.5'lbertao Mathematical Journal. 1"ol. 3!). No. 4, 1998

F R A C T I O N A L F I B O N A C C I G R O U P S A N D M A N I F O L D S t) A. Yu. Vesnin and A. C. K i m UDC 515.162+512.54

w 1. F rac t iona l F ibonacc i G r o u p s

The present article is devoted to generalization of Fibonacci groups and study of geometric and topological properties of the corresponding closed orientable 3-manifolds.

The Fibonacci groups F(2, m) were introduced by J. Conway [1] and have the following presenta- tions:

F(2, m) = (Xl, x2 , . . . , xrn I xi X i + l = Xi+2, i = 1 , . . . , m),

with all indices taken mod m. The nomenclature is due to the analogy between the group relations and the inductive definition of the Fibonacci numbers

ai + ai+l = ai+2,

with initials al = a2 = 1. The study of Fibonacci groups was inspired by Conway's question: are the groups F(2, m) finite? The answer was given in [2-5]: the group F(2, m) is finite if and only if m = 1,2,3,4,5,7.

In [6] D. L. Johnson introduced the following generalization of the Fibonacci groups, called the generalized Fibonacci groups:

F(r, m) = (xl, x2 , . . . , Xm [ X i ' ' ' X i + r - 1 = Xi+r, i = 1 , . . . , m),

where r > 2 is a given integer and all indices are taken mod m. A survey of the results on the groups F(r , m) can be found in [7], where in particular facts about the groups F(r, m), 2 < r < ]0 and 2 < n < 10, are summarized.

Another interesting generalization of the Fibonacci groups was introduced by C. Maclachlan [8]. For integers k > 1 he considered the groups

Fk(2, m ) = ( x l , x 2 , . . . , z m l x i x ~ + l = z i + 2 , i = l , . . . , m ) , m

with all indices taken mod m, and constructed a series of closed orientable 3-manifolds M(k ,n ) whose fundamental groups are isomorphic to Fk(2,2n). Moreover, it was shown in [9] that the manifolds M(k, n) are Haken and hyperbolic for all k >_ 2 and n > 3.

One can see a certain resemblance between this generalization of the Fibonacci groups and the following generalization {a/k } of the Fibonacci numbers:

+ kah, = oL2,

with initials a~ = 1 and a~ = k and the inductive law dependent on a given integer parameter k. It seems natural to generalize this inductive law to a fractional parameter. More exactly, let us consider

the "fractional" analog {a~ It} of the Fibonacci numbers with the recursive law

ak/t k/l _k/t i + k/lai+l = ui+2

t l The authors are thankful for support to the Korean Science and Engineering Foundation KOSEF (Grant 96-0701- 03-01-3} and the Russian Foundation for Basic Research (Grants 95-01-01410 and 98-01-00699).

Novosibirsk. Pusan. Translated from Sibirskff Matemalicheskif Zhurnal, Vol. 39. No. 4, pp. 765-775, July-August, 1998. Original article submhted November 14, 1996.

0037-4466/98/3904-0655 $20.00 @ 1998 Plenum Publishing Corporation 655

_kit = kll Continuing the analogy with groups, for a fractional number k/l with initials a~/t = 1 and "'2 we should naturally consider the groups with presentation

Ixk l FlU~t(2,m) = ('Tl,X2,' ' ' ,Xm ] Zi i+1 = Xi+2, i - - - - I , . . . ,m)

for coprime integers k and l and an integer m _> 3, with all indices taken mod m. The group Fk/I(2, m) is said to be a fractional Fibonacci group.

The aim of the present article is to demonstrate that the generalization of the Fibonacci groups F(2, m) to the fractional Fibonacci groups Fk/l(2, m) is natural from a topological point of view.

We recall that a new geometric stage in the study of Fibonacci groups has started with the article [10] (see also [11]) by H. Helling, A. C. Kim, and J. Mennicke in which it was shown that the groups F(2.2n) arise as the fundamental groups of closed orientable 3-manifolds. More exactly, the group F(2, 2n), n _> 4, is isomorphic to the fundamental group of a closed orientable hyperbolic 3-manifold. The group F(2, 6) is isomorphic to the fundamental group of the Euclidean Hantzsche-W'endt manifold studied in [12]. The group F(2, 4) is finite and isomorphic to the fundamental group of the lens space L(5,2). Thus, the group F(2,2n), n >_ 2, is realizable as a co-compact discrete group of isometries acting without fixed points on the space Xn, where X2 = S 3 is the spherical 3-space, X3 = E 3 is the Euclidean 3-space, and Xn = H 3 is the LobachevskiT 3-space for n _> 4.

