A SPLITTING THEOREM FOR GROUPS ACTING ON QUASI-TREES.

A SPLITTING THEOREM FOR GROUPS ACTING ON QUASI-TREES. Michael Batty Department of Mathematics, National University of Ireland, Galway. e-mail: [email protected] May 4, 1999 AMS Subject Classi cation: 20E08

Abstract

It is well known that a group is free if and only if it acts freely without inversions on a tree. We prove a generalisation of this fact by de ning a quasi-tree to be a graph with a bound on the size of its simple loops. It is shown that a nitely generated group acting freely on such a graph is isomorphic to a free product of free groups and nite groups.

1 Introduction In this paper a graph is de ned to be the data (V; E; L) where V and L are sets, called respectively the set of vertices and the set of labels and E is a subset of V  V  L known as the set of edges. The projections i and t from the set of edges to the rst and second co-ordinates are called the initial and terminal vertex functions and the projection  onto the third is the label function. We allow edges e with i(e) = t(e) and call them single edge loops. Graphs have been de ned in this way in order to include Cayley graphs 1

of groups. Let X = (V; E; L) be a graph. A pair of bijections

 = (V : V ! V; L : L ! L) is called a (graph) automorphism if for all v1 and v2 in V and if for all l in L, (v1 ; v2 ; l) 2 E if and only if (V (v1 ); V (v2 ); L (l)) 2 E . The set of all automorphisms of X is a group under composition, which we denote by Aut(X ). An action of a group on X is a homomorphism G ! Aut(X ). We call an action of a group G on X free if G acts freely on V in the usual sense and for all (v1 ; v2 ; l) 2 E and for all g 2 G we have g((v1 ; v2 ; l)) 6= (v2 ; v1 ; l). The latter condition is known as action without inversions. It is shown in [5], page 20, that if X is a tree and G is a nontrivial group acting freely on X then X is in nite, since a group acting on a nite tree always has a xed point. In fact the following is true. (See [5] p.27. theorem 4.) Theorem 1.1 A group G acts freely on a tree T if and only if G is free. We are going to generalise the notion of a tree and prove a corresponding version of the above result. We shall consider the set of vertices of a graph as a discrete metric space by giving every edge length 1 and de ning the distance d between two points as the length of a shortest path between them. The following trivial lemma shows us that there is no loss of generality in assuming that an action on a graph is by isometries. Lemma 1.2 Let X = (V; E; L) be a graph. If (V ; L) is an automorphism of X then V is an isometry. We recall the Bass-Serre theorem for groups acting on trees. This is covered in the rst ve chapters of [5]. Theorem 1.3 Let G be a group which acts on a tree T without inversions. Then G is isomorphic to the fundamental group of the graph of groups T=G. Note that theorem 1.1 can easily be recovered from theorem 1.3, since if G is a group which acts freely on a tree, the Bass-Serre theorem tells us that G is the fundamental group of a graph of trivial groups, and is therefore free.

2 Cut Points and Blocks. We recall some notions of connectivity of graphs from [4]. 2

De nition 2.1 Let X be a graph with nitely many connected components. A cut point of a graph X is a vertex of X whose removal, along with the edges

adjacent to it, increases the number of connected components of X . In particular, if X is a connected graph then removal of a cut point along with the edges adjacent to it will disconnect X . If a graph has no cut points we say that it is cut connected. De nition 2.2 A graph is said to be nonseparable if it is connected, nontrivial and cut connected. A block of a graph X is a maximal nonseparable subgraph of X . For example, every vertex of a tree T with no leaves is a cut point. The blocks of T are adjacent vertices in T along with the edge joining them. Proposition 2.3 The intersection of two distinct blocks of a graph is either empty or a single vertex. For a proof, see chapter 5 of [4]. We now prove a lemma which is used in the next section (see gure 1). When we refer to a path in a graph we mean an unrestricted sequence of adjacent edges. If the sequence of terminal vertices of edges consists of distinct vertices then we call the path simple. A loop is a path such that the terminal vertex of the nal edge is the initial vertex of the rst. Thus a simple loop is a loop which is simple as a path. Let Y be a subgraph of a graph X . Then a path p in Y is called a Y -path (respectively a Y -loop) if p is a simple path (respectively a simple loop) such that V (p) \ V (Y ) = fi(p); t(p)g. Lemma 2.4 Let X be a cut connected graph and let Y and Z be connected subgraphs of X , each having at least two vertices, such that Y \ Z is a single vertex v. Then there exists a (Y [ Z )-path p in X from Y to Z such that i(p) 6= v and t(p) 6= v. Proof. Suppose that there is no path in X from Y to Z which doesn't meet v. Then if we remove v and edges adjacent to it then there is no path in X from Y to Z . However since Y and Z are connected and meet at a point, Y and Z are in the same connected component of X . Removal of v hence increases the number of connected components of X and v is thus a cut point of X , which contradicts our assumption of cut connectedness of X . So a path p in X exists from Y to Z which avoids v. Any minimal such path is a (Y [ Z )-path. 2 Note that the path p in lemma 2.4 has length at least 1.

