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 fundamental 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 corresponding 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 discussed 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. We further say that γ is closed if α(e1) = ω(ek). We denote the Betti number of a graph A, i.e. the number of edge pairs {e, e−1} outside a maximal subtree of A by b(A). A graph of groups A is a tuple

(A, {Av|v ∈ VA}, {Ae|e ∈ EA}, {αe|e ∈ EA}, {ωe|e ∈ EA}) where A is a connected graph, the Av and Ae are groups called the vertex groups and edge groups with Ae = Ae−1 for all e ∈ EA, αe is a monomorphism αe : Ae → Aα(e) for all e ∈ EA and ωe = αe−1 is a monomorphism ωe : Ae → Aω(e) = Aα(e−1) for all e ∈ EA. We call the maps αe and ωe boundary monomorphisms.

An A-path of length k ≥ 0 from v ∈ VA to v0 ∈ VA is a sequence

p = a0, e1, a1, . . . , ek, ak

0 where k ≥ 0 is an integer, e1, . . . , ek is a path in A from v ∈ VA to v ∈ VA, 0 where a0 ∈ Av, ak ∈ Av and ai ∈ Aω(ei) = Aα(ei+1) for 0 < i < k. We will call k the length of p and denote it by |p|. Note that we allow k = |p| to be equal to 0 zero, in which case v = v and p = a0 ∈ Av. If p is an A-path from v to v0 and q is an A-path from v0 to v00, then the concatenation pq of p and q is deﬁned in the obvious way and is an A-path from v to v00 of length |p| + |q|.

Let A be a graph of groups. Let ∼ be the equivalence relation on the set of all A-paths generated (modulo concatenation) by the elementary equivalences

−1 a, e, ωe(c), e , aˆ ∼ aαe(c)ˆa where e ∈ EA, c ∈ Ae and a, aˆ ∈ Aα(e) and

−1 a, e, aˆ ∼ aαe(c), e, ωe(c )ˆa where e ∈ EA, c ∈ Ae, a ∈ Aα(e) anda ˆ ∈ Aω(e).

If p is an A-path, we will denote the ∼-equivalence class of p by p. Note that p ∼ p0 implies that p and p0 have the same initial vertex and the same terminal vertex in VA.

5 Let v0 ∈ VA be a vertex of A. We deﬁne the fundamental group π1(A, v0) as the set of ∼-equivalence classes of A-paths from v0 to v0. It can be shown that G is in fact a group with multiplication corresponding to concatenation of paths. Throughout this article we will attempt to stick to the following convention: Whenever a graph of groups is denoted by the letter A or its derivatives like A¯ ¯ or Ai we will refer to its underlying graph by A, A and Ai. We further refer to vertex and edge groups by Av and Ae as in the deﬁnition above. If the graph of groups is however denoted by the letter B or its derivatives then we assume the underlying graph to be the appropriate derivative of B and denote the vertex and edge groups by Bv and Be.

1.2 The free decomposition of a graph of groups If a graph of groups has an edge with trivial edge group then its fundamental group decomposes as a free product. This simple observation gives rise to the notion a the free decomposition of a graph of groups. Let A be a ﬁnite graph of groups, E ⊂ EA the set of edges with trivial edge group and A¯ = A − E. Let A1,...,Ar ⊂ A be those components of A¯ whose corresponding subgraphs of groups Ai have non-trivial fundamental group. We call the Ai the free factors of A. We will say that a free factor is cyclic if its fundamental group is cyclic. Let further n be the number of edges outside a maximal tree of the graph Aˆ obtained from A by contracting the subgraphs Ai to a point. We call n the free rank of A. It is easy to see that ∼ r π1( ) = ∗ π1( i) ∗ Fn. A i=1 A

Note that this decomposition does not necessarily coincide with the canonical decomposition of π1(A) into free factors and a free group as we do not exclude that some π1(Ai) is inﬁnite cyclic or a proper free product. We call the pair

cf (A) := (r, n) the free complexity of A.

1.3 Propagation in a graph of groups

Let B be a graph of groups. Suppose that g ∈ Bv for some v ∈ VB and thatg ¯ ∈ Bw for some w ∈ VB. We say thatg ¯ is a propagate of g via the edge path e1, . . . , ek with α(e1) = v and ω(ek) = w if there exists elements g = g0, g1, . . . , gk−1, gk =g ¯ with gi ∈ Bω(ei) = Bα(ei+1) for 1 ≤ i ≤ k − 1 such that for 1 ≤ i ≤ k we have

αei (hi) = gi−1 and ωei (hi) = gi.

for some hi ∈ Bei . It is clear that any two elements that are propagates are also propagates via a reduced edge path.

6 This deﬁnes an equivalence relation on ∪ Bv and we denote the equiva- v∈VB p lence class of all propagates of g by [g] . For a subset S of ∪ Bv we denote v∈VB the set ∪ [s]p by Sp. If for each v ∈ VB we have s∈S

p Bv = hS ∩ Bvi and for each e ∈ EB we have

p αe(Be) = hS ∩ αe(Be)i then we say that S is a p-generating set. Note that the second condition means that a generating set of Be propagates through e. Note that we can always replace an element of a p-generating set S by a propagate, i.e. (S − {s}) ∪ {s0} is a p-generating set if S is a p-generating set, s ∈ S and s0 is a propagate of s.

1.4 A-graphs

In this section we introduce A-graphs. A-graphs represent a subgroup U of π1(A). To any A-graph there is an associated graph of groups which can be thought of as an approximations of the induced splitting of U. In good situations, namely when an A-graph is folded, an induced splitting can be directly read off the A-graph. A-graphs were first introduced in [KMW]. We closely follow the exposition in [KMW] but give a slightly different definition. A-graphs essentially encode a morphism from an H-tree Y to the π1(A)-tree T , where T is the Bass-Serre tree corresponding to A. Let A be a graph of groups. An A-graph B consists of an underlying graph B with the following additional data:

1. A graph-morphism [ . ]: B → A.

2. For each u ∈ VB there is a group Bu with Bu ≤ A[u].

3. For each f ∈ EB there is a group Bf with Bf = Bf −1 ≤ A[f].

4. To each edge f ∈ EB there are two associated group elements fα ∈ A[α(f)] −1 −1 and fω ∈ A[ω(f)] such that (f )α = (fω) for all f ∈ EB.

−1 5. For each f ∈ EB we have fα · α[f](Bf ) · fα ≤ Bα(f).

Remark 1.1 In the deﬁnition of A-graphs as given in [KMW] neither 3 nor 5 appear. Instead the group Bf is is implicitly given as the largest subgroup of −1 −1 −1 −1 A[f] such that 5 holds, i.e. Bf := α[f] (fα Bα(f)fα) ∩ ω[f] (fωBω(f)fω ) ≤ A[f].

If f ∈ EB and u ∈ VB, we shall refer to e = [f] ∈ EA and v = [u] ∈ VA as the type of f and u accordingly. Also, especially when representing A-graphs by pictures, we will sometimes say that a vertex u ∈ VB has label (Bu, [u]). Similarly we will say that an edge f of an A-graph B has label (fα, [f], fω). Note that to give the complete information we would need to speak of edge labels of type (fα, [f],Bf , fω) but we will usually not do this.

7 We will visualize an A-graph B in the obvious way by drawing the underlying graph B with the appropriate labels next to its vertices and edges. For every geometric edge we choose the label of either edge of the corresponding edge pair {f, f −1}. For convenience we will further orient the edge by attaching an arrow such that for an edge with label (a, e, b) one travels from a vertex with label (B, α(e)) to a vertex with label (B0, ω(e)) if one follows the direction of the arrow. It follows that reversing the orientation of an edge and replacing the label (a, e, b) by (b−1, e−1, a−1) yields another diagram of the same A-graph.

(B3, v3)(B3, v3) (a0, e0, b0)( a0, e0, b0) q q > > (B , v )(−1 −1 −1 ? B , v ) (a, e, b)(6 Z 1 1 b , e , a ) Z 1 1 Z~ Z q Z q Z −1 −}1 −1 (a, e, b)(ZZ b , e ,Z aZ) (B2, v2)(B2, v2) q q Figure 2: Two distinct diagrams associated to the same A-graph

To any A-graph we can then associate in a natural way a graph of groups. Let B be an A-graph. The associated graph of groups B is deﬁned as follows:

1. The underlying graph of B is the graph B.

2. The vertex and edge groups are the groups Bu for u ∈ VB and Bf for f ∈ EB.

3. For each f ∈ EB we deﬁne the boundary monomorphism αf : Bf → Bα(f) −1 as αf (g) = fα α[f](g) fα and ωf = αf −1 .

Let B be an A-graph deﬁning a graph of groups B. Suppose u, u0 ∈ VB and p is a B-path from u to u0. Thus p has the form:

p = b0, f1, b1, . . . , fs, bs

0 where s ≥ 0 is an integer, f1, . . . , fs is an edge path in B from u to u , where 0 b0 ∈ Bu, bs ∈ Bu and bi ∈ Bω(fi) = Bα(fi+1) for 0 < i < s. Recall that each edge fi has a label (gi, ei, ki) in B, where ei = [fi], gi = (fi)α and ki = (fi)ω. We associate to the B-path p the A-path µ(p) from [u] to [u0] as follows:

µ(p) = (b0g1), e1, (k1b1g2), e2,..., (ks−1bs−1gs), es, (ksbs)

Notice that |p| = |µ(p)|. The following important Proposition is easily veriﬁed.

Proposition 1.2 Let B be an A-graph, u0 ∈ VB and v0 = [u0]. Then: 1. If p ∼ p0 as B-paths, then µ(p) ∼ µ(p0) as A-paths.

2. The map µ restricted to the set of B-paths from u0 to u0 factors through to a homomorphism φ : π1(B, u0) → π1(A, v0).

8 We say that B represents the subgroup U = φ(π1(B, u0)) of π1(A, v0). Let A be a graph of groups and U be a subgroup of G = π1(A, v0) given by a generating set S ⊂ G. We construct a simple A-graph B0 representing U. For each s ∈ S we choose a reduced -path p = as, es, as, . . . , es , as from A s 0 1 1 ks ks 0 v0 to v0 such that ps = s. Put PS = {ps|s ∈ S} and PS = {ps | |ps| = 0}. 0 Clearly s ∈ Av0 if and only if ps ∈ PS , i.e. if ks = |ps| = 0. We construct an A-graph B0 as follows:

1. The underlying graph B0 has base-vertex called u0 of type v0 and for each ps ∈ PS with ks ≥ 1 we attach a circle subdivided into ks edges to u0 where the ﬁrst k −1 edges have labels (as, es, 1),..., (as , es , 1) and s 0 1 ks−2 ks−1 the last edge has label (as , es , as ). (Note that B is either a single ks−1 ks ks 0 vertex or a wedge of circles).

2. The graph morphism [.]: B0 → A is given by the types of the edges.

3. For every edge e ∈ EB we put Be = 1 and for every vertex u ∈ VB with u 6= u0 we put Bu = 1. 0 4. We put Bu0 = hs | ps ∈ PS i.

This completely describes the A-graph B0. We call such an A-graph a S- wedge. Clearly a S-wedge represents the subgroup U = hSi of π1(A, v0). Remark 1.3 Note that if no element of S is represented by an A-path of length 0 then all vertex and edge groups of the S-wedge are trivial. If we drop the requirement that the A-paths ps are reduced it is easy to see that we can always construct an S-wedge with these properties unless the graph A underlying A consists of the single vertex v0.

Later we will study sequences of A-graphs that are related by folds. The S-wedges will always be the ﬁrst A-graphs in such a sequence while the last ones are folded A-graphs. Let B be an A-graph. We will say that B is folded if the following hold:

(F1) For any two distinct edges f1, f2 ∈ EB with α(f1) = α(f2) = z and labels 0 0 (a1, e, b1), (a2, e, b2) we have a2 6= a a1αe(c) for all c ∈ Ae and a ∈ Bz. −1 −1 (F2) If f ∈ EB is an edge with label (a, e, b) then Bf = αe (a Bα(f)a) = −1 −1 ωe (bBω(f)b ). It is easy to see that if B is folded then any reduced B-path translates into a reduced A-path. This means that the maps µ and φ in Proposition 1.2 are injective if B is folded. Thus we have the following:

Proposition 1.4 Let B be a folded A-graph deﬁning the graph of groups B. Let u0 ∈ VB be of type v0. Put G = π1(A, v0) and U = φ(π1(B, u0)) ≤ G. Then the following hold:

1. For any reduced B-path p the corresponding A-path µ(p) is A-reduced.

2. The epimorphism φ : π1(B, u0) → U is an isomorphism. Proposition 1.4 essentially says that if B is a folded A-graph deﬁning a subgroup U ≤ G then B is the induced splitting of U.

9 1.5 Foldings of A-graphs Again we follow [KMW]. We ﬁrst introduce three moves that can be applied to A-graphs without substantially changing their structure. They can can be applied to any A-graph. Conjugation move A0: Let B be an A-graph. Suppose u is a vertex of B and that g ∈ A[u]. Let B0 be the A-graph obtained from B as follows:

−1 1. Replace Bu by gBug .

2. For each non-loop edge f with α(f) = u replace fα with gfα.

−1 3. For each non-loop edge ω(f) = u replace fω with fωg .

4. For each loop edge f with α(f) = ω(f) = u replace fα with gfα and fω −1 with fωg .

We will say that B0 is obtained from B by an auxiliary move of type A0. If u0 ∈ VB with u0 6= u we will say that this A0-move is admissible with respect to u0.

(B3, v3)(B3, v3) 0 0 0 (a0, e0, b0) (ga , e , b ) q q > > (ga, e, bg−1) (a, e, b) (B , v )(- gB g−1, v ) 6 Z 1 1 6 Z 1 1 q Z q Z Z} Z} (a, e, b) ZZ (a, e, bg−1)ZZ (B2, v2)(B2, v2) q q

Figure 3: A move of type A0 with g ∈ Av1

Bass-Serre move A1: Let B be an A-graph. Suppose f is an edge of B, that e = [f] and that c ∈ A[f] = Ae. 0 −1 Let B be the A-graph obtained from B by replacing Bf with cBf c ≤ Be, −1 fα with fααe(c ) and fω with ωe(c)fω. We say that B0 is obtained from B by an auxiliary move of type A1.

−1 (a,- e, b)(- aαe(c -), e, ωe(c)b) (B1, v1)(B2, v2)(B1, v1)( B2, v2) q q q q

Figure 4: A move of type A1 with c ∈ Ae

Simple adjustment A2: Let B be an A-graph. Suppose f is an edge of B and 0 that a ∈ Bα(f). 0 0 Let B be the A-graph obtained from B by replacing fα with a fα. We say that B0 is obtained from B by an auxiliary move of type A2.

In all three cases it is easy to see that B and B0 represent the same subgroup unless the base vertex is aﬀected by a move of type A0:

10 0 (a,- e, b)(- a-a, e, b) (B1, v1)(B2, v2)(B1, v1)( B2, v2) q q q q 0 Figure 5: A move of type A2 with a ∈ B1

Lemma 1.5 Let A be a graph of groups with base vertex v0 and B be an A- 0 graph with base vertex u0 of type v0. Suppose that B is obtained from B by an auxiliary move of type A1, A2 or an auxiliary move of type A0 that is admissible 0 with respect to u0. Then φ(π1(B , u0)) = φ(π1(B, u0)) ≤ π1(A, v0). If the move is of type A0 and is not admissible with respect to u0 then 0 0 −1 φ(π1(B , u0)) = g · φ(π1(B , u0)) · g for some g ∈ Av0 .

We now introduce folding moves for A-graphs. They can be applied to an A-graph B if and only if B is not folded. These folds are combinatorial versions of the Stallings folds discussed by M. Bestvina and M. Feighn [BF1] combined with vertex morphisms as introduced by M. Dunwoody [D4],[D2].