The 3-manifold M'n = Xn/F(2,2n), n _> 2, uniformized by a Fibonacci group, is referred to as a Fibonacci manifold.

We recall that the Fibonacci manifolds Mr,, n >_ 2, can be characterized by the following topolog- ical properties:

(1) the manifold Mn can be obtained as the n-fold cyclic covering of the 3-sphere branched over the figure-eight knot (see [13]);

(2) the manifold M,, can be obtained by Dehn surgeries with parameters 1 and - 1 on the com- ponents of the chain of 2n linked circles in S 3 (see [14]);

(3) the manifold Mn can be obtained as the two-fold covering of the 3-sphere, branched over the Turk's head link Thn which is the closure of the 3-string braid (awe-l)" (see [15]).

We will consider generalization of Fibonacci manifolds. In Section 2 we define a family of 3- dimensional closed orientable manifolds M~/t with the fundamental group 7rl (A[~/l) = Fk/l(2,2n). These manifolds are referred to as fractional Fibonacci manifolds. In particular, for k/l = 1 we receive the Fibonacci manifoldsMn of [10], whereas the case of l = 1 and k an integer corresponds to the Maclachlan manifolds M(k, n) of [8].

In Section 3 we show that the fractional Fibonacci manifolds M~/l have the following topological properties which are direct generalizations of the corresponding properties of the Fibonacci manifolds Mn:

(1") the manifold M 1/i can be obtained as the n-fold cyclic covering of the 3-sphere branched over the two-bridge knot K(21 + 1/2l) (see Corollary 3.3);

(2*) the manifold M~/l is obtained by Dehn surgeries with parameters k/l and - k / l on the components of the chain of 2n linked circles in the 3-sphere (see Definition 2.1);

(3*) the manifold Af~/l can be obtained as the two-fold covering of the 3-sphere branched over [ k/! -k/l\n

the link Thkn/t which is the closure of the rationaL3-string braid l~r I o" 2 ) (see Corollary 3.2).

Moreover, the fractional Fibonacci manifolds M~/l are hyperbolic for all but finitely many pairs of coprime integers k and l and for every integer n > 3 (see Corollary 2.1).

w 2. F rac t iona l F ibonacc i Mani fo lds

lfk/l The construc- In this section we construct the family of the fractional Fibonacci manifolds :,-n �9 tion generalizes the approach of A. Cavicchioli and F. Spaggiari to describing the Fibonacci manifolds .tin in terms of the Dehn surgery [14].

656

Given an integer n > '2. we consider the link L2. with 2n components each of which is unknotted and linked with exactly two adjacent components. The diagram of the link L6 is depicted in Fig. 1.

i i

;_ . . .-_ -_ .....

Fig. 1 It was shown by W. Thurston [16] that for n > 3 the link L2n is hyperbolic and the hyperbolic

volume of the complement S3 \ L~.,~ is given by the formula

(4 vol(S 3 \ L 2 . ) = 8 n h r + + A

where A(x) is the Lobachevski~ function:

A(x) = - f log 12 sin 0[

o dO.

We remark that there is an obvious asymptotic formula for these volumes:

vol (S 3 \ L2n) '~ 16A(~'/4) �9 u = 7.3277... . n, n ---* oc,

where 16A(r/4) = 7.3277... is the volume of the Borromean rings complement. Let us consider the closed manifolds M(pl/ql,...,p2n/q2n) obtained by Dehn surgeries on the

cusps of the manifold S 3 \ L2n with surgery parameters pi/qi, i = 1 , . . . , 2 n , where pi and qi are coprime integers.