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Figure 1: Cut Point Lemma. v X Y

Z p

3 Quasi-Trees. In this section we take the word \path" to mean a path which is not a loop. A graph X is said to be of bounded valency if there exists an integer N such that the valency of every vertex of X is at most N . De nition 3.1 Let Q be a connected graph of bounded valency. We call Q a K -quasi-tree if every simple loop in Q has length at most K . If there exists a nonnegative integer K for which Q is a K -quasi-tree then we call Q a quasitree. So a graph Q is a 0-quasi-tree if and only if it is a tree. Note that a quasi-tree is more than just a graph which is quasi-isometric to a tree. For example, the Cayley graph of Z  Zn with the standard generating set is quasi-isometric to a tree but contains arbitrarily long simple loops. As is the case with trees, we have the following property, which is trivial to prove. Proposition 3.2 A connected subgraph of a K -quasi-tree is itself a K -quasitree. Every graph admits a unique block decomposition, i.e. expression as a union of blocks. The following lemma shows that for a quasi-tree the situation is somewhat special. Lemma 3.3 A connected graph X of bounded valency is a quasi-tree if and only if there exists an integer M such that for every block B of X , B is nite with jB j < M . Note that lemma 3.3 shows that, with the usual graph metric, the vertex set of a quasi-tree is a hyperbolic metric space, since it follows easily that the vertex 4

set of a graph as in the conclusion of the lemma satis es a linear isoperimetric inequality. Proof. Let X be a connected graph such that for all v 2 V (X ), val(v) 6 N . It is clear that if there exists an integer M such that each block B of X is nite with jB j 6 M then X is a quasi-tree, since M is a bound for the length of simple loops in X . For the converse it is sucient to show that if a connected graph Q is a cut connected K -quasi-tree then Q is nite and there is a bound, depending only on N and K , on the number of vertices of Q. Let Y be a subgraph of Q. For the purposes of this proof we write Pr (Y ) for the set of Y -paths in Q of length greater than or equal to r and de ne an r-thickening Lr (Y ) of Y to be a subgraph of Q of the form Y [ PrM (Y ), where PrM (Y ) is a subset of Pr (Y ) which is maximal subject to the condition that if p1 and p2 are paths in PrM (Y ) then V (p1 ) \ V (p2 )  V (Y ). Since the conclusion of the lemma is trivial for trees and 1-quasi-trees we assume that K > 2 and that Q contains a simple loop L of length K (i.e. Q is not a K 0 -quasi-tree for any integer K 0 < K ). Now de ne subgraphs Li of Q inductively by L0 = L and for all i with 1 6 i 6 K , Li = L K2 +1?i (Li?1 ). (Note that we have to make some choices when constructing the subgraphs Li but this is irrelevant for the purposes of the next claim).

Claim 3.4 Q = L K2 .

Given that the above claim is true, it is easy to see that Q is nite. Suppose that N is a bound on the valency of vertices of Q. Then each Li is nite with jV (Li )j bounded above by a constant only depending on N and i. Thus Q = L K2 is nite with jV (Q)j bounded by a constant only depending on N and K . The rest of the proof is devoted to proving claim 3.4, which is equivalent to proving claim 3.5 below. The reason that the claims are equivalent is that we only need to consider paths, and not loops, in the formation of thickenings, provided that we are in a cut connected graph. For suppose that Y is a subgraph of a cut connected graph and that we add a Y -loop  to Y at the point y. Then since X is cut connected we may apply lemma 2.4 to obtain a (Y [ ^ )-path p from Y to ^ . (Recall that if q is a loop or path in a graph X then we write q^ for the subgraph induced by q.) Thus we can decompose ^ [ p^ into two paths p1 and p2 as in gure 2. Hence in X thickenings can be constructed by only adding paths. 5

Figure 2: No Loops in Thickenings. X Λ p1

Y p

p2

Figure 3: Base Case. t(p)

p

L

i(p)

In what follows we use the notation l(p) to denote the length (number of edges) of a path p.

Claim 3.5 For all i > 0 with i 6 l(p) > K + 1 ? i.