For the remainder of this section let (B, u0) be an A-graph with base vertex u0. Suppose that B is not folded. We distinguish the case that (F1) is not satisﬁed and the case that (F2) is not satisﬁed.

If (F1) is not satisﬁed then there exist distinct edges f1 and f2 with x = 0 α(f1) = α(f2) and labels (a1, e, b1) and (a2, e, b2) such that a2 = a a1αe(c) for 0 some c ∈ Ae and a ∈ Bx. Suppose further that ω(f1) = y and ω(f2) = z. Clearly y and z are of the same type v ∈ VA. We also denote the type of x by w ∈ VA.

By applying a move of type A2 to the edge f2 we can change the label of f2 0−1 0−1 0 to (a a2, e, b2) = (a a a1αe(c), e, b2) = (a1αe(c), e, b2). A move of type A1

then yields the label (a1, e, ωe(c)b2) on f2 and replaces the edge group Bf2 with −1 0 cBf2 c . We denote the resulting A-graph by B .

(By, v)( By, v) (a1, e, b1)( a1, e, b1) 1 q 1 q A2 - PP PP (Bx, w)(P Bx, w) P PqP P qP q PP q PP 0 PPP PPP (a a1αe(c), e, b2)((Bz, v)(a1αe(c), e, b2) Bz, v) q q

(a , e, b ) (By, v) 1 1 ¨ 1 q ¨ ¨¨ A1 PP (Bx, w) P PqP q PP PPP (a1, e, ωe(c)b2) (Bz, v) q We will use B0 as an intermediate object before deﬁning the main folding moves on B. Note that B0 is obtained from B by moves that only alter labels of edges and one edge group (by conjugation) but no vertex groups.

11 It is possible that two or more of the vertices (that are drawn as distinct ver- ¯ ¯ −1 tices) coincide. To simplify notations we put b2 := ωe(c)b2 and Bf2 = cBf2 c . We now introduce four diﬀerent types of folds. They are distinguished by the topological type of the subgraph f1 ∪ f2 in B. Each of these moves will be deﬁned as a sequence of several transformations, exactly one of which will correspond to performing a Stallings fold identifying the edges f1 and f2 in B. That particular portion of a move will be called an elementary move of the respective type.

Fold of type IA: Suppose f1 and f2 are two distinct non-loop edges and that y = ω(f1) 6= ω(f2) = z. Possibly after exchanging f1 and f2 we can assume that z is not the base vertex u0 of B. We ﬁrst perform a move of type A0 on B0 at the vertex z making the label −1¯ ¯−1 of f2 to be (a1, e, b1) and the label of z to be (b1 b2Bzb2 b1, v). Now both f1 and f2 have label (a1, e, b1). Next we identify the edges f1 and f2 into a single edge f with edge group ¯ Bf = hBf1 , Bf2 i and label (a1, e, b1), as illustrated in Figure 6 by the arrow labeled F . The label of the vertex ω(f) is set to be

−1¯ ¯−1 (hBy, b1 b2Bzb2 b1i, v). The other labels do not change. We call this last operation an elementary move of type IA and say that the resulting A-graph is obtained from the original A-graph B by a move of type IA.

(By, v)( By, v) (a1, e, b1)( a1, e, b1) 1 q 1 q A0 - PP PP (Bx, w)(PP Bx, w) PP q qPP q qPP ¯ P P (a1, e, b2)(PPP a1, e, b1) PPP (Bz, v) −1¯ ¯−1 (b1 b2Bzb2 b1, v) q ¨ q ¨ ¨¨ F

(a , e, b ) -1 1 −1¯ ¯−1 (Bx, w)(hBy, b1 b2Bzb2 b1i, v) q q Figure 6: A move of type IA

Fold of type IB Suppose now that f1 is a loop edge and that f2 is a non-loop edge. (The opposite situation is analogous). This implies that e is a loop-edge in A based at the vertex v = w. 0 We ﬁrst perform move A0 on B making the label of f2 to be (a1, e, b1). Next ¯ we fold the edges f1 and f2 into a single loop-edge f with edge group hBf1 , Bf2 i and label (a1, e, b1), as shown in Figure 7. The label of α(f) = ω(f) is set to be

−1¯ ¯−1 (hBx, b1 b2Byb2 b1i, v). We call this operation, called F in Figure 7 an elementary move of type IB.

12 If y = ω(f2) = u0 we then perform the auxiliary move A0 corresponding to ¯−1 the element b2 b1. We will say that the resulting A-graph is obtained from B by a folding move of type IB. (a-1, e, b1)(a-1, e, b1)

A0 -

¯ (a-1, e, b2)(a-1, e, b1) B , v)((B , v)( B , v) −1¯ ¯−1 x y x (b1 b2Byb2 b1, v) qq(a-1, e, b1) qq F +

−1 −1 (hBx, b ¯b2By¯b b1i, v) 1 q 2 Figure 7: A move of type IB

Fold of type IIIA Suppose that f1 and f2 are both non-loop edges such that y = ω(f1) = ω(f2) = z. 0 We identify the edges edges f1 and f2 of B into a single edge f with edge ¯ group hBf1 , Bf2 i and label (a1, e, b1). We set the label of ω(f) to be −1¯ (hBy, b1 b2i, v). We call this last operation an elementary move of type IIIA and say that the resulting A-graph is obtained from B by a folding move of type IIIA.

(a1-, e, b1)

F - (a1-, e, b1) −1 (By, v) ¯ (Bx, w) (Bx, w) (hBy, b1 b2i, v) q- q q q

(a1, e, ¯b2)

Figure 8: A move of type IIIA

Fold of type IIIB Suppose f1 and f2 are distinct loop edges, so that x = y = z, v = w and e is a loop-edge at v = w in A. 0 We identify the edges f1 and f2 in B into a single loop edge with edge group ¯ hBf1 , Bf2 i and label (a1, e, b1), as shown in Figure 9. The new label of z is set to be −1¯ (hBx, b1 b2i, v). We call this last operation an elementary move of type IIIB and say that the resulting A-graph is obtained from B by a folding move of type IIIB. Suppose now that B is not folded because condition (F2) is not satisﬁed. We introduce two folding moves that can be applied to such an A-graph. Note that for these folds there is no distinction between folds and elementary folds.

13 −1¯ (Bx, w) (hBx, b1 b2i, w) ? ? - ? (a , e, b ) ¯ (a , e, b ) 1 1 (a1, e, b2) F 1 1 q q

Figure 9: A move of type IIIB

It follows that there is an edge f ∈ EB with label (a, e, b), with x = α(f) −1 −1 labeled (Bx, w) and y = ω(f) labeled (By, v) such that αe (a Bxa) 6= Bf or −1 −1 −1 ωe (bByb ) 6= Bf . Possibly after replacing e with e this implies that there −1 −1 exists an element g ∈ αe (a Bxa) with g∈ / Bf . We distinguish the cases x = y and x 6= y.

Fold of type IIA: Let B be an A-graph. Suppose that x 6= y, i.e. that f is a non-loop edge of B with the label (a, e, b) and the edge group Bf . 0 Let B be the A-graph obtained from B by replacing the the edge group Bf −1 by the group hBf , gi and replacing the vertex group By by hBy, b ωe(g)bi. We say that B0 is obtained from B by a move of type IIA.

(a,- e, b)(IIA- a,- e, b) −1 (Bx, w)(By, v)(Bx, w)( hBy, b ωe(g)bi, v) q q q q

Figure 10: A move of type IIA with Bf being replaced by hBf , gi

Fold of type IIB: Suppose that x = y, i.e. that f is a loop edge of B with the label (a, e, b). 0 Let B be the A-graph obtained from B by replacing Bf with hBf , gi and Bx −1 0 by hBx, b ωe(g)bi. In this case we say that B is obtained from B by a move of type IIB.

¯ (Bx, v)(? - Bx, v) ? a, e, b) (a, e, b)( IIB q q

−1 B¯x := hBx, b ωe(g)bi

Figure 11: A move of type IIB with Bf being replaced by hBf , gi

Note that each of the folding moves corresponds to a graph-morphism between the underlying graphs which preserves types of vertices and edges. In case of moves IA, IB, IIIA and IIIB this morphism reduces the number of edge- pairs by one. For moves A0-A2, IIA, IIB the morphism is the identity map. Moreover, the moves IIIA and IIIB decrease the rank of the fundamental group of the underlying graph B by one, while IA and IB do not change it. The following important proposition states that folding moves preserve the subgroup deﬁned by an A-graph.

14 Proposition 1.6 Let A be a graph of groups with a base-vertex v0. Denote 0 G = π1(A, v0). Let B be an A-graph obtained from B by one of the folding 0 moves IA, IIA, IIIA, IB, IIB or IIIB. Let u0 be a vertex of B and let u0 be the 0 0 image of u0 in B . Suppose the type of the vertices u0, u0 is v0 ∈ VA. 0 0 Then φ(π1(B , u0)) = φ(π1(B, u0)) ≤ G = π1(A, v0).

2 Graphs of dihedral groups

In this chapter we compute the rank of graphs of dihedral groups, i.e. we give a proof of Theorem 0.1. This solves in particular the rank problem for groups 2 of type π1(S (∞, (h1, . . . , hn))). Recall the the groups Ui in the statement of Theorem 0.2 are of this type. As two types of Fuchsian groups appear repeatedly throughout this chapter, we give them special names. Both of them are graphs of dihedral groups. We put

2 Λ(n; h1, . . . , hn−1) := π1(S ((h1, . . . , hn−1, ∞))) and ∞ 2 ∆ (n; h1, . . . , hn) := π1(S (∞, (h1, . . . , hn))). Note that Section 2.1 is strictly speaking not necessary for the proofs of Theorem 0.1 and Theorem 0.2. The discussion of simple generating sets does however make the assertion of Theorem 0.1 more explicit in the important case ∞ of groups of type Λ(n; h1, . . . , hn−1) and ∆ (n; h1, . . . , hn). This discussion also serves as motivation for the deﬁnition of the simple rank in Section 2.2

2.1 Simple generating sets

In this section we discuss classes of groups Λ = Λ(n; h1, . . . , hn−1) for which rank Λ < n. A thorough understanding of these generating sets is extremely helpful for the understanding of the proof of Theorem 0.1. Note that we simply write Λ(h1, . . . , hn−1) instead of Λ(n; h1, . . . , hn−1) if the value of n is implicitly given by the presentation, i.e. we write Λ(p, q) instead of Λ(3; p, q) but we cannot omit the n in Λ(n; 2,..., 2, p). Some of these classes were already discussed in [We0]. They are obtained by a somewhat involved iteration of a phenomenon exhibited by R. Kaufmann and H. Zieschang [KZ]. The main observation of [KZ] is the following; we give the very short proof as it is prototypical for simple generating sets as deﬁned below. Lemma 2.1 The group Λ(2, 2l + 1) is of rank 2.

2 2 2 2 2l+1 Proof Recall that Λ(2, 2l + 1) = hs1, s2, s3|s1, s2, s3, (s1s2) , (s2s3) i. We l show that the group is generated by g1 = s2 and g2 = s1(s3s2) . Note ﬁrst that −1 l l l l g2 g1g2 = (s2s3) s1s2s1(s3s2) = (s2s3) s2(s3s2) = s3. l l l Thus s2, s3 ∈ hg1, g2i and therefore g2(s2s3) = s1(s3s2) (s2s3) = s1 ∈ hg1, g2i. It follows that hg1, g2i = hs1, s2, s3i = Λ(2, 2l + 1). 2 In order to see how this phenomenon can be iterated we have to formulate it in a more general way. Almost the same calculation as in the proof above yields the following simple but important fact.

15 Lemma 2.2 Let Λ = Λ(n; h1, . . . , hn−1) and U < Λ. Suppose that s ∈ Λ is a reﬂection and that x1, x2, x3 ∈ U such that the following hold:

−1 1. x1 is a reﬂection and x2 = gx1g for some g ∈ Λ.

2. x3 ∈ hx1, si and |hx1, si : hx1, x3i| is odd.

Then there exists an element x4 such that

hU, x4i = hU, s, gi.

Proof We show that the assertion holds with x4 = gs. Clearly

−1 −1 −1 −1 2 x1x4 x2x4 = x1s g gx1g gs = x1sx1s = (x1s) .

As hx1, x3i is of odd index in the (possibly inﬁnite) dihedral group hx1, si it 2 follows that hx1, x3, (x1s) i = hx1, si, in particular s ∈ hx1, . . . , x4i. Thus also x4s = gss = g ∈ hx1, . . . , x4i. Thus hs, gi ⊂ hx1, . . . , x4i ⊂ hU, x4i and therefore hU, s, gi ⊂ hU, x4i. The opposite inclusion is obvious. 2 Iterating the observation made in Lemma 2.2 we are able to describe a larger class of groups of type Λ = Λ(n; h1, . . . , hn−1) which are (n−1)-generated. This gives then rise to the deﬁnition of simple generating sets. Recall that

2 2 h1 hn−1 Λ(n; h1, . . . , hn−1) = hs1, . . . , sn | s1, . . . , sn, (s1s2) ,..., (sn−1sn) i. Instead of starting with the most complicated and general examples we describe a special case ﬁrst and proceed with more involved cases.

Lemma 2.3 Suppose that Λ = Λ(n; h1, . . . , hn−1) where

1. hi = 2li + 1 with li ≥ 1 for 1 ≤ i ≤ n − 2 and

n−2 2. hn−1 = 2 p with p ≥ 1.

Then there exist elements {g1, . . . , gn−1} such that

p hg1, . . . , gn−1i = hs1, . . . , sn−1, (sn−1sn) i.

In particular Λ is (n − 1)-generated if p = 1.

Proof We show by induction on k that there exist elements g1, . . . , gk such that

(2n−k−1p) Uk := hg1, . . . , gki = hsn−k, sn−k+1, . . . , sn−1, (sn−1sn) i.

For k = n − 1 this is the assertion of the lemma. n−2 For k = 1 the assertion holds with g1 = sn−1 as hn−1 = 2 p by assumption (2n−k−1p) (2n−2p) and therefore (sn−1sn) = (sn−1sn) = 1.

Suppose now that the claim holds for k. By assumption sn−k ∈ Uk and (2n−k−1p) (2n−k−2p) −(2n−k−2p) (sn−1sn) sn−1 = (sn−1sn) sn−1(sn−1sn) ∈ Uk. Note that (2n−k−2p) −(2n−k−2p) −1 (sn−1sn) sn−1(sn−1sn) = gsn−kg

n−k−2 (2 p) ln−2 l with g = (sn−1sn) (sn−2sn−1) · ... · (sn−ksn−k+1) n−k .

16 −1 Put s = sn−k−1, x1 = sn−k, x2 = gx1g and x3 = 1. Clearly x1, x2, x3 ∈ Uk and hx1, x3i = hsn−ki is of odd index in hsn−k−1, sn−ki as hn−k−1 is odd by assumption. Thus we can apply Lemma 2.2. Choose gk+1 = x4 as in the conclusion of Lemma 2.2. It follows that

n−k−1 hUk, gk+1i = hUk, sn−k−1, gi = hsn−k−1, . . . , sn−1, (sn−1sn) , gi.

n−k−2 (2 p) −l −ln−2 As (sn−1sn) = g(sn−k+1sn−k) n−k ··· (sn−1sn−2) this implies that (2n−k−2p) Uk+1 = hUk, gk+1i = hsn−k−1, . . . , sn−1, (sn−1sn) i. 2

7 3 22 q qq

Figure 12: Λ(4; 3, 7, 22) is 3-generated

Suppose now that Λ and g1, . . . , gn−1 are as in Lemma 2.3. Recall that p p hsn−1, (sn−1sn) i ⊂ Un−1. If p = 2l + 1 is odd then hsn−1, (sn−1sn) i = l l hsn−1, (snsn−1) sn(sn−1sn) i is a subgroup of odd index in hsn−1, sni. Thus l l Lemma 2.2 can be applied to Un−1 where hsn−1, (snsn−1) sn(sn−1sn) i plays the role of hx1, x3i. This observation leads to the following more general class of groups of type Λ(n; h1, . . . , hn−1) which can be generated by less than n elementes. Note that we call an element primitive if it is an element of a minimal generating set.

