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The Rank Problem for Sufficiently Large Fuchsian Groups

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The Rank Problem for Sufficiently Large Fuchsian Groups The rank problem for sufficiently large Fuchsian groups Richard Weidmann Fachbereich Mathematik Johann Wolfgang Goethe-Universit¨at 60054 Frankfurt Germany [email protected] October 18, 2005 Abstract We give a complete solution for the rank problem of sufficiently large Fuchsian groups, i.e. we determine their minimal number of generators. This class of groups includes most Fuchsian groups. Contents Introduction 2 1 Bass-Serre theory and approximations of graphs of groups 4 1.1 Graphs and graphs of groups . 5 1.2 The free decomposition of a graph of groups . 6 1.3 Propagation in a graph of groups . 6 1.4 A-graphs . 7 1.5 Foldings of A-graphs . 10 2 Graphs of dihedral groups 15 2.1 Simple generating sets . 15 2.2 Graphs of dihedral groups and their normalization . 20 2.3 The simple rank of a graph of dihedral groups . 23 2.4 Simple rank equals rank . 30 3 Planar groups 39 3.1 Decomposing planar groups . 39 3.2 The proof of the rank formula . 41 3.3 Comments on small Fuchsian groups . 47 Bibliography 49 1 Introduction Planar groups act properly discontinuously by isometries on S2, E2 or H2 and are classified by their orbifold, i.e. by the quotient space of the respective plane modulo the group action where one needs to record rotation orders of images of fixed points and reflections. The groups can then be recovered as the fundamental group of the orbifold. Details can be found in the the beautiful article of P. Scott [Sc]. An orbifold O can easily be described in the form O = F (p1, . , pk, (h11, . , h1n1 ),..., (hm1, . , hmnm )) where the following holds: 1. F is a closed surface. 2. pi ∈ N≥2 ∪ {∞}. 3. hij ∈ N≥1 ∪ {∞} where hij = 1 implies that j = ni = 1. The pi are the rotation orders of primitive rotations that are not the product of two reflections, if pi = ∞ then pi corresponds to a boundary component of O that contains no reflections. Each tuple (hi1, . , hini ) stands for a boundary component of O containing ni reflections. hij is the order of the element obtained by multiplying two adjacent reflections. A tuple of type (1) means that the reflection line is closed. 3 4 3 7 2 r r r 8 r r 4 r r Figure 1: The orbifold O = T 2(3, 4, (2, 3, 4), (7, 8), (1)) For any planar group G with orbifold O the Euler characteristic is defined as k X 1 1 X 1 χ(G) = χ(O) = χ(F ) − m − (1 − ) − (1 − ). pi 2 hij i=1 1≤i≤m 1≤j≤ni The Euler characteristic is well-behaved when passing to subgroups of finite index, namely if |G : U| = k then χ(G) = k · χ(U). There also is a covering 2 space theory for orbifolds that has all the nice properties the standard covering space theory has; again we refer to [Sc] for details. It is an important fact that one can read off from the Euler characteristic whether a planar group acts on the sphere, the Euclidean plane or the hyperbolic plane, namely we have the following: 1. If χ(G) > 0 then G is spherical. 2. If χ(G) = 0 then G is Euclidean. 3. If χ(G) < 0 then G is hyperbolic. We will be mostly interested in hyperbolic groups. For those groups we will occasionally distinguish two types of infinity, namely ∞ and∞ ¯ , where we use ∞ if the element corresponding to the respective boundary component is parabolic and∞ ¯ if it is hyperbolic. The rank problem for Fuchsian groups without reflection was solved in [Z1], [PRZ] by H. Zieschang, G. Rosenberger and N. Peczynski; see also [We5]. They show the following: Theorem [Zieschang, Peczynski, Rosenberger] Let G be a Fuchsian group with orbifold O = F (p1, . , pk). Then the following hold: 1. If k = 0 then rank G = −χ(F ) + 2 2 2. If χ(F ) = 2, i.e. F = S , k ≥ 4 is even, pi is odd for some i, pj = 2 for j 6= i then rank G = −χ(F ) + k = k − 2. 3. In all other cases rank G = −χ(F ) + k + 1. For Fuchsian groups containing reflections there are plenty of examples where the rank is significantly smaller than what one might expect from looking at the standard presentation, first examples were given in [KZ], more complicated ones in [We0]. The most substantial result regarding the rank of Fuchsian groups with reflections is due to E. Klimenko and M. Sakuma [KlS] who give a complete classification of 2-generated Fuchsian groups. In this article we give a solution to the rank problem for sufficiently large Fuchsian group where we say that a Fuchsian group is sufficiently large if m ≥ 2 or k ≥ 2 and if additionally one of the following holds: 1. There are 1 ≤ i < j ≤ k such that pi 6= 2 and pj 6= 2. 