Institute of Mathematics, University of Gdańsk, Gdańsk, Poland b Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia

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1 This article was downloaded by: On: 15 January 2010 Access details: Access Details: Free Access Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Algebra Publication details, including instructions for authors and subscription information: On generalized fibonacci groups with an odd number of generators Andrzej Szczepański a ; Andrei Vesnin b a Institute of Mathematics, University of Gdańsk, Gdańsk, Poland b Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia To cite this Article Szczepański, Andrzej and Vesnin, Andrei(2000) 'On generalized fibonacci groups with an odd number of generators', Communications in Algebra, 28: 2, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 COMMUNICATIONS IN ALGEBRA, 28(2), (2000) ON GENERALIZED FIBONACCI GROUPS WITH AN ODD NUMBER OF GENERATORS Andrzej Szczepariskit Andrei Vesnint Institute of Mathematics Sobolev Institute of Mathematics University of Gdarisk Russian Academy of Sciences ul. Wita Stwosza 57 pr. Koptyuga Gdarisk Novosibirsk Poland Russia matas@paula.univ.gda.pl vesnin@math.nsc.ru Abstract The geometrical properties of cyclically presented groups of Fibonacci type F(r,m,lc) and H(r,m,k) are discussed. It is shown that for even r and odd m some infinite family of generalized Fibonacci groups F(r, m, k) cannot be fundamental groups of hyperbolic 3-orbifolds of finite volume. 1 Introduction In the present paper we study geometrical properties of a class of cyclically presented groups of Fibonacci type. In 1965 Conway defined a series of groups F(2, m), rn >_ 3, with the presentation where all the suffices are reduced mod m. The natural generalization of these groups was introduced and studied in [8]. The groups F(r, m), r 2 2, m 2 3, with the presentation tsupported by the Polish grant (BW ). $Supported by the Russian Foundation for Basic Research (grant number ). 959 Copyright by Marcel Dekker, Inc.

3 960 SZCZEPANSKI AND VESNIN are said to be Fibonacci groups. The first question on these groups was: is the group F(r, m) finite? A review of results can be found in (71. The study of 3-manifolds uniformized by Fibonacci groups F(2, m) with an even number of generators was started by Helling, Kim and Mennicke (51. We recall that a 3-manifold M is said to be hyperbolic if it can be obtained as the quotient-space M = E13/r, where E13 is the Lobachevsky (also called hyperbolic) 3-space, and r c 1som(IH3) is a discrete, torsion-free group of isometries. From [5] the group F(2,6) is the fundamental group of the Hantzsche-Wendt flat 3-manifold, and F(2,2n), n 2 4, is the fundamental group of a closed orientable hyperbolic 3-manifold. The groups F(2, m) with an odd number of generators have completely different geometrical properties. It is easy to see [7, p. 841 that they have a torsion and so cannot be fundamental groups of Euclidean or hyperbolic 3-lnanifolds. We recall that a hyperbolic 3-orbifold C3 is the quotient space is torsion- H3/r, where r is a discrete group of isometries (in particular, if I? free we get a manifold). It was shown in [9] that if m is odd, then the group F(2, m) cannot be the fundamental group of a hyperbolic 3-orbifold of finite volume. However in [6] it is proved that F(2,9) is automatic. Moreover, if r is even, and m is odd and coprime with r + 1, then the Fibonacci group F(r, m) cannot be the fundamental group of a hyperbolic 3-orbifold of finite volume [12]. In the present paper we consider two series of groups of Fibonacci type. The first series are the generalized Fibonacci groups F(r, m, k), r > 2, m 2 3, k 2 1, with the presentation F(r, m, k) = (xl,..., xm ( xi xi+^. +. xi+,-l = ~~+,-~+k, i = 1,..., m), where t,he suffices are reduced mod m. The second series are the groups H(r, m, k), r > 2, m > 3, k 2 1, with the presentation H(r, m, k) = (xl,..., x, I x,xi+~... xi+,-l = x,+~.. xi+,-l+k, i = 1,..., m), where the suffices are reduced mod m. These groups were introduced in [2]. We have F(r,m, 1) = F(r, m) and H(r,m, 1) = F(r,m). Hrre we generalize the method from [9] and [12] for the groups F(r, 7n, k) Wr show that for triples of integrrs r, m, and k with r evrn, m odd and coprime with r + 2k - 1, the groups F(r,m, k) cannot be the fundamental gro~~ps of hyperbolic 3-orbifolds of finite volume. Moreover, we discuss some rxtrcnial cascs with rn = r +2k particular our groups are fundamental groups of Neuwirth manifolds (31, [ll], of Sieradski man~folds [4], and of