Lemma and Deﬁnition 2.4 Let Λ(n; h1, . . . , hn−1). Suppose that there exist k1, . . . , km (possibly m = 0, i.e. no such ki) such that the following hold:

1. 2 ≤ k1 < k2 < . . . < km ≤ n − 2.

n−1−km 2. hn−1 = 2 .

ki−ki−1 ¯ ¯ 3. hki = 2 hki with hki = 2lki + 1 ≥ 3 for 2 ≤ i ≤ m.

k1−1¯ ¯ 4. hk1 = 2 hk1 for some hk1 = 2lk1 + 1 ≥ 3. ¯ 5. hj = hj = 2lj + 1 ≥ 3 otherwise.

Then Λ is of rank n − 1 and s1 is primitive and sn is not. We say that Λ is of type A.

Before we proceed with the proof we give an example that illustrates the condition spelled out in Lemma 2.4. The group Λ(7; 3, 5, 22 · 7, 3, 22 · 5, 2) falls into this class with n = 7, m = 2, k1 = 3 and k2 = 5. Proof of Lemma 2.4 Note ﬁrst that rank Λ ≥ n − 1 as there exists a surjective ¯ ¯ homomorphism φ :Λ → Λ(n − 1; h1,..., hn−2) deﬁned by φ(si) = si for 1 ≤ ¯ i ≤ n − 1 and φ(sn) = 1. Recall that all hi are odd and greater than 2. The ¯ ¯ group Λ(n − 1; h1,..., hn−2) can be rewritten as an amalgamated product with

17 22 · 7 3 5 22 · 5 q q 3 q q 2 q q

Figure 13: Λ(7; 3, 5, 22 · 7, 3, 22 · 5, 2) is 6-generated n − 2 factors and malnormal amalgam which implies that rank H ≥ n − 1 by Corollary 1 of [We1]. Thus rank Λ ≥ n − 1. The generating set of cardinality n − 1 is constructed by induction on m, at each step we use the generating sets provided by the proof of Lemma 2.3. In order to facilitate the statement of the inductive statement we put km+1 = n−1 n−1−km ¯ ¯ and hkm+1 = hn−1 = 2 hn−1 with hn−1 = 1. We show that there exists a generating set g1, . . . , gn−1 such that for each i ∈ {1, . . . , m + 1} we have

¯ hki hg1, . . . , gki i = hs1, . . . , ski , (ski ski+1) i. (Ai)

The proof of (Ai) is by induction on i. Note that (A1) is nothing but the statement of Lemma 2.3 and that (Am+1) immediately yields the assertion of ¯ ¯ Lemma 2.4 as hkm+1 = hn−1 = 1.

We assume that (Ai) holds. Let ¯ ¯ ¯ Λ = Λ(ki+1 − ki + 2; hki , hki+1, . . . , hki+1−1, hki+1 /hki+1 ).

Note that by Lemma 2.3 the group Λ¯ is (ki+1 − ki + 1)-generated and any element conjugate to s1 ∈ Λ¯ is primitive. Thus there exists a generating set ¯ g¯ki ,..., g¯ki+1 of Λ such thatg ¯ki = s1. ki−ki−1−1 2 lki Put g = (ski+1ski ) (ski+1ski ) ∈ Λ. Clearly there exists a homomorphism φ : Λ¯ → Λ deﬁned by ¯ hki lki −lki 1. φ(s1) = ski (ski ski+1) = (ski+1ski ) ski+1(ski+1ski ) ∈ hg1, . . . , gki i,

−1 2. φ(sj) = gski+j−1g for 2 ≤ j ≤ ki+1 − ki ¯ hki+1 −1 3. φ(ski+1−ki+1) = gski+1 (ski+1 ski+1+1) g . It is a simple exercise to see that

¯ ¯ hki+1 hg1, . . . , gki , φ(Λ)i = hs1, . . . , ski+1 , (ski+1 ski+1+1) i.

As φ(¯gki ) = φ(s1) ∈ hg1, . . . , gki i this implies that (Ai+1) holds with gj = φ(¯gj) for ki + 1 ≤ j ≤ ki+1.

Clearly s1 ∈ Λ is primitive as s1 is conjugate to the element g1 of the described generating set. To see that sn is not primitive it suﬃces to observe that sn lies in the kernel of the map φ deﬁned above. 2

18 Lemma 2.4 implies that the rank of a group of type Λ = Λ(n; h1, . . . , hn−1) is diﬀerent from n if the some sub-tuple (hi, . . . , hi+k) (or its inverse (hi+k, . . . , hi)) corresponds to a group of type A. If more such subtuples occur it is clear that the generating set can be even smaller, the group Λ(6; 2, 3, n, 2, 3) is clearly 4- generated for any n as the generators s1, s2, s3 can be replaced by two generators by Lemma 2.4 and so can the generators s4, s5, s6. If the generating sets corresponding to two such subgroups overlap, the situation is less obvious. The group Λ(5; 3, 2, 3, 2) for example can be seen to be 3-generated as s1, s2, s3 can be replaced by two generators and the s4, s5 can be replaced by a single generator as s3 is part of a 2-element generating set of hs3, s4, s5i.

2 3 3 2 q q q q

Figure 14: Λ(5; 3, 2, 3, 2) has a simple generating set of cardinality 3

The group Λ(5; 3, 2, 2, 3) however cannot be shown to be 3-generated by using an iteration of the above observation as s3 is neither primitive in hs1, s2, s3i nor in hs3, s4, s5i. It is in fact easy to see that this group is not 3-generated as the factor group Λ(5; 3, 2, 2, 3)/N(s3) is isomorphic to D6 ∗ D6 and therefore of rank 4 by Grushko’s theorem.

2 2 3 3 q q q q

Figure 15: Λ(5; 3, 2, 2, 3) has no simple generating set of cardinality 3

The above examples give rise to the deﬁnition of simple generating sets of groups of type Λ(n; h1 . . . , hn−1). They are generating sets that are obtained from iterations of the above phenomena.

To any tuple τ = τ(k, l) := (hk, . . . , hk+l−1) with 1 ≤ k ≤ k + l ≤ n we associate the subgroup Uτ = hsk, . . . , sk+li of Λ(n; h1, . . . , hn−1). This is a group of type Λτ := Λ(l + 1; hk, . . . , hk+l−1) as the map

ητ :Λτ → Λ(n; h1, . . . , hn−1) deﬁned by ητ (si) = si+k−1 is a monomorphism with ητ (Λτ ) = Uτ . We proceed with the deﬁnition of simple generating sets.

Let Λ = Λ(n; h1, . . . , hn−1). Suppose that m ≥ 0 and that there are are tuples τi = (ki, li) with 0 ≤ ki ≤ ki + li ≤ n for 1 ≤ i ≤ m such that the following hold:

1. Λτi is of type A for 1 ≤ i ≤ m.

19 2. For all i there exists at most one j < i such that Uτi ∩ Uτj 6= 1.

3. If j < i and Uτi ∩ Uτj 6= 1 then Uτi ∩ Uτj = hsti for some t ∈ {1, . . . , n}

and st is primitive in Uτi .

We can then equip Λ with a generating set M of cardinality n − m in the following way. We call such generating sets simple.

1. M0 = ∅

2. If Uτi ∩ Uτj = 1 for all j < i then put Mi = Mi−1 ∪ ητi (Si) where Si is a

generating set of Uτi of cardinality li − 1 as provided by Lemma 2.4. ¯ ¯ 3. If Uτi ∩ Uτj 6= 1 for j < i then put Mi = Mi−1 ∪ Si where Si is a set of ¯ ¯ cardinality li − 2 such that hSi ∪ (Uτi ∩ Uτj )i = Uτi . The existence of Si is guaranteed by Lemma 2.4 and by assumption 3 above.

4. M is obtained from Mm by adding all reﬂections si not contained in

any Uτi .

Note that the standard generating set {s1, . . . , sn} is simple by taking m = 0. This implies in particular that two distinct simple generating sets of a given group do not need to be of the same cardinality. However for any group of type Λ(n; h1, . . . , hn−1) it is a trivial ﬁnite problem to compute the minimal cardinality of simple generating sets. This enables us to make the assertion of Theorem 0.1 more precise, at least ∞ in the case of groups of type Λ(n; h1, . . . , hn−1) and ∆ (n; h1, . . . , hn). Note that this enables us to compute the rank of any planar group using Theorem 0.2.

Theorem 2.5 (Corollary 2.11) Any group of type Λ(n; h1, . . . , hn−1) has a minimal generating set that is simple.

∞ For groups of type ∆ (n; h1, . . . , hn) the formulation is slightly more complicated. Here Σ ≤ Sn is the group of cyclic permutations of {1, . . . , n}.

∞ Theorem 2.6 (Corollary 2.12) Let G = ∆ (n; h1, . . . , hn). Then

rank G = 1 + min rank Λ(n; hσ(1), . . . , hσ(n−1)). σ∈Σ

2.2 Graphs of dihedral groups and their normalization

We will assume unless stated otherwise that all dihedral groups are ﬁnite and of order at least 4. We further say that D2n is even if n is even and odd if n is odd. We will say that a graph of groups A is a graph of dihedral groups if is satisﬁes the following conditions:

1 1. Every vertex group Av is a dihedral group generated by reﬂections sv 2 and sv.

2. Every edge group Ae is of order two generated by the element se.

20 3. The element αe(se) is a reﬂection of Aα(e) for every edge e ∈ EA, i.e. 1 2 αe(se) is in Aα(e) conjugate to either sα(e) or sα(e).

Thus it makes sense to call an element of π1(A) a reflection if and only if it is conjugate to a reflection of one of the dihedral groups and a rotation if it conjugate to a rotation of some vertex group. In the following we will associate to a graph of dihedral groups A another graph of groups N(A) which we call the normalization of A. This normalization is done in four steps: Step 1: We change the boundary monomorphisms by multiplication with an 1 2 inner automorphism of the vertex group such that αe(se) ∈ {sα(e), sα(e)} if 1 1 2 Aα(e) is even and that αe(se) = sα(e) if Aα(e) is odd, i.e. if sα(e) and sα(e) are conjugate in Aα(e). Step 2: We subdivide all edges. The vertex groups of the emerging vertices and edges are cyclic of order two and the boundary monomorphism are the obvious ones. Step 3: This step is repeatedly performed as long as there are two distinct edges e and f with α(e) = α(f) = v such that Av is a dihedral group and αe(Ae) = αf (Af ). The modification we perform depends on whether ω(e) = ω(f) or ω(e) 6= ω(f):

1. If ω(e) 6= ω(f) we simply identify e and f and ω(e) and ω(f) and take the obvious graph of group structure. 2. If ω(e) = ω(f) we also identify e and f but add a new loop of length two at ω(e) = ω(f) with edge and vertex groups of order two. We call the vertex that is added a loop vertex.

Step 4: The ﬁnal modiﬁcation is performed at every vertex v of valence 1 for which Av is an even dihedral group. We introduce a new vertex w with Aw of order 2 and an edge e with α(e) = v and ω(e) = w such that Ae = hsei i is of order two. We put αe(se) = sv where i ∈ {1, 2} is chosen such that the k generator of the other edge group incident to v get mapped onto sv with k 6= i.

Lemma 2.7 The normalization N(A) is uniquely determined by the original ∼ graph of dihedral groups A. Moreover π1(A) = π1(N(A)). Proof To see that the above normalization process yields a unique graph of groups it clearly suffice to show that the order of modifications done in Step 3 does not affect the outcome. To see this it suffices to observe that if A1 and A2 are both obtained from A by one of those modifications from Step 3 then there exists a graph of groups A3 which is obtained from both A1 and A2 by one such modification. The uniqueness then follows from the diamond argument. The second assertion follows as any modification made during the normalization does not change the isomorphism type of the fundamental group. 2 We illustrate the normalization with ∆∞(3; 3, 3, 3) and ∆∞(3; 3, 3, 4). In the figures small balls stand for vertex groups with vertex group of order two and fat balls for vertices with dihedral vertex group. We label a vertex with vertex group D2n with the label n. We further think of vertices with vertex groups of order 2 to have label 1 but will not draw these labels.

21 ∞ The case ∆ (3; 3, 3, 4): As all reflections in D6 are conjugate we have to identify the two edges emerging at any vertex with label 3. However the two edges adjacent to the vertex with vertex group D8 = D2·4 do not get identified as the images of the boundary monomorphisms of the two adjacent edges are not conjugate in D8. Thus we only apply two modifications of the first type.

4 4 %e % ve v % e % e 1 + 1 - % r r e 3 % e 3 3 r 3 v r v v v Figure 16: Normalizing ∆∞(3; 3, 3, 4)

∞ The case ∆ (3; 3, 3, 3): As all vertex groups are of type D2·3 = D6 we have to identify the two edges emerging at any vertex with dihedral vertex group. At the ﬁrst two vertices this puts us into the ﬁrst situation of Step 3 above but eventually we are in the second situation and have to introduce a new loop vertex.

3 3 %e e % ve ve % e 1 - e % e e % r r e r e 3 % e 3 3 r e 3 v r v v 1 v ? 3 3

v v r 2

3 r 3 3 r 3 v v v v Figure 17: Normalizing ∆∞(3; 3, 3, 3)

Thus we have associated to any normalization N(A) (and therefore to any graph of dihedral groups A) in a canonical way a graph Θ = Θ(A) = Θ(N(A)) with a labeling l : V Θ → N − {0} with the following properties, where we say that a vertex v ∈ V Θ is loop vertex if v is of valence 2, v is adjacent to exactly one vertex w and l(w) = l(v) = 1: 1. If v and w are adjacent vertices then either l(v) = 1 or l(w) = 1. 2. If v is not a loop vertex, l(v) = 1 and w is adjacent to v, then either l(w) ≥ 2 or w is a loop vertex.

22 3. If l(v) ≥ 3 is odd then v ∈ V Θ is of valence 1. 4. If l(v) is even then v ∈ V Θ is of valence 2. We call a labeled graph with the above properties admissible. We will refer to vertices v with label 1 that are not loop vertices as small vertices and to vertices with label ≥ 2 as fat vertices. Thus every vertex is either a loop vertex or a small vertex or a fat vertex. The map assigning to any normalization N(A) the admissible labeled graph Θ(A) is clearly injective. It is however not surjective as an admissible labeled graph Θ is isomorphic to Θ(A) for some graph of dihedral groups if and only Θ has a fat vertex. An admissible labeled graph Θ without a fat vertex must consist of one small vertex v and a ﬁnite number of loop vertices w1, . . . , wk connected to v by two edges. To make the above map bijective we will also admit those graphs of groups whose underlying graph is the wedge of circles of length two such that all edge and vertex groups are of order two. We call the class of graphs of groups that consists of these newly introduced graphs of groups and those that occur as normalizations of graphs of dihedral groups normalized graphs of dihedral groups. We then have a bijection between the class of admissible labeled graphs and the class of normalized graphs of dihedral groups. Thus we can assign to any admissible labeled graph Θ a normalized graph of dihedral groups A(Θ) such that Θ(A(Θ)) and Θ are isomorphic labeled graphs and that A(Θ(A)) and A are equivalent graphs of groups. It is clear that the conjugacy classes of reﬂections of π1(A) are in a 1-to-1 correspondence with the small vertices. The fourth step in the normalization procedure is essential for this. ∞ For a group G of type Λ(n; h1, . . . , hn−1) or ∆ (n; h1, . . . , hn) we denote by Θ(G) the labeled graph obtained from the decompositions

Λ(n; h1, . . . , hn−1) = hs1, s2i ∗hs2i hs2, s3i ∗hs3i ... ∗hsn−1i hsn−1, sni and ∞ −1 ∆ (n; h1, . . . , hn) = hΛ(n + 1; h1, . . . , hn), t|ts1t = sn+1i as graphs of dihedral groups. A moment’s thought reveals that an admissible labeled graph Θ occurs as Θ(G) for some group G = Λ(n; h1, . . . , hn−1) if and only if Θ is a tree and the subtree spanned by all fat vertices with even labels is a segment, i.e. contains no vertex of valence more than 2. This segment is drawn horizontally in the example given in Figure 18; see also Figure 13.