L e m m a 2.1 [17]. With the above notations, the following group presentation holds:

-q2k ' l ' la - -P2k+2 ~_ a q2k+3 ~ l ( M ( p l / q x , . . . ,p2n/q2n)) - - ( a l , . . . , a 2 n I '%k+1 2k+2 2k+3,

aq2k,,P2k+l ~ q 2 k + 2 k = 1 . , n ) . 2k'~2k+l ='~2k+2, . ' ' "

In virtue of Lemma 2.1, the following relation, observed in [14], holds between the manifolds M(pl /q l , . . . , P2n/q2n) and the Fibonacci groups F(2, 2n). Suppose that qi = 1 and Pi = ( -1 ) i+I for i = 1 , . . . ,2n . Then the group

~rl ( M ( 1 , - 1 , . . . , 1 , - 1 ) ) = (al,... ,a2n [ aiai+l = ai+2, i - - 1 , . . . , 2n ) ,

where the indices are taken rood 2n, is the Fibonacci group F(2, 2n) and the manifold M ( 1 , - 1 , . . . , 1 , -1 ) is the Fibonacci manifold Ma. Thus, the" Fibonacci manifolds Mn can be obtained by Dehn surgeries with parameters 1 and - 1 on the components of the link L2n.

DEFINITION 2.1. For any coprime integers k and I and for any integer n _> 2, the closed 3-manifold

Itk/t M(k/l. - k / l . . . . . k/l, - k / l ) obtained by Dehn surgeries with parameters k/l and - k / l on the zw I n

components of the link L2n is said to be a fractional Fibonacci manifold. The following lemma demonstrates that the definition is natural.

L e m m a 2.2. The fundamental group of the fractional Fibona~ci manifold M kit is isomorphic to the fractional Fibonacci group Fk/t(2.2n).

657

PROOF. Apply Lemma 2.1 with qi = I and Pi = ( -1 ) i+ lk for i = i . . . . ,2n. Then for the manifold Mr, "/t = M(k / l , - k / l , . . . , k / l , - k / l ) we obtain the group

I') ,a2.1 ' = d = , . . . aiai+l i+2, i = 1 , . . . , 2n> ,

with all indices taken ,nod 2n, which is the fractional Fibonacci group Fk/l(2,2n). Using Lemma 2.1, we can study geometric structures that correspond to the fractional Fibonacci

groups. Since the manifolds M}n/l are obtained by Dehn surgeries on the hyperbolic links L2n; accord- ing to the Thurston-Jcrgensen theory of hyperbolic surgery [16], we arrive at the following

C o r o l l a r y 2.1. For every integer n _> 3 and all but finitely many pairs of coprime integers k and l, the fractional Fibonacci manifolds M k/t n are hyperbolic manifolds.

Considering the fundamental groups of these manifolds, we obtain

C o r o l l a r y 2.2. For every integer n >_ 3 and all but t~nitely many pairs of coprime integers k and l, the fractional Fibonacci groups

F } / t ( 2 , 2 n ) ( a , , . ,a2, , [ t k = a t ,2n> = "" a i a i + l i + 2 , i = I , . . .

are hyperbolic groups.

Since the manifolds Mkn/l are defined in terms of the Dehn surgery, the computer program SnapPea [18] can be used for studying their hyperbolic structures. Considering the link L6 in Fig. 1, whose complement has hyperbolic volume 14.6554.. . , we in particular obtain the following table of volumes

of the hyperbolic manifolds M~/I for small k and 1:

l\k I 2 3 4 5

l Euclidean 5.3348.. . 9.1036,. . 11.1990.., 12.3607., ,

2 5.2870.. . - - 10.5996... - - 12.6621.. .

3 9.0210.. . 10.5968... - - 12.4427,., 12.9847...

4 11.1506... - - 12.4425... - - 13.2737.. . 5 12.3406... 12.6572... 12.9840... 13.2736...

Moreover, calculations by SnapPea prompt us to conjecture that the manifolds M~ II, n >__ 3, are always hyperbolic with M31 the only exception.

The recursive definition of the fractional Fibonacci groups allows us to find the first homology group of the fractional Fibonacci manifolds. It is easy to see that, by analogy to [8, 11], the commuta tor

quotient group Fk/t(2, 2n) ab can be expressed in terms of the fractional Fibonacci numbers a~/t and a fractional analog of Lucas numbers. In particular, the following simple formula holds for the manifolds Mk31l:

Hi (M~/t, Z) = Fk/t(2, 6) ab = Zka+Skl2 ~ Zka+3kl2.

w 3. Covering Properties of Manifolds

In this section we study some topological properties of the fractional Fibonacci manifolds M~ ft. We demonstrate that these properties can be considered as a direct generalization of the corresponding properties of the Fibonacci manifolds Mn.