4 FIBONACCI GROUPS WITH ODD NUMBER OF GENERATORS 96 1 Briclskorr~ rr~nrlifoltls [lo]. Wr also.;how that thrrr is a relation t)rtwc.cn thc groups H(r, ~n, k) and the generalized Sieratlski groups S(k, n) (cf. [4]). 2 The Groups F(r, m, k) Applying the methods from [9] and [12] for the groups F(r, m, k) with an odd number of generators, we get the following result. Theorem. Sl~ppose that r is cvc11, m is odd and coprinw with r + 2k - 1. Then the group F(r, m, k) cannot be the fundamental group of a hyperbolic 3-orbifold of finite volume. Proof: Let p : F(r, m, k) 4-1som(lH3) be a faithful representation such that F(r, m, k) = p(f(r, m, k)) is a discrete group of finite covolume. Obviously, the group F(r,m, k) admits an automorphism a: permuting the generators cyclically. By the Mostow rigidity theorem there exists an isometry t E 1som(JH3) such that a(y) = tyt-' for each y E F(r,m, k). Let us consider the natural extension r(r, m, k) = (F(r, m, k), t). That is the fundamental group of a hyperbolic 3-orbifold of finite volume. Since tm does commute with all elements of the non-elementary group F(r, m, k), we get tm = 1 (cf. [I]). So, t is of degree ml, where ml is a divisor of m. Using xi+l = txit-l, from the relation ~1x2... x, = x,+k, we get Obviously, Consider the group r(2)(~, m, k) = (r2 ( E r(r, m, k)). Since ml is odd, t E I'(2)(r, m, k). Moreover, since r is even, we get.rl E r(2)(r, m, k) and r(*)(r, m, k) = r(r, m, k). Therefore T(r, m, k) is a group of orientation preserving isometries of IH3, and r(r, m, k) c PSL(2, C). Without loss of generality we can assume that t = P ( ) where ( is a 2ml'th root of unity, and P ( :) denotes the image in PSL(2, C) of ) E SL(2, F). the matrix ( : :

5 962 SZCZEPAI~SKI AND VESNIN Let sl = P ( ) with xw - yr = 1. Remark that yr # 0, because z w - F(r, m, k) has finite covolume. From (1) we get By induction on j it follows that From (2), using (3), we get Since yz # 0, we have and so, <2r+4k-2 = 1 But ( is the primit,ive root of degree 2ml and m is coprime with r + 2k - 1. Therefore, the group F(r, m, k) cannot be the fundamental group of a finite volume hyperbolic 3-orbifold. 0 3 Some Non-hyperbolic Manifolds In this section we discuss some groups F(r, m, k) which do not satisfy of the conditions of the theorem. Suppose that