3 5 3 J vJ v v J J 22 · 7 22 · 5 2 v rrv rv r Figure 18: Θ(Λ(7; 3, 5, 22 · 7, 3, 22 · 5, 2)

2.3 The simple rank of a graph of dihedral groups We assign to any admissible labeled graph Θ = (Θ, l) the complexity srank Θ which we call the simple rank of Θ. It is apparent from the deﬁnition that

23 the simple rank is easily computed by a simple algorithm. We further define srank A :=srank Θ(A). The rank problem for graphs of dihedral groups is then solved by showing that rank π1(A) = srank Θ(A). Let Θ0 be the admissible graph consisting of a single vertex with label 1. In the definition of srank two types of modifications are essential, α-moves and β-moves. They both transform an admissible labeled graph into a new admissible labeled graph. Suppose that v is a small vertex of the admissible labeled graph Θ1.

α-moves: We say that the admissible labeled graph Θ2 is obtained from Θ1 by an α-move relative v if Θ1 = Θ2 or one of the following holds

1. Θ2 is obtained from Θ1 by adding a fat vertex w with odd label p and an edge joining v and w.

2. There exists a vertex w adjacent to v such that w has label m ≥ 2 and Θ2 is obtained from Θ1 by replacing the label of w by mp for some odd p.

4 4 QQ QQ Q Q r v Q v - r v Qv p r r v 7 7 v v

vm 3 - v mp 3 r v r v r v r v Figure 19: The two types of α-moves

β-moves: We further say that the admissible labeled graph Θ2 is obtained from Θ1 by a β-move relative v if Θ1 = Θ2 or one of the following holds

1. Θ2 is obtained from Θ1 by adding a fat vertex w with label 2 joined to v by an edge and a small vertex y joined by an edge to w. 2. There exists a vertex w that is adjacent to v and to a small vertex z m of valence 1 such that w has label 2 p and Θ2 is obtained from Θ1 by replacing the label of w by 2m+1p.

Remark 2.8 Note that the small vertex is added in the β-move of the ﬁrst type to ensure that the resulting labeled graph is admissible. It is further important to note that in the α move of the second type (see Figure 19) v could be either the vertex to right or the vertex to the left of the vertex with label m while in the β-move (see Figure 20) of the second type only the vertex to the right is permitted, i.e. z is not.

We now deﬁne the complexity srank Θ for an admissible labeled graph Θ. 0 We call srank Θ the simple rank of Θ. The labeled graphs Θ1,Θ2 and Θ below are also assumed to be admissible.

24 4 4 QQ QQ Q Q r v Q v- r v Q v 2 r r v r 7 7 v v

m m+1 z2 p v3 - z 2 p v 3 r v r v r v r v Figure 20: The two types of β-moves

∼ 1. srank Θ = 1 if and only if Θ = Θ0.

2. srank Θ = k + 1 if srank Θ > k and there exist Θ1 and Θ2 such that srank Θ1 + srank Θ2 = k + 1 and one of the following holds:

(a) Θ is obtained from Θ1 and Θ2 by identifying two small vertices v1 ∈ V Θ1 and v2 ∈ V Θ2 and applying an α-move relative the new vertex.

(b) Θ is obtained from the disjoint union of Θ1 and Θ2, a vertex v∗ with even label and two edges joining v∗ with a small vertex of Θ1 and a small vertex of Θ2, respectively. 3. srank Θ = k + 1 if srank Θ > k and there exists some Θ0 such that srank Θ0 = k and one of the following holds:

0 0 (a) There are small vertices v1 6= v2 of Θ and Θ is obtained from Θ by identifying v1 and v2 and performing an α-move relative the new vertex. (b) There are small vertices v and w of Θ0 and Θ is obtained from Θ0 by adding a vertex y with even label and two edges joining y to v and w. (c) There is a small vertex v of Θ0 and Θ is obtained from Θ0 by an α- move relative v and by adding a loop vertex w with two edges joining v and w. 4. srank Θ = k + 1 if srank Θ > k and there exists some Θ0 such that srank Θ0 = k and there is a small vertex v of Θ0 such that Θ is obtained from Θ0 by a α-move relative v followed by a β-move relative v.

Note that this is a proper definition as it enables us to produce a list of all admissible labeled graphs of simple rank k + 1 provided we have lists of admissible labeled graph of simple rank m for all m ≤ k. As (1) gives the list of admissible labeled graphs of simple rank 1 this is well-defined. Note that the simple rank of an admissible labeled graph does not change if a vertex label 2n(2l + 1) with l ≥ 1 and n ≥ 0 is replaced by the label 2n(2l0 + 1) with l0 ≥ 1. It follows from the definition of the simple rank that any admissible labeled graph Θ can be built up starting with graphs of type Θ0 such at any stage we are in one of the situations spelt out in the definition. We call each such way a minimal construction of Θ. Clearly there can be multiple minimal constructions of a given graph Θ; however some are more convenient to work with than others.

25 Lemma and Definition 2.9 Let Θ be an admissible labeled graph. Then there exists an minimal construction of Θ such that the following hold: 1. The label of any fat vertex that emerges in as in part 2(b) or 3(b) of the definition of the simple rank does not change at a later stage of the construction of Θ. 2. Each vertex v with even label that does not emerge as in part 2(b) or 3(b) is built up by β-moves relative some small vertex w possbily followed by a single α-move relative the other small vertex z adjacent to v that is also not identified with w at a later stage of the construction. 3. Each vertex with odd label is affected by a single non-trivial α-move. We call such a minimal construction an efficient construction of Θ.

Proof To see that the first assertion holds note that any further changes would be α- or β-moves which we could replace by trivial moves provided that we give the fat vertex the right (final) label when it first appears in this process. Note that this implies that any fat vertex with even label that does not emerge as in part 2(b) or 3(b) is built up by α-moves and β-moves. The same argument proves that in 2 and 3 we can assume that the first α-move that affects a given vertex v leaves the vertex with label 2n(2l + 1) if 0 the final label is of type 2n (2l + 1). To see part two note first that we can assume that whenever in the construction there is an admissible labeled graph Θ¯ that consists of a single fat vertex v with even label and one or two adjacent small vertices (joined by 2 or 1 edges with v) then we can assume that the label of v does not further change during the construction. This holds as srank Θ¯ ≥ 2 since Θ¯ 6= Θ0 and we can therefore assume (after replacing the minimal construction with another minimal construction) that Θ¯ emerged as in situation 2(b) from two copies of Θ0 or as in 3(b) from one copy of Θ0 yielding the final label by the same argument as before. It follows that we can assume that all β-moves that change the label of some fat vertex v (or create it) are relative the same small vertex w (see Remark 2.8). Note next that we can assume that no α-move relative w affects the label of v. Indeed we could then swap the effect of this α-move with the effect of some α-move (also relative) w that comes in conjunction with one of the β-moves that builds up the label of v (recall that β-moves only appear in situation 4 of the definition of srank). After this change in the construction we are in the situation that at some stage we are in situation 4 and the α-move and the β- move are both relative w and both affect the label of v. Then we can again change the construction of Θ and replace this occurrence of situation 4 by an occurrence of situation 2(b) where one of the admissible labeled graphs is Θ0. We further replace all other β-moves affecting v with the trivial β-move as we can assume as before that no changes to the label of v are done after v emerges as in situation 2(b). We now show that the only α-move (relative the small x vertex different from w) affecting the label of v occurs after all β-moves affecting v. This is clear if x is no longer of valence 1 after this modification as then no more β-moves relative w are possible. Note that this is always the case unless the modification is of

26 type 2(a) with one admissible labeled graph of type Θ0 or the modification if of type (4) with a trivial β-move. In both of these cases however we can change the minimal construction and replace all β-moves affecting v by trivial β-moves and the move of type 2(a) or 4 by a single move of type 2(b) with one admissible labeled graph being Θ0 that gives v its final label. Suppose finally the label of v is affected by an α-move relative z and that w and z get identified at a later stage of the construction, necessarily as in situation 3(a). Then the α-move coming along this modification can be swapped with the α-move relative x that affected the label v which means that we can drop the β-moves and replace this occurence of 3(a) with an occurence of 3(b) that gives v its final label. 2 To illustrate the definition of srank we look at the two graphs of dihedral groups already considered in Section 2.1, namely the group Λ(5; 3, 2, 3, 2) which is of rank 3 and Λ(5; 3, 2, 2, 3) which is of rank 4. In the case of Λ = Λ(5; 3, 2, 3, 2) it is easy to see that srank Θ(Λ) = 3. Note first that srank Θ(Λ) ≤ 3 as Θ(Λ) can be obtained by starting with Θ0 and applying an α-move and an β-move with respect to the single small vertex v of Θ0 and then repeating the same for the new small vertex w. To see that srank Θ(Λ) > 2 it suffices to observe that Θ(Λ) can neiter be constructed from two copies of Θ0 as in part (2) nor from a single copy of Θ0 as in part (3) of the definition of the simple rank. w w 2 2 2 v r v r v

αv, βv- αw, βw- v v v 3 r r r v r

3 3 v v Figure 21: srank Θ(Λ(5; 3, 2, 3, 2)) = 3

For Λ = Λ(5; 3, 2, 2, 3) the argument is more involved. As before we see that srank Θ(Λ) ≥ 3 and we easily see that srank Θ(Λ) ≤ 4 as we can follow the ﬁrst part of the above construction. One way of seeing that srank Θ(Λ) = 4 is to make the complete list of admissible labeled graphs of simple rank 3 and observing that Θ(Λ) is not one of them. However it also follows from the fact that rank Λ ≥ 4 (see Section 2.1) and Lemma 2.13 below.

3 2 2 3

rv rv rv v Figure 22: srank Θ(Λ(5; 3, 2, 2, 3)) = 4

The following theorem gives a complete answer to the rank problem for graphs of dihedral groups:

Theorem 2.10 (Theorem 0.1) Let A be a graph of dihedral groups. Then

rank π1(A) = srank Θ(A) = srank A

27 We postpone the proof of Theorem 2.10 to Section 2.4 and conclude this section by discussing those consequences of Theorem 2.10 that are essential to the solution of the rank problem of planar groups.

Corollary 2.11 (Theorem 2.5) Let Λ = Λ(n; h1, . . . , hn−1). Then Λ has a minimal generating set that is simple. Proof In view of Theorem 2.10 we have to show that Λ has a simple generating 2 ∼ set of cardinality srank Θ(Λ). We also admit the group Λ(1; −) = hs1 | s1i = Z2. Clearly Θ(Λ(1; −)) = Θ0. The proof is by induction on srank Θ(Λ). For n = 1 there is nothing to show as 1 = srank Θ0 and the single non-trivial element that generates Z2 constitutes a simple generating set. A trivial but important observation is that in the definition of the simple 0 rank no loop ever disappears when going from Θ1 and Θ2 or Θ to Θ. This implies that we do not have to worry about part 2 of the definition of srank as the graph Θ = Θ(Λ) contains no loops. It is further clear that each of the 0 graphs Θ1 and Θ2 or Θ is a tree and the subtree spanned by the fat vertices with even labels is a segment. Thus they are the graphs associated to groups 0 0 0 Λi = Λ(ni; h1, . . . , hni−1) for i = 1, 2 or Λ = Λ(n ; h1, . . . , hn −1) such that 0 0 Θi = Θ(Λi) for i = 1, 2 or Θ = Θ(Λ ). Thus we can assume that Θ emerges as in part 2 or 4 of the definition of srank. We deal with one case at a time. Without loss of generality we can assume that the construction of Θ is efficient, i.e. that the conclusion of Lemma 2.9 holds.

2(b): By induction we know that Λi has a simple generating set of cardinality srank Θi for i = 1, 2. We further have

Λ = Λ1 ∗Z2 D2m ∗Z2 Λ2 with even m. It follows from the normalization process that after a permutation of the odd hi (which does not affect the simple rank) we can assume that Λ1 = (k; h1, . . . , hk−1) and Λ2 = Λ(n−k; hk+1, . . . , hn−1) and that m = hk. It is easy to see that the union of the minimal simple generating sets of Λ1 and Λ2 give a simple generating set of Λ which is clearly of cardinality srank Θ1+srank Θ2 = srank Θ = rank Λ. 2(a): If the α-move is of the first type we argue as in case 2(b) with odd n. In the remaining case Θ is obtained from Θ1 and Θ2 by identifying vertices v1 and v2 and then applying an α-move relative the new vertex that affects the m label 2 of a fat vertex v that was in say Θ1 adjacent to v1 and replaces it with 2mp for some odd p. As we assume that the construction is efficient it follows from Lemma 2.9 m that the label 2 of v is built up by m β-moves relative a vertexv ¯ 6= v1 0 adjacent to v. Let now Θ1 be the maximal connected (admissible labeled) 0 subgraph of Θ1 that containsv ¯ but not v. Let further Θ2 be the admissible labeled graph obtained from Θ2 by applying relative v2 all modifications that m were applied to Θ1 relative v1 after the creation of the label 2 of v. It is clear 0 0 that srank Θ1 + srank Θ2 ≤ srank Θ1 + srank Θ2 and that Θ is obtained from 0 0 Θ1 and Θ2 by a modification of type 2(b) creating a fat vertex with label of type 2mp. Thus we are back in situation 2(b). 4: In this case the labeled graph Θ is obtained from Θ0 by applying an α-move and a β-move relative a small vertex v. Let Λ0 be the group corresponding

28 to Θ0. By induction there is a simple generating set of Λ0 that is of cardinality srank Θ0 = srank Θ − 1. Note that we can embed Λ0 in Λ in the obvious way. If both moves are of the first type then it is easy to see that Λ has a simple generating set of cardinality srank Θ as we can simply take the simple generating 0 set of Λ and add one more tuple τj (see definition of simple generating sets) to take care of the two new generating reflections. Note that in this case the groups Λτj is of type Λ(2, 2l + 1). In the remaining cases we see in a similar way that the simple generating set for Λ0 can be modified into a simple generating set of Λ by increasing the cardinality by one. In these cases however the number of tuples τi does not increase but one of the τi is modified. 2 ∞ Corollary 2.12 (Theorem 2.6) Let G = ∆ (n; h1, . . . , hn) and t be the generator corresponding to the boundary component. Then t is a primitive element of G and rank G = 1 + min rank Λ(n; hσ(1), . . . , hσ(n−1)). σ∈Σ Proof Note first that the fact that t is a primitive element of G is an immediate consequence of the second part of the claim as for any choice of σ ∈ Σ the canonical image of Λ = Λ(n; hσ(1), . . . , hσ(n−1)) in G together with t generates G as G can be considered as an HNN-extension of Λ with stable letter t.

This argument also shows that rank G ≤ 1+rank Λ(n; hσ(1), . . . , hσ(n−1)) for all σ ∈ Σ. It remains to verify that rank G = 1 + rank Λ(n; hσ(1), . . . , hσ(n−1)) for some σ ∈ Σ. As ∞ ∼ ∆ (n; hσ(1), . . . , hσ(n−1), ∞) = Z ∗ Λ(n; hσ(1), . . . , hσ(n−1)) it follows from Grushko’s theorem that it suﬃces to show that rank G = ∞ rank ∆ (n; hσ(1), . . . , hσ(n−1), ∞) = 1 + rank Λ(n; hσ(1), . . . , hσ(n−1)) for some σ ∈ Σ.