First of all, we define some series of links. We recall [19] that any link can be obtained as the

closure of some braid. Given coprime integers p and q, denote by aft q the rational p/q-tangle whose incoming arcs are the ith and (i + l)th strings.

658

T h e o r e m ;3.1. Every closed manifold M(pl/qa . . . . . p2n/q2~) can be obtained as the two-fold covering of the 3-sphere branched over the closed rational 3-strings braid cr~ l/q~ ... a~ ~''dq~'"

P a o o v . By definition, the manifold M(pl/ql,....P2n/q'tn) is obtained by Dehn surgeries with parameters pl/ql, ... p2n/q'2~ on the components of the link L2n. It is obvious from Fig. 1 that there is an involution p of the 3-sphere whose axis r, indicated by a dashed line, intersects each component of the link L2n in two points. Thus, the above manifold is obtained by Dehn surgeries on a link which is strongly-invertible according to the terminology of [20]. By the Montesinos theorem [20], the manifold M(pl/ql . . . . . p2n/q2n) can therefore be obtained as the two-fold covering of the 3-sphere branched over some link. Moreover, the Montesinos algorithm [20] provides a description for this link. Denote by Ni, i = 1 , . . . , 2n, a tubular neighborhood of the ith component of the link L2n. The image of Ni under the canonical projection p : S 3 ~ $3/p is the trivial tangle consisting of a 3-ball Bi with two interior arcs formed by the image p(r N Ni). For i = 1 , . . . , 2n, denote by B~ the pi/qi-tangle with the underlying 3-ball Bi. According to the Montesinos algorithm, the manifold M(pl/ql, .. . , P2n/q2n) is the two-fold branched covering of

2n

- i =1 "

where the branch set is a link formed by-arcs of tangles B I and by the images p(r N ($3\ UNi)). Using the Reidemeister moves, it is easy to see that the branch set is the link that can be described as the

closure of the rational 3-strings braid a~ l/q~ a~2/q2 ... ap2,-l/q2,-ll . 2~zP2"/q2"" [2]

Coro l l a ry 3.1. Given a closed 3-stings braid K, let the manifold M(K) be the two-fold covering of the 3-sphere branched over K. Then M(K) can be obtained by Dehn surgeries on the components of the link L'2~.

Given an integer n _> 1, we denote by Th~/q the n-periodic link. i.e., the closure of the rational .3-

strings braid (a~/q~2P/q) n. The diagram of the knot Th~/2 arising in the case of n = 2 and p/q = 3/2 is depicted in Fig. 2.

Fig. 2

In particular, for p/q = 1 we obtain the Turk's head link Th,~ which is the closure of the 3-strings braid (axcr.; "1 )".

It was shown in [15] that the Fibonacci manifolds Mn, n > 2, can be obtained as the two-fold coverings of the 3-sphere branched over the Turk's head links Thn. As a particular case of Theorem 3.1

~lk/l appears the fact that the fractional Fibonacci manifolds ~, , have an analogous property.

Coro l l a ry 3.2. For any coprime integers k and 1 and for any integer n >_ 2 the fractional Fibonacci manifold M~/l can be obtained as the two-fold covering of the 3-sphere branched over the link Th~/I

Since the manifolds M(k, n) with the fundamental groups

= F k ( 2 . 2 n ) = (a l . . . . . I = i = 1 , . . . . 2 n ) .

659

conslructed in [8]. can be obtained as the two-fotd coverings of the 3-sphere branched over the links Th~ [9]. we see that for each pair k, m the manifold ~ ( k , n ) is homeomorphic to the fractional

~k Fibonacci manifold . 1,~.

Now, let us consider the particular case of the link Thk,/l with k = 1 and n = 2. Obviously, the

links Th~/l are knots and have a simple structure.

L e m m a 3.1. For every integer l, the link Th~/I is the two-bridge knot K(21 + 1/21).