2. There are 1 ≤ i < j ≤ m such that (hi1, . , hini ) 6= (1) 6= (hj1, . , hjnj ). 3. F 6= S2 or F = S2, k ≥ 2 and there are 1 ≤ i ≤ k and 1 ≤ j ≤ m such that pi 6= 2 and (hj1, . , hjnj ) 6= (1). We further say that a Fuchsian group is small if it is not sufficiently large. In some sense the sufficiently large case is the generic case while the small case is exceptional. Note that class of small Fuchsian groups contains the planar re- 2 flection groups, i.e. those with corresponding orbifold O = S ((h11, . , h1n1 )). 3 2 Other simples examples are those with orbifold O1 = S (2, (2, 2l + 1)) and 2 O2 = S (2, 2l + 1, (1)). 2 It is easy to see that the groups with orbifold S (∞, (h1, . , hn)) are fun- damental groups of graphs of dihedral groups (see Section 2.2 for precise def- initions) and it turns out that the computation of the rank of a fundamental group of a graph of dihedral groups is the first of two steps in the computation of the rank of sufficiently large Fuchsian groups. This first step is achieved by assigning to any graph of dihedral groups A a labeled graph Θ(A) and then defining a simple complexity for those labeled graphs which we call the simple rank of Θ(A) and denote by srank Θ(A). The solution of the rank problems for fundamental groups of graphs of dihedral groups is then given by the following theorem. Theorem 0.1 (Theorem 2.10) Let A be a graph of dihedral groups. Then rank π1(A) = srank Θ(A). Suppose now that G is a sufficiently large Fuchsian group with correspond- ing orbifold O = F (p1, . , pk, (h11, . , h1n1 ),..., (hm1, . , hmnm )). Put F = 2 π1(F (p1, . , pk, ∞1,..., ∞m) and Ui = π1(S (∞, (hi1, . , hini ))) for 1 ≤ i ≤ m. It is easy to see that G admits a graph of group decomposition with infinite cyclic edge groups and vertex groups F and the Ui for 1 ≤ i ≤ m. The rank problem for the group F follows from [PRZ] if m = 0 and is a trivial consequence of Grushko’s theorem otherwise as for m ≥ 1 F is a free product of cyclic groups. The rank problem for the groups Ui is solved by Theorem 0.1. The following therefore provides an effective solution for the the rank problem for sufficiently large Fuchsian groups. Theorem 0.2 Let G be a sufficiently large Fuchsian group with rank G ≥ 3 and orbifold O = F (p1, . , pk, (h11, . , h1n1 ),..., (hm1, . , hmnm )). Then X rank G = rank F − m + rank Ui. 1≤i≤m It turns out that for the groups corresponding to the orbifolds O1 and O2 described above the formula of Theorem 2.10 does not hold. Those and other classes of small groups for which the conclusion of Theorem 2.10 fails are dis- cussed in Section 3.3. The paper is organized as follows: In the first chapter we recall some basic notions from Bass-Serre theory and foldings. We discuss how foldings can be used to approximate a given graph of groups starting with a very simple one. In the second chapter we study graphs of dihedral groups and prove The- orem 2.10 using folding sequences. In the third chapter we then prove Theo- rem 0.2 using similar ideas. 1 Bass-Serre theory and approximations of graphs of groups In this Chapter we review the notion of graphs of groups and how they can be approximated using A-graphs and foldings. 4 1.1 Graphs and graphs of groups We recall some basic notions about graphs of groups. For a detailed account of Bass-Serre theory we refer to the article of H. Bass [Ba] or the book of J.P. Serre [Ser]. A graph A is a tuple (V A, EA, α, ω,−1 ) where −1 : EA → EA is an involution and α, ω : EA → VA are maps such that α(e) = ω(e−1) for all e ∈ EA. We say that VA is the vertex set and that EA is the edge set. We further refer to α(e) as the initial vertex of e and ω(e) as the terminal vertex of e. A path γ in A is a sequence of edges e1, . , ek such that ω(ei) = α(ei+1) for 1 ≤ i ≤ k − 1. We call α(e1) the initial vertex of γ and ω(ek) the terminal −1 vertex of γ. The path γ is called a reduced path if ei 6= ei+1 for 1 ≤ i ≤ k − 1.
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