Note that if all hi are odd then Θ(G) consists of a single small vertex v, one loop vertex adjacent to v and some fat vertices with odd labels also adjacent to v. If some hi is even then Θ(G) consists of a loop consisting of fat vertices with even labels alternating with small vertices and a number of fat vertices with odd labels that are attached to the small vertices of the loop. Choose an efficient construction of Θ(G). As Θ(G) contains a loop at some point in the construction we must be in situation 3 of the definition of srank. After this modification of type 3 there can be no more β-move in the construction but only moves of type 2(a) and 4 (with trivial β move). It is clear that all of these subsequent modifiations could have been done earlier thus we find another efficient construction of Θ(G) such that the single modification of type 3 occurs last. Thus there exists an admissible labeled graph Θ0 with srank Θ0 = srank Θ(G)−1 such that Θ(G) is obtained from Θ0 by a modification of type 3. Unless the modification is of type 3(a) and the α-move affects the label of an already existing fat vertex it follows from the normalization procedure of graphs of dihedral groups that for some σ ∈ Σ the graph Θ(Λ(n; hσ(1), . . . , hσ(n−1))) 0 can be obtained from Θ after a permutation of the odd hi. As this operation does not affect the simple rank it follows that

0 rank G = srank Θ = srank Θ + 1 = srank Θ(Λ(n; hσ(1), . . . , hσ(n−1))) + 1

29 = rank Λ(n; hσ(1), . . . , hσ(n−1)) + 1. Thus we are left with the case that the modification is of type 3(a), i.e. identifies two small vertices v1 and v2, and then applies an α-move relative the new vertex m 0 that affects the label 2 of a fat vertex v that was in Θ adjacent to say v1. The same argument as in part 2(a) of the proof of Corollary 2.11 shows that we find another minimal construction such that this final modification is of type 3(b). This proves the claim. 2

2.4 Simple rank equals rank This section is dedicated to the proof of Theorem 2.10. The easy part is to show that rank π1(A) ≤ srank Θ(A), i.e. to show that there exists a generating set of cardinality srank Θ(A).

Lemma 2.13 Let A be a normalized graph of dihedral groups. Then

rank π1(A) ≤ srank Θ(A).

Proof Put Θ = Θ(A). The proof is by induction on n = srank Θ. For n = 1 there is nothing to show as srank Θ0 = 1 = rank Z2. Suppose now that 0 srank Θ ≥ 2. Choose Θ1 and Θ2 or Θ such that Θ is obtained as in one of the cases of the deﬁnition of the simple rank. 0 Note that we can embed π1(A(Θ1)) and π1(A(Θ2)) or π1(A(Θ )) in to π1(A) in an obvious way. If Θ is obtained from Θ1 and Θ2 as in 2(a) or 2(b) then the union of the generating sets of π1(A(Θ1)) and π1(A(Θ2)) yields a generating set of π1(A) which proves the assertion. In all other cases a generating set of 0 π1(A) can be constructed from a generating set of π1(A(Θ )) by adding one more generator as explained in Lemma 2.2. 2 The remainder of this section contains the proof of the opposite inequality. Let A be a normalized graph of dihedral groups that is not a single vertex group. Suppose that B is an A-graph with underlying graph B. We call a vertex v ∈ VB fat if [v] is fat and small if [v] is small. We say that B is regular if for every vertex v ∈ VB one of the following holds:

1. Bv is trivial. ∼ 2. Bv = Z2 and Bv is generated by a reﬂection of A[u]. ∼ 3. Bv = D2n and Bv is generated by two reﬂections of A[u].

Let B be a regular A-graph with associated graph of group B. Let n be the free rank of B and B1,..., Br be the free factors of B. Clearly Bi is a regular A-graph without any trivial edge or vertex groups. Note that we can ¯ construct for every Bi a normalized graph of dihedral groups Bi by collapsing maximal subtrees whose vertices all have vertex groups of order 2, applying the normalization moves and subdividing possible loop edges. We deﬁne r X ¯ srank B = srank B := n + srank Bi. i=1

30 Now let S be a minimal generating set of G = π1(A) = π1(A, v0). Represent every element s ∈ S by a (not necessarily reduced) A-path of positive length; this is possible as we assume that A contains an edge. Let B0 be the corresponding S- wedge of G. Clearly B0 is a regular A-graph representing G as all vertex groups of the associated graph of groups B0 are trivial and b(B) = #S = rank G. This implies that B0 has no free factors and that the free rank of B0 equals b(B). It follows that srank B0 = rank G. We ﬁrst show that it is not hard to a avoid non-regular A-graphs in a folding sequence approximating A. We could avoid this technicality by dealing with non-regular A-graphs in the proof itself; this would however have added more subcases.

Lemma 2.14 There exists an S-wedge B0 representing G as above and a sequence of regular A-graphs B1, B2,..., Bm such that Bm is folded and that the following hold:

1. Bi+1 is obtained from Bi by a fold of type IA, IIA or IIIA for 0 ≤ i ≤ m−1. 2. Folds of type IA or IIIA are only applied if no fold of type IIA is possible. Proof Note first that the existence of such a sequence without the assumption that all Bi are regular follows from the fact that the edge groups of A are noetherian which implies that there is no infinite sequence of folds of type IIA and the fact that A contains no loop edge which implies that folds of type IB, IIB and IIIB cannot occur. The only way a fold can turn a regular A-graph Bi into a non-regular A-graph Bi+1 is that a fold of type IIIA adds a rotation to a formerly trivial vertex group of a fat vertex. As however another folding move must add a reflection to this vertex group at some later point we could just as well have added a reflection in the first fold as for any rotation r and reflection s in a dihedral group there exists a second reflection s0 such that hr, si = hs, s0i. The appropriate modificatioins to the sequence of A-graphs are easily made. 2 The main step is to prove the following:

Proposition 2.15 Let B be a regular A-graph and suppose that B0 is obtained from B by a fold of type IIA or if no fold of type IIA is applicable by a fold of type IA or IIIA. Then srank B ≥ srank B0. Before we give the proof we show that it implies Theorem 2.10. Proof of Theorem 2.10 If A consists of a single vertex the assertion is trivial. Otherwise choose a sequence B0, B1,..., Bm as in Lemma 2.14. As Bm is folded it follows that Bm is the induced splitting of G itself, in particular we have srank Bm = srank A. It then follows from Proposition 2.15 that

rank G = srank B0 ≥ srank B1 ≥ ... ≥ srank Bm = srank A = srank Θ(A). Together with Lemma 2.13 this proves the assertion of Theorem 2.10. 2 Proof of Proposition 2.15 We need to show that

r r0 X ¯ 0 X ¯0 n + srank Bi ≥ n + srank Bi. i=1 i=1

31 0 0 0 where n and n are the free ranks of B and B and the Bi and Bi are their free ¯ ¯0 factors with associated normalized graphs of dihedral groups Bi and Bi. As the auxiliary folds do clearly not change the simple rank, we only need to consider elementary folds of type IA, IIA or IIIA. We deal with each type of fold individually; there are many subcases. We assume that notations are as in the deﬁnition of the folding operations, i.e. that there are vertices x, y and z and edges f1 and f2 or f of the graph B underlying B with corresponding groups Bx,By,Bz and Bf1 ,Bf2 ,Bf . Folds of type IA: We distinguish the cases that y and z are small vertices and that y and z are fat vertices. Recall that we can assume that Bf1 = Bf2 as the fold is elementary and no fold of type IIA is applicable. Case 1: y and z are small vertices. This case is harmless.

(a) If Bf1 = Bf2 6= 1 then the fold preserves the free rank and only aﬀects one free factor but not its associated normalized graph of groups because of step 3 (ﬁrst case) of the normalization process. Thus srank B = srank B0.

(b) If Bf1 = Bf2 = 1 then also By = 1 and Bz = 1 as otherwise a fold of type IIA could be applied. In this case the free rank and all free factors are preserved. Thus srank B = srank B0.

Case 2: y and z are fat vertices and Bf1 = Bf2 6= 1. If By or Bz are of order ¯ two then the normalized graphs of dihedral groups Bi does not change and we are done. Thus we can assume that By and Bz are dihedral subgroups of A[y] of order ≥ 4. As the fold is elementary By ∩ Bz contains at least one reflection, ∼ ∼ say the reflection s. Choose k, k1, k2 such that A[y] = D2k, By = D2k1 and ∼ ∼ Bz = D2k2 . It is easy to see that hBy,Bzi = D2k∗ with k∗ = scm(k1, k2). We can choose reflections sy, sz and s∗ ∈ hs, sy, szi such that By = hs, syi, Bz = hs, szi and hBy,Bzi = hs, s∗i.

(D2k , v) 1 v (D , v) IA - 2k∗ PP PP rPP r v P (D2k2 , v) v Figure 23: The eﬀect of an elementary fold of type IIA on B

So far we have only looked at how the fold affects the A-graph B. In order to see how the fold affects the simple rank we do however need to study how ¯ the normalized graph of dihedral groups Bi corresponding to Bi changes. In ¯ the figures below we show how the corresponding labeled graph Θi = Θ(Bi) changes. Sometimes we will denote the image of a vertex v under the normalization by p(v). We always assume that p(v) then plays in the remainder of the construction of the labeled graph the role of v. Note that k1, k2 > 1 We need to distinguish a variety of cases:

(a) Suppose that k1 and k2 are odd. In this case two fat vertices with odd label (i.e. vertices of valence 1) attached to a small vertex x get replaced by one fat vertex with odd label attached to the same vertex. This clearly does not increase (but possibly decrease) the simple rank.

32 l¯ ¯ (b) Suppose that k1 is odd and k2 = 2 k2 with l ≥ 1 and k2 odd. Clearly we l¯ ¯ have k∗ = 2 k∗ for some odd k∗.

¨¨ k1 ¨ 2lk¯ ¨¨ v IA - ∗ HH H l¯ r HH 2 k2 rv r v r l¯ Here the simple rank does no increase. Indeed, if the label 2 k2 emerged by l¯ move of type 2(b)/3(b) we can use this move to create the label 2 k∗ and replace the α-move that created the vertex with label k1 by a trivial one, otherwise we ¯ l¯ can use it to produce the odd factor k∗ for the fat vertex with label k∗ = 2 k∗. l¯ l¯ ¯ ¯ (c) Suppose that k1 = 2 k1 and k2 = 2 k2 with k1, k2 odd and l ≥ 1. As y and z both have even dihedral vertex groups, their fat vertices in Θi must be of valence 2, i.e. they must be adjacent to vertices v1 and v2, respectively. It is easy to see that in D2k∗ sy and sz are both conjugate to s∗ but not to s. Suppose first that v1 6= v2. We discuss the case that x 6= v1 and x 6= v2. The case where either x = v1 or x = v2 requires no new arguments, only the figures look slightly different. Because of the fold we must identify the two edges l¯ l¯ joining x and the fat vertices and replace the vertices with labels 2 k1 and 2 k2 l with a vertex with label 2 k∗. The normalization requires that we identify edges adjacent to the new fat vertex that correspond to conjugate reflections. Thus we have to identify the two edges joining the fat vertex with v1 and v2 respectively.

¨ l v1 ¨ 2 k¯1 ¨ 2lk¯ ¨¨ v r IA - ∗ x H H p(x) p(v1) = p(v2) H l r H 2 k¯2 rv r H v2 v r To see that the simple rank does not go up recall first that by Lemma 2.9 we can assume that all fat vertices with label 2lq with odd q and l ≥ 1 either emerge as in situation 2(b) or 3(b) of the definition of simple rank or are build up by l β-moves relative one small vertex and at most one α-move. If the labels of both y and z emerge as in situation 2(b)/3(b) then the simple rank of the new graph is not greater as we can obtain it by a move of l¯ type 2(b)/3(b) that inserts the fat vertex with label 2 k∗ between x and v1 and a move of type 2(a)/3(a) that identifies v1 and v2 and applies a trivial α-move. l¯ If one the labels, say 2 k1, emerges as in situation 2(b)/3(b) and the other is built up by β-moves then we can use these β-moves to build up the factor 2l l¯ of 2 k∗ and then replace the 2(b)/3(b) move by a 2(a)/3(a) move that identifies l¯ v1 and v2 and applies an α-move to the new fat vertex yielding the label 2 k∗. If both labels emerge by a sequence of β-moves relative x possibly followed l¯ by an α-move then a subset of these moves can be used to yield the label 2 k∗ in the new graph. This clearly shows that the simple rank does not increase.

If the label ki emerges by a sequence of β-moves relative vi for i = 1, 2 possibly followed by an α-move then the vertex x must emerge by a move of

33 type 2(a) or 3(a) from the vertices created by these moves. Without loss of l¯ l¯ generality we can assume that the label 2 k1 emerges before 2 k2 if the efficient construction imposes such an order. We then replace all β-moves relative v2 (that build up k2) by trivial ones and also omit one of the α-moves relative v2 that came in conjunction with one of these β-moves. (note that this saves 1 0 in the computation of the simple rank.) The graph Θi is then obtained by a move of type 2(a)/3(a) that identifies v1 and v2 such that the α-move makes up for the omitted α-move relative v2 followed by a move of type 2(a) with a copy of Θ0 that makes up for the α-move relative x, here we spend what we saved before. The assertion follows. l¯ Suppose finally that the label 2 k1 emerges by a sequence of β-moves rela- l¯ tive v1 and the label 2 k2 emerges by β-move relative x (the opposite case is l¯ analogous). Then arguments as before show that the lable 2 k∗ can be built up using the β-moves of the label that emerges first.

If v1 = v2 then the normalization after the fold yields a new loop vertex. The same arguments as in the case before show that the simple rank does not l 2 k¯1 ¨¨HH ¨¨ H ¨ v HH IIA - r x H ¨ H ¨ v1 p(x) H ¨ l¯ rH ¨ r 2 k∗ H¨l¯ 2 k2 p(v1) v rv r go up. The only diﬀerence is that we need to replace a move of type 3(a) by a move of type 3(c).

l1 ¯ l2 ¯ ¯ ¯ (d) Suppose that k1 = 2 k1 and k2 = 2 k2 with k1 and k2 odd and l1 > l2 ≥ 1 (the case l2 > l1 ≥ 1 is analogous). The situation is similar as in case (c) however this time sy is conjugate to s∗ and sz is conjugate to s. Thus the normalization identiﬁes the edge joining the fat vertex with v2 with the edge joining the fat vertex with x. We distinguish the cases x = v2 and x 6= v2.

Suppose ﬁrst that x 6= v2. We discuss the case where v1 6= x and v1 6= v2. No new arguments are needed to deal with the remaining cases. There are four diﬀerent cases.

¨ l v1 ¨ 2 1 k¯1 ¨ 2l1 k¯ ¨ v r IA ∗ x H¨ - H p(x) = p(v2) p(v1) H l r H 2 2 k¯2 rv r H v2 v r If both of the labels of the fat vertices y and z emerge as in situation l1 ¯ 2(b)/3(b) we can use a move of type 2(b)/3(b) to create the label 2 k∗ of the fat vertex between x and v1 and then a move of type 2(a)/3(a) to identify x and v2 and apply a trivial α-move. l1 ¯ l2 ¯ If the label 2 k1 occurs as in situation 2(b)/3(b) and the label 2 k2 is build up using β- and α-moves, we use the 2(b)/3(b) move responsible for the label l1 ¯ l1 ¯ 2 k1 to create the label 2 k∗ and replace the α- and β-moves by trivial ones. Thus the simple rank does not increase.