PROOF. We recall that the two-bridge knot K(21 + 1/21) is the closure of the rational (2l + 1/2/)-

tangle. Both knots Th~/l and K(21 + 1/2l) are alternating knots of order 41, and their equivalence follows from the definition of the rational tangle operations [21]. []

W'e note that, according to Lemma 3.1, the knot Th~ is the figure-eight knot K(5/2) = 41 and

Th~/2 is the knot K(4 + 1/4) = 83 in the notations of [19].

Given an integer l, consider the link s whose diagram with 21 + 6 cross-points is depicted in Fig. 3.

I 'v #

! ;

2[-

Fig. 3

L e m m a 3.2. For every integer I, the link E. 1/l is two-component, and its components are equiv- alent. Moreover, s is the two-bridge link K(p/q) with p = 812 + 2 and q = 4l 2 - 21 + 1.

PROOF. It is obvious from Fig. 3 that E ill is a two-component link, with every component unknotted. It easy to see by using the Reidemeister moves that the components of the link s are equivalent. Moreover, using the rational tangle operations, we see that s is the two-bridge link K(p/q), where p = 8l 2 + 2 and q is an odd number such that 21q = +1 mod (4l 2 + 1) with q <_ p. Hence, q = 4 1 2 - 2 1 + 1 . 0

Observe that s is the 2-component link 6~, i.e., the two-bridge link K(10/3), while/;1/2 is the 2-component link ~ ~5~, i.e.. the two-bridge link K(34/13).

We now consider orbifolds whose underlying space if the 3-sphere and singular sets are the above

links. Denote by Th~/l(2) the orbifold whose underlying space is the 3-sphere and whose singular set is

the link Th~/t with branch index 2. Similarly, denote by Th~/t(n) the orbifold whose singular set is the

link Th~/l (which the 2-bridge knot K(21 + 1/21) by Lemma 3.1) with branch index n. Also, consider the orbifold s n) whose singular set is the two-component link s with branch indices 2 and n corresponding to its components (which are equivalent by Lemma 3.2). Then we have the following

660

T h e o r e m 3.2. With the above notations, for all integers l >_ 1 and n >_ 2 the following covering

diagram is commut.ative: M1/,

The/t(2) Th~/t(n)

s n) _ _ ffl/t(2, n)

PROOF. Consider the orbifold s n) whose singular set is the link s in Fig. 3. Let a, ~3. "). 5, e, a, v, and t~ be the loops indicated in Fig. 3. To find the presentation of the fundamental group of the link s we will use the following observation. Given an integer k, consider a k-tangle with loops a and b corresponding to incoming arcs and with loops at~ and bk corresponding to outcoming arcs. Then ak and bk can be expressed through a and b by some elementary formulae that are proven directly by induction using the Wirtinger algorithm.

L e m m a 3.3. (i) I l k < 0 then b~ = 9~(a,b) and a~ = 9~_~(a,b) with

gk(a, b) = x l l x 2 1 . XklXk+lZk . . . X2Xl,

b, i = 1 mod2, and xi =

a, i = 0 mod2; (ii) f f k > 0 then a~ = f~(a,b) and b~ = f~_~(a,b) with

fk(a, b) yly2 YkYk+IY'k ~ -1 -1 = . . . . . . Y2 Yl

a, i = l mod2, a n d y i = b, i = 0 mod2.

According to Lemma 3.3 and Fig. 3, we have

Since

we have

and

Now, consider th~ word Wl(a, fl) = ~- Ip f lv . Using the above expressions for v and tL, we obtain

wt(c,, gt_l yt_l a - l O, a ) )

Therefore. we come to the following presentation for the link group:

7 r 1 ( $ 3 \ / ~ 1 / I ) = (ot,~ l wl(ot,~) = 1);

where wl(a, 3) is given by the above formula. Consider the orbifold s n). Its singular set is the l ink 's with branch indices 2 and n

corresponding to the components indicated in Fig. 3. According to the general properties of the

661

fundamemal group of an orbifold [22], using the presentation for the group 7Q($3 \ s we derive the equality

~,l(Z2/t(2, n)) = (~.~3 I a2 = 3" = 1, wt(a, J) = 1),

where the generators 5 and ;} correspond canonically to c~ and/3. Look at the group

g.2 ,~ g . = (a I ~2 = 1 ) - (b I b" = 1)

and the epimorphism O : rrl(s n)) ~ Z~. G Zn

defined by the conditions 0(&) = a and 0(,t~) = b.

