figure 5.

Notice that any new interior vertices added to K by hex refinement havedegree 6, while the degrees of the original vertices remain unchanged.

Aside from the embedding provided by circle packing, there is anotheruseful embedding of the hex refinement of K. If K is embedded in C in


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Figure 5. Two triangles before (left) and after (right) hexrefinement. Notice that the new interior vertex has degree 6.

such a way that edges correspond to euclidean line segments, then its hexrefinement may be realized by adding line segments joining the mid-pointof each edge. In this case, each face of K will be subdivided into four newfaces, each similar to the original and having edges one-half as long.

This second embedding is generally different from the embedding pro-vided by circle packing. Since the faces are broken into similar faces, nonew geometry emerges. By contrast, the circle packings can make fine ad-

justments to the embeddings as K is repeatedly refined. In fact, the Packing

Lemma implies that after many hex refinements, most triangles embeddedby a circle packing are nearly equilateral. This phenomenon leads to thefollowing result of Bowers and Stephenson [5].

Lemma 5.2. Let K be an abstract triangulation of a compact Riemann sur-face realized by a packingPn on a surface Rn and suppose Kn is the n

th hex

refinement of K. Then {Rn} converges in the Teichmuller metric to thepoint corresponding to the surface formed by gluing together equilateraltriangles in the pattern given by K. In particular, lies in a ball centered

at with radius determined only by the degree of K.

5.3. Trivalent barycentric subdivision. A second very instructive ex-ample is provided by trivalent barycentric subdivision, or starringfrom the barycenter. In this refinement scheme, a face is subdividedby adding a vertex with edges to each of the three existing vertices. SeeFigure 6.

Proposition 5.3. Trivalent barycentric subdivision doubles the degree of

every interior vertex. Notice that the new complex can be embedded usingthe original packing by adding a vertex at the barycenter of each face and

embedding the new edges as geodesic segments.

Since the degree of the complex grows without bound, the Ring Lemmaand Packing Lemma are useless to control the angles in the refined complex(and hence the distortion of discrete maps).


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Figure 6. The effect of trivalent barycentric subdivision ona complex (top left) and its packing (lower left). Notice thesubdivided complex (top right) has changed, but the six large

circles in the new packing (lower right) are unchanged.

Surprisingly, however, trivalent barycentric subdivision does not affectthe location of the packed surface in moduli space at all! A packing forthis complex is formed by simply adding a circle to each interstice withoutmoving any of the original circles.

5.4. Vertex-edge subdivision. Notice that if a vertex is added to an edgeof an existing triangle, this triangle will become a quadrilateral, having four

vertices instead of only three. Thus without additional modifications, ourcomplex will cease to be a triangulation.

This situation can be easily remedied by adding a new edge to the com-plex. See Figure 7. The new edge can obviously be embedded in |K| as aline segment. It was shown in [27] that if the new vertex is bounded awayfrom the old vertices, the angles created by this vertex-edge subdivisionwill be bounded.


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v v

w w

Figure 7. The addition of a vertex w to an edge (left) canbe embedded by simply adding a new edge [v, w] (right).

Lemma 5.4. If a vertex w is added to an edge e of length l and the distancefrom w to the endpoints of e is at least l, then the angles created by vertex-edge subdivision are bounded away from 0 and by a constant dependingonly on .

5.5. Local hex refinement. A final example is provided by local hex re-finement. See Figure 8. Notice that the hex refinement procedure describedin Section 5.2 adds a vertex to each edge of the complex. If we wish to refine

only one triangle, we quickly run into difficulty since adding a vertex to aninterior edge affects the neighboring triangle as well.

Figure 8. A complex (left) can be locally hex refined alongan edge path (right).

It is possible, however, to refine in a neighborhood of an edge path without affecting the remainder of the complex. Let G be the collectionof triangles hitting . We first hex refine triangles in G as before. Notice,however, that this procedure adds a vertex to the midpoint of all triangles


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which share an edge with G. We cannot hex refine these neighboring tri-angles, as that would force us to propagate the refinement throughout theentire complex. Instead, we use vertex-edge subdivision to absorb theseextra vertices.

Notice that all angles created by hex refinement have the same bounds asthe original angles. Similarly, the angles created by vertex-edge subdivisionare bounded by Lemma 5.4. Finally, notice that the degree of the locallyhex refined complex is bounded by the larger of 6 and D + 1, where D isthe original degree of K.

6. Combinatorial welding deformations.

6.1. Correctly cutting a torus. We now describe a combinatorial de-formation analogous to the conformal welding process. Suppose K is anabstract triangulation of a torus with degree D. By Theorem 4.3 (or origi-nally, [4]), there is a circle packing P on a torus R. We would like to cutK along some geodesic and reattach the two copies + and of usinga welding map : + .

Unfortunately, the is an object from the continuous world, not oneof our discrete creations. In general, will pass indiscriminately through

triangles and vertices of|K| with no regard for the underlying combinatorialstructure. Thus we need to construct an edge path ofK corresponding to .

One option is to completely replace with a polygonal edge path em-bedded nearby. This approach is used, for example, in [28] to cut openhyperbolic triangulations and approximate earthquakes. We prefer a dif-ferent construction here, adding edges to |K| so that itself becomes thedesired edge path.