34 l1 ¯ If the label 2 k1 is built up using β-move and possibly an α-move and the l2 ¯ label 2 k2 emerges as in situation 2(b)/3(b) then we use the β-moves to build l1 l1 ¯ up the factor 2 of the label 2 k∗ and replace then replace the 2(b)/3(b) move by a 2(a)/3(a) move identifying x and v2 and applying the α-move relative the ¯ l1 ¯ new vertex yielding the factor k∗ of the label 2 k∗. Thus the simple rank does not increase. If both labels emerge by a sequence of β-moves possibly followed by an α- l2 ¯ move we can simply omit the β-moves responsible for the label 2 k2 and use the l1 l1 ¯ remaining β-moves to build up the factor 2 of 2 k∗ and possibly an α-move ¯ to get the factor k∗.

Suppose now that x = v2. In this case we need to add a loop vertex in the normalization. Again the same arguments as in the case x 6= v2 show that the simple rank does not increase.

v1 ¨¨ l1 ¯ ¨ 2 k1 ¨ v r IA r x ¨ - HH l1 ¯ H l2 ¯ 2 k∗ r HH 2 k2 p(x) p(v1) r v rv

Case 3: y and z are fat vertices and Be1 = Be2 = 1. If By = 1 or Bz = 1 then it is obvious that neither the free rank nor any of the free factors is aﬀected; it follows that the simple rank does not change. Thus we only need to consider the case that By 6= 1 and Bz 6= 1.

Suppose that Bz and By are both cyclic of order 2 and that y and z belong to the same free factor Bi. It is clear that the free rank decreases by one and that 0 Θi is obtained from Θi as in case 3(a), 3(b) or 3(c). This proves the assertion. Suppose that Bz and By are both cyclic or order 2 and that y and z belong to diﬀerent free factors Bi and Bj. In this case the free rank is preserved and 0 the graphs Θi and Θj are replaced by a graph Θi obtained from Θi and Θj as in situation 2(a) or 2(b). Thus the simple rank does not increase. If one or both of the groups are proper dihedral groups then a case by case study yields the desired result directly. It is however also easy to see that this case can be reduced to cases already checked; this is what we will do. There is no need to distinguish whether y and z belong to the same free factor or not. We show that there exist A-graphs Bˆ and Bˆ0 with srank Bˆ = srank B and srank Bˆ0 = srank Bˆ0 such that Bˆ0 is obtained from Bˆ by a fold of type IA where

Be1 = Be2 = 1 and By, Bz are cyclic of order two and two folds of type IA with involved edge groups of order two. As we have already dealt with these types of folds it follows that srank Bˆ ≥ srank Bˆ0 and therefore srank B ≥ srank B0. Recall ﬁrst what happens going from B to B0, the dotted lines stand for trivial edge groups in the associated graph of groups; after applying some auxiliary moves we can assume that the edge labels are (1, e, 1) for some e ∈ EA. We now modify B to obtain a new A-graph Bˆ. It will be clear that Bˆ and B represent the same subgroup and that Θ(B) = Θ(Bˆ), in particular they have the same simple rank. As By and Bz are assumed to be non-trivial, they contain reﬂections ry ∈ Av and rz ∈ Av. Thus there exist edges ey, ez ∈ EA with α(ey) = α(ez) = v and

35 (1, e, 1) (By, v) 1 (Bx, u)(v IA - Bx, u) (1,- e, 1) (hBy,Bzi, v) q r(1, e, 1) r v (Bz, v) v

−1 −1 elements gy, gz ∈ Av such that ry = gyαey (sey )gy and rz = gzαez (sez )gz

where sey and sez are the generators of Aey and Aez , respectively. y z ˆ Put Z2 = hωey (sey )i and Z2 = hωez (sez )i and construct B from B by removing the edges with labels (1, e, 1) and replacing them by segments of length three with labels as in the ﬁgure below.

(B , v) (B , v) (Zy , u ) y ¨ y 2 y ) ¨¨ J y J (Z , uy ) t y ¨ t r 2 P (gy , ey , 1) (Z , uy ) ¨ J PP 2 ¨ (gy , ey , 1) iP H (gy , ey , 1) J r PP (hry i, v) ] HH 1 r YH J (B , u) IA- (1,- e, 1) HH IA+IA- (1,- e, 1) J x (1, e, 1) t ¨¨ q (Bx, u) (Bx, u) (hBy ,Bz i, v) ¨(hry , rz i, v) r (hrz i, v) r ¨ t r t ¨ ) ¨ (g , e , 1) (gz , ez , 1) z z H z z (Z , u ) (Z2 , uz ) 2 z P (gz , ez , 1)t H PP r HY iP H r PP (Bz , v) HH (Bz , v) z (Z2 , uz ) t t r The new vertices are inessential in Bˆ, in particular srank B = srank Bˆ. After applying 3 folds as in the ﬁgure we obtain the A-graph Bˆ0. The vertices with 2 2 0 ˆ0 vertex groups Zy and Zz become inessential; thus srank B = srank B ; this proves the assertion. Folds of type IIA: As all vertex groups are cyclic of order 2 this type of fold can only happen if the edge group Bf of the associated graph of groups is trivial before and cyclic of order 2 after the move. Case 1: x is a fat vertex and y is a small vertex.

(a) If By = 1 then this fold preserves the free rank and all but one free factor Bi. It is however clear that the associated normalized graphs of dihedral group ¯ 0 Bi is unchanged, thus srank B = srank B. (b) If By 6= 1 then we could apply instead a fold of type IIA with y taking the role of x and x taking the role of y having the same eﬀect on Bi. This is case 2 below. Case 2: x is a small vertex and y is a fat vertex.

If By = 1 then we can argue as in (a) of Case 1. Thus we can assume that By 6= 1. Note that if x and y belong to the same free factor Bi then the free rank decreases by one and all other free factors are unchanged. Thus we need to show that the simple rank of the free factor that is being changed increases by at most one. If x and y belong to distinct free factors Bi and Bj then the free rank does not change, all other free factors are unchanged and Bi and Bj 0 are replaced by a new free factor Bi. We then have to check that the simple 0 rank Bi does not exceed the sum of the simple ranks of Bi and Bj.

36 (a) Suppose first that By is of order two. After the move the group is dihedral as a reflection gets added. 0 If x and y belong to the same free factor Bi then Θ(Bi) is obtained from Θ(Bi) as in part 3(a), 3(b) or 3(c) of the definition of the simple rank. This 0 implies that srank Θ(Bi) ≤ srank Θ(Bi) + 1 which proves the assertion. 0 If x and y belong to distinct free factors Bi and Bj then Θ(Bi) is obtained 0 from Θ(Bi) and Θ(Bj) as in part 2(a) or 2(b). It follows that srank Θ(Bi) ≤ srank Θ(Bi) + srank Θ(Bj) which proves the assertion. (b) Suppose that By is a dihedral group. As in case 3 for folds of type IA we reduce this situation to cases already handled. Recall how the fold of type IIA looks like, again an edge with trivial edge group is drawn as a dotted line.

(1,- e, 1) - (1,- e, 1) (Brx, u)(By v, v) (Brx, u)(hBy, ωe(s ve)i, v)

Choose an edgee ¯ ∈ EA with α(¯e) = v, an element g ∈ Av and a reﬂection −1 r ∈ By such that gαe¯(se¯)g = r and replace the dotted line in B with a segment of length three to obtain Bˆ. The argument then goes as before where se is the generator of Ae, se¯ the generator of Ae¯, u0 = ω(¯e) and Z2 = hωe¯(se¯)i.

(Z2, u0) (Z2, u0) (Z2, u0) L L rL IIA- rL IA- r K K (g, e,¯ 1) ( g, e,¯ 1)L ( g, e,¯ 1)L 6 (1, e, 1) (1, e, 1) (1, e, 1) - LL (By , v) - LL (By , v) - (Bx, u) (hri, v) (Bx, u) (hr, ωe(se)i, v) (Bx, u) (hBy , ωe(se)i, v) r tt r tt r t Folds of type IIIA: We distinguish two cases:

1. case: Suppose that Be1 = Be2 = 1. Here the free rank decreases by one.

(a) Suppose that By is trivial. Then one generator gets added to By. As we assume that B0 is regular this element must be a reﬂection. Thus one new free factor emerges whose associated labeled graph is Θ0. As all other free factors are unchanged this implies that the simple rank is unchanged.

(b) Suppose that By is of order 2. As one element gets added to By the group is a dihedral group after the fold. It follows that all but one free factors Bi are 0 unchanged and that Θi is obtained from Θi and Θ0 by a move of type 2(a)/2(b). 0 This implies that srank Bi ≤ srank Bi + 1, the assertion follows. (c) Suppose that By is a dihedral group; thus y is fat. This case can be handeled by expanding the graph of groups as in the last case of fold IIA and then applying a fold of type IIIA as in (b) followed by a fold of type IA.

2. case: Suppose that Be1 6= 1 6= Be2 . Clearly By 6= 1. This implies that the simple rank and all but one free factor Bi are preserved. Note first that if y is 0 small that Θi is obtained from Θi by removing one loop edge. This clearly does not increase the simple rank. Thus we assume that y is fat. Recall the effect of a move of type IIIA. −1 We first show that By is a subgroup of hBy, b1 b2i of index at most 2. Let −1 −1 se be the generator of Ae. Clearly b1 ωe(se)b1 ∈ By and b2 ωe(se)b2 ∈ By. −1 Thus b1 b2 conjugates one reflection of By onto another reflection of By. By is

37 (a-1, e, b1) IIIA- (a-1, e, b1) (B , v) −1 (Bx, w) y (Bx, w) (hBy, b1 b2i, v) p- p p p (a1, e, b2)

a subgroup of the dihedral group A[y]. As By is of order 2 or dihedral is is easy −1 to see that the index of By in hBy, b1 b2i is at most 2. We need to record the eﬀect on the associated labeled graph to the see that the simple rank does not increase. We distinguish two cases:

(a) Suppose that By is an odd dihedral group. In this situation a loop vertex adjacent to a small vertex v disappears and a vertex with odd label 2l + 1 adjacent to v obtains the new label 2(2l + 1). This is a β-move relative v. This implies that the simple rank does not increase as situation 3(c) is replaced by situation 4 in the process of building up the labeled graph. - 2l + 1- 2(2l + 1) - r- r t rt

−1 (b) Suppose that By is an even dihedral group and that b1 ωe(se)b1 is conjugate −1 −1 to b2 ωe(se)b2 in By or that By is of order 2 and b2 b1 ∈ By. It follows that −1 −1 −1 b1 ωe(se)b1 = b2 ωe(se)b2 after an auxiliary move of type A2. Thus b1 b2 −1 commutes with b1 ωe(se)b1 and lies therefore in By as By is even. It follows that in the labeled graph a loop vertex disappears and no other changes appear. Thus the simple rank does not increase. −1 (c) Suppose that By is an even dihedral group and b1 ωe(se)b1 is in By not −1 −1 conjugate to b2 ωe(se)b2 or that By is of order 2 and b2 b1 ∈/ By. In this situation the change in the labeled graph is illustrated in the figure below. If the vertex with label 2l is created as in situation 2(b)/3(b) of the definition of simple rank then we can create the label 4l in the new labeled graph by a move of type 2(b) with one graph being Θ0. Thus the simple rank does not increase. If the label 2l was created by a sequence of β-moves and possibly one α-move (necessarily relative v), then the loop that is being collapsed by this fold must have appeared as in situation 3(a) of the definition of the simple rank, i.e. the loop was created and an α-move relative v was performed. This however implies that the simple rank does not increase as the new labeled graph can be built up in the same ways except that a β-move relative v is performed instead of the loop being created. This means that situation 3(a) is replaced by situation 4. - 2l - - 4l r- t r t

This completes the proof of Theorem 2.10. 2

38 3 Planar groups

In this chapter we give a proof of Theorem 0.2 using the results established in Chapter 2. We ﬁrst discuss how to decompose Fuchsian groups as fundamental groups of graph of groups. Then we prove Theorem 0.2; the proof uses again a sequence of approximations of this splitting. We conclude by showing that more pathologies occur if the assumption that the groups are suﬃciently large is dropped.

3.1 Decomposing planar groups

We show how to write a planar group as the fundamental group of a graphs of groups. Let G be a planar group with orbifold

O = F (p1, . . . , pk, (h11, . . . , h1n1 ),..., (hm1, . . . , hmnm )). We discuss two diﬀerent types of splittings which we will call the standard free product decomposition and the standard Z-splitting. As our interest is purely algebraic, we do not distinguish ∞ and∞ ¯ . The standard free product decomposition This splitting is trivial whenever pi < ∞ and hij < ∞ for all i and j. Thus we can assume that after a permutation of the boundary components and cyclic permutations of the tuples

(hi1, . . . , hini ) that either p1 = ∞ or that pi 6= ∞ for all i and that h11 = ∞.

If p1 = ∞ then there is a boundary component γ of O containing no re- ﬂections and we can cut up O along properly embedded arcs that have both endpoints in γ such that we obtain an orbifold of type F (p1, . . . , pk) and orb- 2 ifolds of type S (∞, (hi1, . . . , hini )) for 1 ≤ i ≤ m. The Seifert-van Kampen theorem then yields

m ∼ ∞ G = π1(F (p1, . . . , pk)) ∗ F ∆ (ni; hi1, . . . , hini ) i=1 which we call the standard free product decomposition of G.

If pi 6= ∞ for all i and h11 = ∞ then we can ﬁnd a properly embedded 2 arc cutting O into two orbifolds where one is of type S ((h11, . . . , h1n1 )) and one of type F (∞, p1, . . . , pk, (h21, . . . , h2n2 ),..., (hm1, . . . , hmnm )). The second orbifold can now be cut up as before and we get

m ∼ ∞ G = π1(F (∞, p1, . . . , pk)) ∗ Λ(n1; h12, . . . , h1n1 ) ∗ F ∆ (ni; hi1, . . . , hini ), i=2 which we again call the standard free product decomposition of G. Note that the nature of the free factors of the standard free product decomposition shows that the rank problem for non-cocompact planar groups is solved by Grushko’s theorem in conjunction with Theorem 2.5 and Theorem 2.6. The standard Z-splitting This splitting is defined for all planar groups con- 2 taining reflections if O is neither of type S ((h1, . . . , hn)) nor of type 2 S (p, (h1, . . . , hn)). 2 If O is not of type S ((h11, . . . , h1n1 ), (h21, . . . , h2n2 )) we define the standard Z-splitting of G to be the graph of group decomposition AG of G corresponding

39 to the decomposition of O obtained by cutting O along simply closed curves (dotted lines) that cut oﬀ annuli such that one boundary component contains reﬂections.

3 4 3 7 2 r r r 8 r r 4 r r

Figure 24: Cutting O = T 2(3, 4, (2, 3, 4), (7, 8), (1)) along simple closed curves

Thus AG has inﬁnite cyclic edge groups, a vertex group v0 with vertex group π1(F (p1, . . . , pk, ∞,..., ∞)), vertices v1, . . . , vm with vertex groups Avi = ∞ ∆ (ni; hi1, . . . , hini ) for 1 ≤ i ≤ m. Furthermore there are edges ei for 1 ≤ i ≤ m with α(ei) = v0 and ω(ei) = vi for 1 ≤ i ≤ m. 2 If O = S ((h11, . . . , h1n1 ), (h21, . . . , h2n2 )) we deﬁne the standard Z-splitting to be the graph groups corresponding to the amalgamated product decompo- ∞ ∞ sition G = ∆ (h11, . . . , h1n1 ) ∗hti ∆ (h21, . . . , h2n2 ). Note that this avoids the cyclic vertex group corresponding to an annulus. Thus we only have vertices v1 ∞ and v2 with vertex groups Avi = ∆ (ni; hi1, . . . , hini ) for i = 1, 2.