6 FIBONACCI GROUPS WITH ODD NUMBER OF GENERATORS 963 If k = 1 then from (4) we get m = r + 1. The corresponding groups F(m - 1, m, 1) are the Fibonacci groups F(m - 1, m) which were discussed in the Neuwirth paper [ll]. From [ll] and [3] the following property holds. Proposition 1. For a given m the group F(m - 1, m, 1) is the fundamental group of the Siefert fibered space Em = ( (2, I), (2, I),..., (2, 1) ). \, m times In the particular case, r = 2, according to (4) we get groups Let us consider the Sieradski groups S(n) with the cyclic presentation S(n) = (all..., a, I aiai+2 = ai+l, i = 1,..., n), where all the suffices are reduced mod n. The topological properties of the groups S(n) were studied in 141. It is easy to see that the groups F(2,2k+l, k) and S(2k + 1) are isomorphic under the correspondence From the results on the Sieradski groups [4], we get Proposition 2. For k 2 1 the Sieradski group F(2,2k + 1, k) 2 S(2k + 1) is the fundamental group of a closed 3-manifold which can be obtained as a (2k + 1)-fold cyclic covering of the 3-sphere branched over the trefoil knot. We should mention (cf. 1101) t,hat the group F(2,2k + 1, k) is the fundamental group of the Brieskorn manifold B(3,2,2k + 1) which is spherical for 2k + 1 < 6 and of the E(2,IR)-manifold for 2k + 1 > 6. Summing up, we observe in both cases described in Propositions 1 and 2 that t,he groups F(r, m, k) cannot be the fundamental groups of hyperbolic 3-orbifolds of finite volume. 4 The Groups H(r, m, k) In this section we discuss properties of the groups H(r, m, k) in some particular cases. The generalized Sieradski groups S(k, n) were defined in (41: S(k, n) = (a,,..., a, ( aiai+z... ar+2k-2 = ai+iai+3... ai+zk-3, i = 1... n).

7 964 SZCZEPA~SKI AND VESNIN We have S(2, n) = S(n). It is easy t,o see that groups H(k, 2k - 1, k - I) and S(k, 2k - 1) are isomorphic under the correspondence Due to results on the generalized Sieradski groups [4], we get Proposition 3. For k > 2 the group H(k, 2k - 1, k - 1) r S(k, 2k - 1) is the fundamental group of a closed 3-manifold. This manifold can be obtained as a (2k - 1)-fold cyclic covering of the 3-sphere branched over the torus (2k - 1,2)-knot. For k = 2 we get the 3-fold cyclic covering of the 3-sphere branched over the trefoil knot. We remark that in terms of [lo] the group H(k, 2k- 1, k - 1) is the fundamental group of the Brieskorn manifold B(2,2k - 1,2k - 1) which is spherical for k = 2, and of the g(2, JR-manifold for k > 3. ACKNOWLEDGMENTS Boths authors wish to thank the referee for his many helpful comments and suggestions concerning the presentation of the paper. References [I] A. Beardon, The geometry of discrete groups, Springer-Verlag, New York - Heidelberg - Berlin, [2] C. M. Campbell, E. F. Robertson, A class of finitely presented groups of Fibonacci type, J. London Math. Soc. 11 (1975), [3] A. Cavicchioli, Neuwirth manifolds and colourings of graphs, Aequationes Math. 44 (1992), (41 A. Cavicchioli, F. Hagenbarth, A. C. Kim, A geometric study of Sieradski groups, Preprint, [5] H. Helling, A. C. Kim, J. L. Mennicke, A geometric study of Fibonacci groups, Journal of Lie Theory 8 (1998), [6] D. F. Holt, An alternative proof that the Fibonacci group F(2,9) is infinite, Experimental Mathematics 4 (1995),

8 FIBONACCI GROUPS WITH ODD NUMBER OF GENERATORS 965 [7] D. L. Johnson, Topics in the theory of group presentations, London Math. Soc. Lecture Notes Series 42, Cambridge Univeersity Press, [8] D. L. Johnson, J. W. Wamsley, D. Wright, The Fibonacci Groups, Proc. London Math. Soc. 29 (1974), [9] C. Maclachlan, Generalizations of Fibonacci numbers, groups and manifolds, Combinatorial and Geometric Groups Theory, Edinburgh 1993, ed. by A. J. Duncan, N. D. Gilbert and J. Howie, LMS Lecture Notes 204 (1995), (101 J. Milnor, 011 the 3-ditnensional Brieskorn manifolds M(p.q.r), Knots, Groups and 3-Manifolds, ed. by L.P.Neuwirth Ann. of Math. Studies 84, Princeton Univ. Press, Princeton, N. J., 1975, (111 L. Neuwirth, An algorithm for the construction of 3-manifolds from 2- complexes, Proc. Cambridge Phil. Soc. 64 (1968)) [12] A. Szczepanski, The euclidean representations of the Fibonacci groups, Preprint, University of Gdarisk, Received: October 1998