By Theorem 3.1 the fractional Fibonacci manifold M~/1 is the two-fold covering of the 3-sphere branched over the link Th~n/t. In other words, M~n/t is the two-fold covering of the orbifold The~t(2). According to the definition of the link s the orbifold Thl/t(2) is the n-fold cyclic covering of the orbifold s n). Hence, we arrive at the following diagram of coverings

,M~/z _~ Th~/l(2) -~ s

which induces the sequence of group embeddings

FUr(2, 2n) ,a rl(Th~/t(2)) ,~ ~rl(s n)),

with [rl(s n)) : ZCl(Th~/t(2))] = n and [Trx(Th~/t(2)) : F'/t(2,2n)] = 2. By the construction of the n-fold covering Thllt(2) --* s n), the loop/3 in rl(s n)) lifts

to a trivial loop in 7rl(Thln/t(2)), and the loop & in rl(s n)) lifts to a loop in rrx(Thln/t(2)) which generates a group of order 2. Hence,

r,l(Th~/t(2)) = o-l((a [a ~" = 1)) = O-~(Z2).

Under the 2n-fold covering Mln It ~ s n), both loops & and ~ in rl(/Zl//(2, n)) lift to trivial loops in F1/l(2, 2n). Hence, FUr(2, 2n) = Ker0.

Denote by F(I, n) the subgroup of ~rl(s n)) given by

r ( t , n ) = 0 - 1 ( 0 1 b" = 1)) = 0 - 1 ( z . ) .

We thus have the sequence of normal subgroups

Fl/t(2,2n) .~ r(l,n) ,~ 7rl(~,l[l(2,n))

where n ) ) : r ( t ,n) ] = 9 and [r(t ,n) : F U t ( 2 , 2 n ) ] = n.

Under the two-fold covering corresponding to the group embedding F(l,n) 4 7rl(s the loop & lifts to a trivial loop, and the loop ~ lifts to a loop which generates a cyclic group of order n. Therefore, this two-fold covering is branched over one of components of the link s In virtue of Lemma 3.2 its components are equivalent. Using Lemma 3.1, we infer that this covering space is the orbifold Th~/t(n) whose singular set is the two-bridge knot K(21 + 1/2/). Considering the embedding

Fl/t(2,2n) 4 F(/, n), we deduce that the manifold M~/1 is the n-fold cyclic covering of the orbifold Th~/t(n) and

M l/t ~ Th~/l(n) -~ s n).

Comparing the above sequences of coverings, we find out that the covering diagram in the s tatement of tile theorem is commutative. []

The particular case of Theorem 3.2 with l = 1 for the Fibonacci manifolds was proven in [15].

6~2

,,1/l Corol lary 3.3. For any integers I and n >_ 2. the fractional Fibonacci manifold ~vl,, is the n-fold cyclic covering of the 3-sphere branched over the two-bridge knot K(2l + 1/2l).

We recall that if l = 1 then we obtain the Fibonacci manifold M,~ and the figure eight knot K(5/2).

Since the two-fold coverings of the 3-sphere branched over two-bridge knots are studied rather well [19, p. 183], we come to the following statement.

Corol lary 3.4. For every integer l, the fractional Fibonacci manifold M~/t is the lens space L(412 + l, 2l) and F1/t(2, 4) = Z4t2+l.

Using the results on hyperbolicity of orbifolds whose singular sets are two-bridge knots [23], we derive

Corol lary 3.5. For any integers l > 2 and n > 3, the fractional Fibonacci manifold M~/~ is hyperbolic and the group F1/t(2, 2n) is intinite.

We recall that arithmeticity of the Fibonacci groups was studied in [10] and [13], where it was shown that the group F(2,2n) is arithmetic if and only if n = 4,5,6,8, 12.

In virtue of the results of [23], Corollary 3.3 gives information on arithmeticity of the fi'actional Fibonacci groups.

Corol lary 3.6. The fractional Fibonacci group FU2(2,2n), n > 3, is arithmetic if and only if n = 4. The fractional Fibonacci group FU3(2,2n), n > 3, is nonarithmetic for any n.

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