Let T be the Bass-Serre tree corresponding to AG. It turns out that in many cases the action of G on T is 1-acylindrical, i.e. that no non-trivial element of G stabilizes a segment of length greater than 1, however there are exceptions:

Lemma 3.1 Suppose that g ∈ G − 1 ﬁxes a segment of length ≥ 2. Then one of the following holds:

2 ∼ 1. Av0 = π1(S (2, 2, ∞)) = D∞ and g corresponds to a power of the boundary curve. 2 ∼ 2. Av0 = π1(P (∞)) = Z and g corresponds to a power of the boundary curve. ∞ 2 ∼ 3. Avi = ∆ (1; 1) = π1(S (∞, (1))) = Z ⊕ Z2 for some i = 1, . . . , m and g corresponds to a power of the boundary curve.

Proof Suppose that g ∈ G ﬁxes edges e1 = [x, y] and e2 = [x, z] with y 6= z. If [x, y] and [x, z] are G-equivalent then there exists some h ∈ Stab x such −1 that he1 = e2. In particular g ∈ Stab e1 ∩ Stab e2 = Stab e1 ∩ h(Stab e1)h . This puts us into one of the above cases as in all other cases Stab x can be realized as a hyperbolic group with parabolic subgroup Stab e1 which implies that Stab e1 is malnormal in Stab x.

40 If [x, y] and [x, z] are not G-equivalent then x must be a vertex corresponding to v0 as all other vertices are of valence one in AG. As we have excluded the annulus in the deﬁnition of the standard Z-splitting it follows that we can assume that Av0 is hyperbolic with adjacent edge groups parabolic. It follows that elements corresponding to diﬀerent boundary components cannot be conjugate; thus this case never occurs. 2

3.2 The proof of the rank formula We need to establish some simple lemmas before we tie everything up and give a proof of Theorem 0.2.

Lemma 3.2 Let G be a Fuchsian group without reﬂections and corresponding orbifold O. Let t be an element corresponding to a boundary component. 2 2 Then t is primitive if and only if O = S (∞, 21,..., 22l, p) with odd p.

2 Proof If O= 6 S (∞, 21,..., 22l, p) then there is nothing to show as the solution to the rank problem for Fuchsian groups (without reﬂections) shows that rank G = rank G/hht2ii which clearly implies that t2 cannot be primitive. 2 If O = S (∞, 21,..., 22l, p) it is easy to see that the unique (up to Nielsen 2 equivalence) minimal generating set {g1, . . . , g2l} of G/hht ii can be lifted to a 2 2 generating set {g˜1,..., g˜2l, t } of G as the relation t is used only once in the 2 computation that shows that {g1, . . . , g2l} generates G/hht ii, see [PRZ]. 2 Occasionally we will replace a k-generated subgroup of a planar group with a maximal (with respect to inclusion) k-generated subgroup. In groups in general such maximal subgroups do not exist; however, as we only need to do this in non-cocompact planar groups, this is not an issue as maximal k-generated subgroups containing a given k-generated subgroup always exist in virtually free groups.

We call a subgroup of a planar group π1(O) peripheral if it is conjugate to a subgroup corresponding to a boundary component of O.

∞ Lemma 3.3 Let G = ∆ (n; h1, . . . , hn). Let U be a maximal (with respect to inclusion) k-generated subgroup of G. Then one of the following holds:

1. There exists a generating set {u1, . . . , uk} of U such that any maximal peripheral subgroup of U is in U conjugate to hu1i (if U contains peripheral subgroups).

∞ 2 2. G = ∆ (1; 1) = π1(S (∞, (1))) and U = hgi where g is a glide reﬂection.

Proof Clearly we can assume that k ≤ n as the maximality of U otherwise implies that U = G and the assertion of the Lemma holds. If G = ∆∞(1; 1) we can therefore assume that U = hgi is cyclic. If g is a reﬂection or a translation we are in situation (1), if g is a glide-reﬂection in situation (2). If G 6= ∆∞(1; 1) we can realize G as a hyperbolic planar group such that the boundary curve corresponds to a parabolic element. A subgroup of G is then peripheral if and only if it is parabolic. Recall that we use ∞ for parabolic and ∞¯ for hyperbolic boundary components.

41 Note first that it suffices to show that any maximal parabolic subgroup is conjugate to huii for some i. Suppose there exist maximal parabolic subgroups C1 and C2 conjugate to huii and huji, respectively. As there is up to conjugacy −1 ±1 only one maximal parabolic subgroup in G it follows that guig = uj for some g ∈ G. Now g must lie in U as otherwise U is a proper subgroup of hu1, . . . , uj−1, uj+1, . . . , uk, gi contradicting the maximality of U. Thus C2 is also conjugate to huii, the assertion follows by exchanging ui with u1. Case 1: : O = U\H2 6= F (∞,..., ∞), i.e. O is not a m-punctured surface with all boundary components being parabolic. Suppose first that U contains no reflections. It follows that either U contains elliptic elements or that O has a hyperbolic boundary component. In both cases it is obvious that we can choose a minimal generating set with the desired properties. Thus we can assume that U contains reflections. We can further assume that U contains a parabolic element u corresponding to a boundary component ω of O as there is nothing to show otherwise.

Let O = F (p1, . . . , pk, (h11, . . . , h1n1 ),..., (hm1, . . . , hmnm )) be the orbifold of U. It follows from the standard free product decomposition of U and Grushko’s theorem that m X ∞ rank U = rank π1(F (p1, . . . , pk)) + rank ∆ (ni; hi1, . . . , hini ). i=1 To prove the lemma it suffices to describe a generating set of cardinality rank U that contains one element for every conjugacy class of parabolic elements. Take the standard Z-splitting of U. Take a minimal generating set S¯ of π1(F (p1, . . . , pk, ∞¯ 1,..., ∞¯ m)) that contains an element for every parabolic boundary component. This is possible as there are m ≥ 1 hyperbolic boundary components and for all but one boundary component we can choose the boundary path in the minimal generating set. Note that π1(p1, . . . , pk, ∞¯ 1,..., ∞¯ m) ∞ contains a primitive element of every vertex group ∆ (ni; hi1, . . . , hini ) by ∞ Corollary 2.12, thus we can find sets Si of cardinality rank ∆ (ni; hi1, . . . , hini )− 1 for 1 ≤ i ≤ m such that S = S¯ ∪ S1 ∪ ... ∪ Sm is a generating set. Clearly S is minimal and has the desired properties. Case 2: : Suppose now that F := U\H2 is a m-punctured surface with only parabolic boundary components. We will show that this is not possible. Note that this implies that F is of finite volume; thus U is of finite index j in G and χ(F ) = jχ(G). Recall that n n 1 1 X 1 1 X 1 χ(G) = 2 − 1 − 1 − − 1 − = − 1 − ∞ 2 h 2 h i=1 i i=1 i and that for any surface F with boundary we have rank π1(F ) = 1 − χ(F ). Note that j ≥ 2hi for all i as G contains a subgroup of order 2hi for any i. In particular we have j ≥ 4. Thus we have n n j X 1 j X 1 −jn χ(F ) = − 1 − ≤ − 1 − = 2 h 2 2 4 i=1 i i=1 which implies that −jn jn 4n k = rank π (F ) = 1 − χ(F ) ≥ 1 − = 1 + ≥ 1 + = 1 + n 1 4 4 4

42 contradicting the assumption k ≤ n. 2 We need an analogous result for planar groups without reﬂections.

Lemma 3.4 Let G = π1(O) be a non-cocompact planar group without reﬂec- tions and U ≤ G be a maximal k-generated subgroup. Then one of the following hold:

1. There exists a generating set {u1, . . . , uk} of U such that any conjugacy class of maximal peripheral subgroups of U has a representative huii. If i 6= j and huii and huji are both peripheral then they are not conjugate in G. 2. U = G is torsion-free, i.e. O is a surface. 3. O = S2(2,..., 2, ∞), U is free and |G : U| = 2.

Proof The second part of (1) follows as in the proof of Lemma 3.3 from the maximality of the subgroup U. We first deal with the cases O = P 2(∞) and O = S2(∞, ∞) and O = S2(2, 2, ∞). In the first two cases G is infinite cyclic which puts us into case 2 ∼ (2). In the last case we have G = π1(S (2, 2, ∞)) = Z2 ∗ Z2. If U is of rank 2 or ∼ U = Z2 we are in situation (1) and if U is infinite cyclic we are by maximality in situation (3). In all other cases G is hyperbolic and we can realize O such that all boundary components are parabolic. Thus peripheral is the same as parabolic. Let O0 be the orbifold corresponding to U. If O0 contains a cone point or a hyperbolic boundary component we are in situation (1). Thus we can assume that U is free and O is a punctured surface with all boundary components parabolic. This implies in particular that |G : U| < ∞ as otherwise O0 must have a hyperbolic boundary component. If O is not of type S2(2,..., 2, ∞) an Euler characteristic argument shows that finite index subgroups of G have at least the same rank as G; thus we are in situation (2) by maximality. Otherwise we are in situation (3) as U must be torsion-free. 2 We now have all tools to give a proof of Theorem 0.2. Proof of Theorem 0.2 As we assume the group to be sufficiently large we only need to be concerned with planar groups whose standard Z-splitting AG of G is non-trivial. The proof of the theorem is by induction on m, the number of boundary components containing reflections. For m = 0 there is nothing to show as the assertion follows from the rank formula for Fuchsian groups without reflections.

Let A be the graph of groups obtained from AG by subdivision, this clearly only introduces cyclic vertex groups. We call these new vertices small and the original ones fat. For an A-graph B we call a vertex u ∈ VB small, resp. fat, if [u] is small, resp. fat. Let B be an A-graph, S be a p-generating set for B and S¯ ⊂ S such that either S = S¯ or S = S¯ ∪ {s∗}. We say that the triple (B, S, S¯) is tame if the following hold:

1. B represents G and S¯ is of cardinality rank G − b(B).

43 2. If S 6= S¯, i.e. if S = S¯ ∪ {s∗} then there exists a vertex u∗ ∈ VB such ¯p that hS ∩ Bu∗ i = Bu∗ = A[u∗] and s∗ ∈ Bu∗ . Furthermore [u∗] = v0

and A[u∗] = Av0 is the fundamental group of a punctured surface. The element s∗ corresponds to a boundary component.

p 3. The following holds for all u ∈ VB. The group Bu = hBu ∩ S i is a maximal (with respect to inclusion) subgroup of A[u] that is generated by p #(Bu ∩ S ) elements. Furthermore every maximal peripheral subgroup of p Bu is in Bu conjugate to hgi for some unique g ∈ S ∩ Bu. 4. No fold can be applied to B that is based at a fat vertex.

5. For every f ∈ EB either Bf = A[f] or Bf = 1

p 6. No element of S propagates via a reduced edge path f1, f2 if ω(f1) = α(f2) is a fat vertex.

We ﬁrst make some simple observations about tame triples: ¯ (a) Every element of S is primitive. This is clear as π1(B) is generated by b(B) elements corresponding to the graph B and the set S¯. If m ≥ 1 then every element of S is primitive as every boundary curve of a surface with more than one boundary component represents a primitive element.

(b) If u ∈ VB is a fat vertex and f1, f2 ∈ EB with α(f1) = α(f2) = v and

Bf1 6= 1 6= Bf2 then [f1] 6= [f2]. Otherwise the maximality 3 implies that

αf1 (Bf1 ) = αf2 (Bf2 ) which contradicts condition 6.

(c) Observation (b) implies in particular that vertices that are of type vi with i ≥ 1 have at most one adjacent edge with non-trivial edge group. (d) There is no element s ∈ S and vertex u ∈ VB such that two distinct prop- p agates of s lie in Bu. Otherwise an element s ∈ S ∩ Bu must propagate from a fat vertex u via an edge path f1, f2 to a small vertex w and back −1 to u. As A is a tree we must have [f1] = [f2] contradicting (b). (e) For any given u ∈ VB we can assume, after replacing elements of S with p propagates, that S ∩ Bu = S ∩ Bu. This follows immediately from (d). (f) We can always ﬁnd a minimal generating set M of G such that no element acts with a ﬁxed point on T , if we then construct the M-wedge relative a small base vertex we clearly obtain a tame triple (B0, ∅, ∅).

(g) The existence of a tame triple (B, S, S¯) with B =∼ A implies the assertion of the theorem. Indeed S¯ must be of cardinality rank G, contain a generating

set of Av0 and contain sets that generate the groups Avi together with the element corresponding to the boundary curve. This clearly proves the assertion of Theorem 0.2.

We will now show that if a tame triple (B, S, S¯) representing G is given then one of the following holds unless B is folded:

(A) G contains a primitive element g that is conjugate to an element of a group 2 ∼ Avi of type π1(S (∞, (1))) = Z ⊕ Z2.

44 (B) There exists a tame triple (B0,S0, S¯0) such that either B0 has fewer edges than B or that B0 has fewer edges with trivial edge groups than B. To see that this implies the theorem note the following: In situation (A) the assertion of the theorem follows from Claim A below. If B is folded then the theorem follows from observation (g). In situation (B) we replace (B, S, S¯) by (B0,S0, S¯0) and repeat the argument. As this must clearly end after finitely many steps we eventually end up in one the first two cases. As we assume that B is not folded it admits a fold which must be based at a small vertex by our tameness assumption. We construct (B0,S0, S¯0) differently depending on which type of fold is applicable. As A is a tree of groups we only need to consider folds of type IA, IIA and IIIA. We apply a fold of type IIIA first whenever possible, otherwise a fold of type IIA. We apply a fold of type IA only if no fold of type IIIA or IIA is possible. We assume that the vertices and edges of B involved in the fold are called as in the definition of the folding operations, i.e. x, y and z and f1 and f2 or f. Fold of type IIIA: Note that by remark (b) above at least one of the two edges f1 and f2 involved in the fold must have a trivial edge group, say Bf1 is trivial. Denote the resulting A-graph by B1. Thus the fold has the effect of removing one edge f1 with trivial edge group and adding one generator g to 1 By, i.e. we have By = hBy, gi ⊂ A[y]. Note that g is primitive in G. Clearly S1 = S ∪ {g} is a p-generating set of the graph of groups B1 and by observation p (e) we can assume that By ∩ S = By ∩ S ⊂ S. We now define B2 to be the A-graph obtained by replacing the vertex group 1 2 By with a maximal subgroup By that is generated by #(By ∩S)+1 elements and 1 2 contains By . Clearly B also represents G. We have to adapt the p-generating set to the new A-graph. By Lemma 3.3 and Lemma 3.4 there are four different situations that can occur: 2 (i) There exists a minimal generating set M of By such that for every conjugacy 2 class of peripheral elements (i.e. images of the cyclic edge groups) in By there is one generator. This implies that after modifying B2 by auxiliary moves we have 2 S = (S − (Bv ∩ S)) ∪ M is a new p-generating set. If S 6= S¯, i.e. S = S¯ ∪ {s∗} 2 2 we further put S¯ = S − {s∗}. 2 (ii) By = A[y] = A[v0] is the fundamental group of a surface H. In this case 2 we need to add one additional element s∗ ∈ By ⊂ Av0 to the p-generating set as we cannot choose all element corresponding to boundary components of H 2 simultaneously primitive. Thus we have S = (S − (By ∩ S)) ∪ M ∪ {s∗}. We 2 further define S¯ = (S − (By ∩ S)) ∪ M. Note that we only need to deal with this situation if S¯ = S. Otherwise S¯ ¯ contains generating sets of two conjugates of Avo in S ∪ {g}. (Remember that rank Av0 ≥ 2 as we assume that G is sufficiently large.) Thus we can replace the generators of both conjugacy factors with the generators of one conjugate and with the conjugacy factor contradicting the minimality of the generating set. 2 2 (iii) Bv is the free subgroup of index 2 of A[y] = Av0 = π1(S (2,..., 2, ∞)). This is not possible as we assume that G is sufficiently large. 2 (iv) A[y] is of type π1(S (∞, (1))) = Z ⊕ Z2. As g is clearly primitive this puts us into situation (A).