If some portions of already correspond to edges of |K|, then we accept

these edges without modification. Elsewhere, however, as enters eachtriangle T, it joins either a vertex to an edge or, more commonly, an edgeto an edge.

In the first case, splits T into two smaller triangles. Thus it would becombinatorially acceptable to merely add a new edge to K corresponding tothis segment of . Unfortunately, we must embed the new edge using thissegment of since we need our welding map : + to act on ournew edge path. This addition might unacceptably alter the embedding |K|.

In particular, if exits T very near an existing vertex v1, one of the newtriangles will have a small angle. See Figure 9.To avoid this difficulty, we will slightly alter our embedding |K| by moving

v1 to lie on . We let =1

3CD, where CD is the constant from the Ring

Lemma. If the distance from v1 to is less than times the minimum sidelength of T, then we will move v1 to the closest point on . Notice that theangles in a euclidean triangle vary continuously with the side lengths, and


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v1

Figure 9. If hits one vertex and passes very near another,the resulting angle will be small.

all the triangles in |K| containing v1 have CD-comparable side lengths bythe Ring Lemma. Thus we maintain a uniform (albeit weaker) bound onthe angles.

In the second case, if the distance from to a vertex is less than timesthe minimum side length of T, we again move that vertex to . Thus wemay assume that is bounded away from the vertices of T. Notice thatadding an edge corresponding to T divides T into a smaller triangle T1and a quadrilateral. We then add another edge to divide the quadrilateral

into triangles T2 and T3. See Figure 10.

T

T

T

1

2

3

Figure 10. Adding an edge corresponding to , we form atriangle T1 and a quadrilateral. A second edge then breaksthe quadrilateral into triangles T2 and T3.

Lemma 6.1. All the angles in T1, T2, andT3 are bounded away from 0 and. This bound depends only on the degree of K.

The proof involves only elementary geometry and is similar to the proofof Lemma 5.4 found in [27].

6.2. Combinatorial re-attachment. Once our torus has been cut open,we have two copies of the edge path , one of which we label + and the


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other which we label . If : + is an orientation reversing home-omorphism, then we can re-attach these two copies using as our guide.A general procedure for performing this re-attachment is described in [28];however, for completeness, we will briefly recount it here.

If maps each edge of + to a corresponding edge in

, then we merelyglue the edges together to form a new abstract triangulation. However, mostwelding maps and certainly those of the type described in Section 3.2 willnot respect the combinatorial structures of + and in this way. Thus itis necessary to modify K so that will respect the modified combinatorics.

A first attempt at a solution might be:

(1) For each v +, add a vertex to corresponding to (v).(2) For each w , add a vertex to + corresponding to 1(w).

Now will respect the combinatorics of + and .Of course, we cannot simply add vertices to K and expect that K will

remain a triangulation. Each triangle to which a new vertex was addedmust be subdivided in some way. The vertex-edge subdivision described inSection 5.4 is precisely the tool we need.

However, since we have a future application of the Ring Lemma in mind,our modifications must maintain a uniform bound on the degree ofK and onthe angles in any new triangles we create. Lemma 5.4 allows us to control

the angles created by the vertex-edge subdivision provided the new verticesare not placed too near either each other or the existing vertices.

A priori, might map vertices of + all over the place, defying anyattempts to control angles or the degree of K. However, if is bilipschitzand the lengths of edges of + and are uniformly comparable, then it isa relatively simple matter to find such bounds [27].

Thus we will hereafter assume that is B-bilipschitz; that is, for allx, y +,

1

B|x y|+ < |(x) (y)| < B|x y|+ ,(6.1)

where |xy|+ denotes the distance from x to y inside + and |(x)(y)|

denotes the distance from (x) to (y) inside . Since bilipschitz mapsare quasisymmetric, Theorem 3.2 will hold.

It follows from the Ring Lemma and the control on our modifications thatthe lengths of adjacent edges of+ and are CD-comparable. Thus edges

which are separated by n vertices are CnD-comparable. Unfortunately, since

this bound grows with n, if we attempt to weld a sequence of finer and finercomplexes (as we will in Section 7), we cannot maintain uniform controlthroughout the entire sequence. We need a bound which depends only onthe degree of K and not on the number of edges in K.

Our remedy is to break edges which are too long into shorter pieces usingthe local hex refinement scheme of Section 5.5. More precisely, let r be the


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minimum edge length of + . Let E1 be the set of all edges in +

with length greater than CDr.Recall that our method for embedding a local hex refinement along an

edge path splits edges in the path in half. Thus we can locally hex refine

each component of E1 and reduce the edge lengths by a factor of 2. Recallthat edges of + which share a vertex with an edge of E1 will also besplit in half.

Now let E2 be the set of all edges in + whose length is still greater

than CDr. Local hex refinement along E2 will again reduce edge lengths bya factor of 2. Repeating this process, we see that after finitely many stepsall the edges in + will have length at most CDr.

Moreover, any edge which is created at step n but is not split at step n+1

will have length at least12CDr; otherwise, it would have been too short to

be split at stage n. This does not yet give a lower bound on the lengths ofall edges after all the splitting, since it is possible such an edge might besplit once more (and only once more, by the Ring Lemma) if it shares avertex with an edge which is in En+1. Thus the length e of any edge formedby splitting an edge in some Ei must satisfy

1

4CDr

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Roger W. Barnard and G. Brock Williams- Combinatorial Excursions in Moduli Space

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