45 If we are in situation (i) or (ii) we further have to modify the A-graph in order to obtain another tame triple. We first apply folds of type IA based at y as long as possible. These folds are harmless as they only identify edges and vertices with maximal cyclic or trivial groups. The sets S and S¯ are being preserved. Then we apply all folds of type IIIA based at y. These folds add one element to the p-generating set S2 and to S¯2, at a small vertex y0 adjacent to y. It follows that the edge groups of both edges and the vertex group corresponding to y0 must be trivial as otherwise we get again a contradiction to the minimality assumption. Denote the resulting A-graph by B3, the new p-generating set S3 3 3 3 3 and put S¯ = S or S¯ = S − {s∗} if s∗ exists. We conclude with performing all moves of type IIA based at y. As this only propagates elements already in the p-generating set this does not change the p-generating set. Denote the resulting A-graph by B0. It follows immediately from the discussion that the triple (B0,S3, S¯3) satisfies conditions (1)-(5). In situation (ii) condition (6) is clearly satisfied as the case 2 Avo = π1(P (∞)) does not occur for sufficiently large groups. In situation (i) 2 it is satisfied except in the case A[y] = π1(S (2, 2, ∞)) = Z2 ∗ Z2 which does not occur in the sufficiently large situation. Fold of type IIA: Note by part (3) of the definition of the tame triple we can p assume that the fold pushes an element of g ∈ S ∩ Bx through the edge. In particular S is also a p-generating set of the resulting A-graph B1. As we assume that no fold of type IIIA is possible this means that no prop- 1 p 1 agate of g is already in By. Thus we can assume that S ∩ By = S ∩ By is a 1 1 generating set of By . We replace the vertex group By = hBy, gi by a maximal 1 2 subgroup of rank #(S ∩ Bv) + 1 containing By to obtain a new A-graph B . We then argue as in the case of a fold of type IIIA that we can further modify B2 by folds based at y such that (A) or (B) holds. Fold of type IA: Thus one can perform a fold of type IA based at a small vertex x that identifies vertices y and z and yields the new vertex [y]. Let x = x1, . . . , xn be all small vertices such that a fold of type IA can be applied to B that identifies y and z. As we assume that no fold of type IIA is possible it follows that each of these folds identifies edges that either have both trivial edge groups or both non-trivial edge groups. Suppose that in k instances the edge groups are non-trivial. Note that this implies that there are precisely k element of S that have propagates in both By and Bz. Thus after replacing S with propagates we can assume that S ∩ By is a generating set of By and that (S ∩ Bz) ∪ M is a generating set of Bz where M are the k propagates of S ∩ By that lie in Bz. We perform the fold of type IA based at x = x1 and then the folds that are based at x2, . . . , xn which are now folds of type IIIA as the first fold has already identified y and z. We obtain a new p-generating set which is S1 = 0 0 S ∪ {g2, . . . , gn} where S is the image of S under the folds and the gi are the elements added to By by the move of type IIIA based at xi. We call the new 1 1 1 A-graph B . We next replace By by a maximal subgroup containing By that is 1 1 generated by #(S ∩ By ) + n − 1 elements and then argue as before. To conclude the proof it suffices to establish the following.

46 Claim A Let G and O be such that there exists a primitive element g ∈ G 2 such that g ∈ Avi for some i ≥ 1 where Avi is of type π1(S (∞, (1))). Then Theorem 0.2 holds for G. Proof We may clearly assume that i = 1, i.e. that

O = F (p1, . . . , pk, (1), (h21), . . . , h2n2 ), (hm1), . . . , h2nm )).

2 Recall that Avi = ha, b |, b , [a, b]i where hai corresponds to the boundary curve of S2(∞, (1)). There are three cases to consider. If g = b then G/hhgii is generated by rank G − 1 elements and of type ∼ G/hhgii = π1(F (p1, . . . , pk, ∞, (h21, . . . , h2n2 ), (hm1, . . . , h2nm ))) a group whose orbifold has only m − 1 boundary components with reflections. The assertion now follows by induction as G/hhgii is again sufficiently large. l 2 l 2l ∼ If g = a b note that ha, b | b , [a, b]i/hha bii = ha | a i = Z2l. It follows that ∼ 2 G/hhgii = π1(S (p1, . . . , pk, 2l, (h21, . . . , h2n2 ), (hm1, . . . , h2nm ))) and as before G/hhgii is generated by rank G − 1 elements. The assertion now follows by induction as G/hhgii is again sufficiently large. In the case g = al we can clearly assume that g = a. If m ≥ 2 or m = 1 and k ≥ 3 then the group

∼ 2 G/hhgii = π1(S (p1, . . . , pk, (h21, . . . , h2n2 ), (hm1, . . . , h2nm ))) ∗ Z2 is generated by rank G − 1 elements and the assertion follows by induction and Grushko’s theorem as G/hhgii is again suﬃciently large. Suppose now that m = 1 and k = 2. We have to show that G has no generating set containing g that is of cardinality 2. As we assume that G is suﬃciently large it follows that p1 6= 2 and p2 6= 2. The proof in this particular 2 case is identical to the proof that the Fuchsian group π1(S (2, 2, p1, p2)) is not 2-generated. 2

3.3 Comments on small Fuchsian groups In this section we show that in Theorem 0.2 the assumption that G is suﬃciently large is necessary to obtain the stated conclusion. We describe three families of groups for which the conclusion does not hold. Theorem 0.2 would in all of these cases give a number that is higher by one. All of these pathologies already occur in the list of 2-generated Fuchsian groups provided by E. Klimenko and M. Sakuma [KlS]. In the following F , k and m are as in Theorem 0.2.

Lemma 3.5 Suppose that O = S2(2,..., 2, 2l + 1, (1),..., (1)) with m ≥ 1. Then G = π1(O) has a generating set of cardinality rank F + m − 1.

Proof Note first that rank F = m+k −1. Put O0 = S2(2,..., 2, ∞, (1),..., (1)), i.e. O0 is obtained from O by replacing 2l + 1 by ∞. 0 0 We first argue that it suffices to show that the group G = π1(O ) has a subgroup U of index 2 that is generated by rank F + m − 1 = 2m + k − 2

47 elements such that U ∩ hti = ht2i where t is the element corresponding to the boundary component of O0. Indeed, once the existence of such a subgroup is established this subgroup 2 clearly projects onto a generating set of G = π1(O) as in G a power of t equals t. We now establish the existence of such a subgroup U. Note first that there 0 are homomorphisms G → Z2 that map non-trivial torsion elements to non- trivial torsion elements. The kernel U of each such homomorhisms is torsion free and therefore free. The Euler characteristic of the covering orbifold O¯ ¯ 0 1 corresponding U is χ(O) = 2χ(O ) = 2(2 − m − ((k − 1) 2 − 1)) = 3 − 2m − k. Thus rank U = k + 2(m − 1) + 1 = 2m + k − 2. It now suffices to show that for some such U the orbifold O¯ has one boundary component as the covering must then be 2-sheeted on the boundary component which implies that U ∩ hti = ht2i. This is equivalent to showing that t gets 0 0 ∼ mapped to the generator of Z2 under the quotient map G → G /U = Z2. 0 Note first that G is generated by elements s1, . . . , sk−1 corresponding to the rotations of order 2, reflections r1, . . . , rm and elements γ1, . . . , γm corresponding to loops parallel to the reflection lines. Any of the above homomorphisms 0 G → Z2 must map the si and ri to 1 and the γi can be mapped to either 0 or 1. As t = s1 · ... · sk−1γ1 · ... · γm and m ≥ 0 we can clearly choose such a homomorphism such that t gets mapped to 1. Note that this is not possible if m = 0 and k is odd. It is this very observation that is responsible for the fact that in the rank formula for Fuchsian groups without reflections the pathology only occurs if the orbifold has an even number of cone points. 2

Lemma 3.6 Suppose that O = S2(2,..., 2, (1),..., (1), (2, 2l + 1)). Then G = π1(O) has a generating set of cardinality rank F + m.

Proof Clearly rank F = m + k − 1 which implies we have to show that G has a generating set of cardinality 2m + k − 1. Put O0 = S2(2,..., 2, ∞, (1),..., (1)), 0 0 0 i.e. O is obtained from O by replacing (2, 2l + 1) by ∞. Put G = π1(O ) and let t be the element of G0 corresponding to the boundary curve of O0. We distinguish two cases. 1. case. m ≥ 2 or m = 1 and k is odd. As in the proof of Lemma 3.5 we see that there exists a free subgroup U ≤ G0 of index 2 and rank 2m + k − 3 such that U ∩ hti = ht2i. Clearly

0 ∞ G = G ∗hti ∆ (2; 2, 2l + 1).

It suffices to show that there exist elements g, h ∈ G such that G = hU, g, hi. Choose g, h ∈ ∆∞(2; 2, 2l+1) such that ∆∞(2; 2, 2l+1) = ht2, g, hi; the existence of such elements is easily verified. As t2 ∈ U this implies that t ∈ ∆∞(2; 2, 2l + 1) ⊂ hU, g, hi. Thus also G0 = hU, ti ⊂ hU, g, hi which shows that G = hU, g, hi. 2. case. m = 1 and k is even. The case k = 0 doesn’t exist as this is a bad orbifold, see [Sc]. Thus we can assume that k ≥ 2. Again we find a free subgroup U ≤ G0 of index 2 and rank 2m + k − 3 = k − 1. In this case however the orbifold covering space corresponding to the subgroup has two boundary component that are both 1-sheeted covers of the boundary component of O0. Thus there exist elements g, h ∈ G0 with g 6= h such that gtg−1, hth−1 ∈ U and gh−1 ∈/ U. After conjugation we can assume that h = 1, in particular g∈ / U.

48 ∞ −1 We next show that ∆ (2; 2, 2l + 1) = ht, s2ts2 , s1i where t is the element corresponding to the boundary component. Recall that

∞ 2 2 2 −1 2l+1 ∆ (2; 2, 2l + 1) = hs1, s2, t|s1, s2, (s1s2) , (s1t s2t) i. We clearly have

−1 2 −1 −1 −1 (s1t s2t) = s1 · t · s2ts2 · s1 · s2t s2 · t ∈ hs1, t, s2ts2 i

−1 −1 2(l+1) −1 which implies that s1t s2t = (s1t s2t) ∈ hs1, t, s2ts2 i. This clearly −1 implies that s2 ∈ hs1, t, s2ts2 i and the assertion is proven. Now recall that 0 ∞ G = G ∗hti ∆ (2; 2, 2l + 1).

We will show that G = hU, gs2, s1i which clearly proves the lemma. Note ﬁrst ∞ −1 −1 −1 −1 that ∆ (2; 2, 2l+1) ⊂ hU, gs2, s1i, as s1, t = hth , s2ts2 = s2g ·gtg ·gs2 ∈ 0 hU, gs2, s1i. Thus also g ∈ hU, gs2, s1i. As U is of index 2 in G and g∈ / U it 0 follows that G = hU, gi ⊂ hU, gs2, s1i which proves the lemma. 2 The last class is in fact a family of planar Coxeter groups.

2 Lemma 3.7 Let G = π1(S ((31,..., 3n−1, p))) with p∈ / 3Z. Then G is generated by n − 1 elements.

Proof Recall that G is given by the presentation

2 2 3 3 p G = hs1, . . . , sn | s1, . . . , sn, (s1s2) ,..., (sn−1sn) , (sns1) i.

We will show that G is generated by the elements g1 = s1 and gi = sis1si+1 for 2 ≤ i ≤ n − 1. Put U = hg1, . . . , gn−1i. We have to show that U = G. As s1s2 is of order 3 it follows that s2s1s2s1s2 = s1 ∈ U. As

−2 2 gi sis1sis1sigi = (si+1s1si)(si+1s1si)sis1sis1si(sis1si+1)(sis1si+1) =

si+1s1sisi+1sisi+1sis1si+1 = si+1s1si+1s1si+1 it follows by induction that sis1sis1si ∈ U for 2 ≤ i ≤ n. It follows in particular 3 that sns1sns1sn ∈ U and therefore (s1sn) ∈ U. As the order p of s1sn is no 3 multiple of 3 it follows that a power of (s1sn) equals s1sn which implies that s1sn ∈ U. Thus sn ∈ U. As si = gisi+1s1 another inductive argument shows that si ∈ U for 2 ≤ i ≤ n − 1 which proves the lemma. 2 If follows from [L], [LM], that these groups are in fact of rank n − 1 and that they are of rank n if p ∈ 3Z.

References

[Ba] H. Bass Some remarks on group actions on trees, Comm. in Algebra 4, 1976, 1091-1126.

[BF1] M. Bestvina and M. Feighn Bounding the complexity of simplicial group actions on trees, Invent. Math. 103, 1991, 449-469.

49 [D2] M.J. Dunwoody Folding sequences Geometry & Topology Monographs. Volume 1: The Epstein Birthday Schrift, 139-158.

[D4] M.J. Dunwoody Groups acting on protrees J. London Math. Soc. 56, 1997, 125- 136.

[G] I. Grushko On the bases of a free product of groups Mat. Sbornik 8, 1940, 169- 182.

[KMW] I. Kapovich, A. Myasnikov and R. Weidmann. A-graphs, foldings and the induced splittings, IJAC 15, 2005, 95–128.

[KZ] R. Kaufmann and H. Zieschang. On the rank of NEC groups In Discrete Groups and Geometry, LMS Lecture Note Series 173, 137–147. Cambridge University Press, 1992.

2 [KlS] E. Klimenko and M. Sakuma, Two-generator discrete subgroups of IsomH containing orientation-reversing elements, Geom. Dedicata 72, 1998, 247–282.

[L] M. Lustig, On the rank, the deﬁciency and the homological dimension of groups: the computation of a lower bound via Fox ideals. In Topology and Combinatorial Group Theory, Lecture Notes in Mathematics 1440. Springer-Verlag, 1990.

[LM] M. Lustig and Y. Moriah, Generalized Montesinos knots, tunnels and N -torsion. Math. Ann. 295, 1993, 167–189.

[PR] N. Peczynski and W. Reiwer, On cancellations in HNN-groups Math. Z., 158, 1978, 79–86

[PRZ] N. Peczynski, G. Rosenberger and H. Zieschang Uber¨ Erzeugende ebener diskon- tinuierlicher Gruppen Invent. Math. 29, 1975, 161–180.

[Sc] G.P. Scott The geometries of 3-manifolds Bull. Lond. Math. Soc. 15, 1983, 401– 487.

[Ser] J.P.Serre Trees Springer 1980.

[St] J.R. Stallings Topology of Finite Graphs Invent. Math. 71, 1983, 551-565.

[We0] R. Weidmann Uber¨ den Rang von amalgamierten Produkten und NEC-Gruppen thesis, Bochum 1997.

[We1] R. Weimdann On the rank of amalgamated products and product knot groups Math. Ann. 312, 1998, 761-771.

[We2] R. Weidmann The Nielsen method for groups acting on trees Proc. London Math. Soc. (3) 85, 2002, 93–118.

[We5] R. Weidmann The rank of planar dicontinous groups Abh. Math. Sem. Univ. Hamburg 72, 2002, 1–8.

[Z1] H. Zieschang, Uber¨ die Nielsensche K¨urzungsmethode in freien Produkten mit Amalgam, Invent. Math., 10, 1970, 4–37

[Z2] H. Zieschang, On Decompositions of Discontinuous Groups of the Plane, Math. Z., 151, 1976, 165–188.

[ZVC] H. Zieschang, E. Vogt, and H. Coldewey. Surfaces and Planar Discontinuous Groups, Lecture Notes in Mathematics 835, Springer-Verlag, 1980.