K there is no Li -path 2

p in Q such that

2

To prove the claim we use induction on i. First we show that there is no L0-path p in Q with l(p) > K2 +1. Suppose for contradiction that there is, as in gure 3. We introduce some notation at this point. If p is an oriented loop and v1 and v2 are two vertices on p^ then we write p(v1 ; v2 ) for the path obtained by travelling round p from v1 to v2 in a positive direction. We also use this notation for paths (here there is no ambiguity about which direction to travel). Now, with p as before, assume without loss of generality that l(L(t(p); i(p))) > K=2. Then de ne the loop  = p  L(t(p); i(p)). Since i(p) 6= t(p) by assumption and p is an L0 -path,  is a simple loop in p with l() > K + 1, which contradicts the fact that Q is a K -quasi-tree. For the inductive step, we assume that there are no Li -paths p in Q with l(p) > K +1?i and show that there are no Li+1 -paths p in Q with l(p) > K ?i. Suppose 2 2 for contradiction that such an Li+1 -path p does exist in Q. By maximality of 6

Figure 4: Inductive Step. Case 1

Case 2 i

L t(r1)

Li i(r1)

t(r1) i(p) i(r1)

i(p)

r1

t(p)

t(p) p Case 3 i(r1) a)

Li

i

b)

L

t(r3 ) r3

t(r3 )

i(r1) p

r1 p

P KM2 +1?i (Li ) and the fact that Li  Li+1 , either t(p) 2= Li or i(p) 2= Li . Assume without loss of generality that t(p) 2= Li . Then t(p) lies on an Li -path r1 such that t(p) 6= i(r1 ) and t(p) 6= t(r1 ). Thus r1 (t(p); i(r1 )) has length at least 1 and is a (Li [ p^)-path since p is an Li+1 -path. If i(p) 2 Li , de ne r2 to be the empty path, as in gure 4, case 1. Otherwise i(p) lies on an Li -path r3 such that i(p) 6= i(r3 ) and i(p) 6= t(r3 ). Now by construction of a loop neighbourhood, either r3 = r1 or r3 \ r1  Li . Suppose that r3 = r1 as in gure 4, case 2. Then de ne r2 = r1 (i(p); t(r1 )). Otherwise assume without loss of generality that i(r1 ) 6= t(r3 ) and de ne r2 = r3 (i(p); t(r3 )) (see gure 4, case 3). Now de ne  = r2  p  r1 (t(p); i(r1 )): Then  is an Li -path with l() > 0+( K2 ? i)+1, which contradicts the inductive hypothesis. 2

4 The Block-Cutpoint Tree. We would now like to de ne a tree which expresses how the blocks of a connected graph X intersect. As a rst attempt at a de nition we might de ne the vertex set of this \tree" to be the set of blocks of X and the edge set to be the pairs of blocks with nontrivial intersection. However this gives rise to a 7

Figure 5: Block-Cutpoint Trees. v3 v2

W2

W1

v1

W3 v4

Wn

B(l)

W4

vn l v8

v5 W5

W7 v7

W6

v6

complete subgraph on n vertices for every instance of n blocks intersecting at a common point. To overcome this problem, we must introduce extra vertices corresponding to the cut points. Given a graph X we de ne a graph T (X ) as follows. Let V be the set of cut points of X and let W be the set of blocks of X . Then T (X ) has as its set of vertices V [ W and as its edges the pairs (v; w) whenever v belongs to the block w. T (X ) is vertex 2-coloured depending upon whether a given vertex of T (X ) is in V or W .

Example 4.1 vertex.

1. Let X be a cut connected graph. Then T (X ) is a single

2. Let X be the Cayley graph of (Z; +) with respect to the standard generating set. Then every point of X is a cut point and every block of X consists of two vertices joined by a single edge. Thus T (X ) is a homogeneous bipartite tree with two edges meeting at each vertex.

Lemma 4.2 If X is a connected graph then T (X ) is a bipartite tree. Proof. The fact that T (X ) is bipartite comes from the de nition of its adjacency relation: an edge is of the form (v; w) so its initial vertex is in V and its terminal

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vertex is in W . Now suppose that T (X ) contains a loop L, which we assume without loss of generality to be simple. Then as T (X ) is bipartite and contains no single edge loops, L has even length. L can't have length 2 by the nature of the adjacency relation so l(L) > 4. Denote the vertices of L by

v1 ; W1 ; v2 ; W2 ; : : : ; vn ; Wn ; v1 as in gure 5 and let B (L) denote the subgraph W1 [


[ Wn of X . Now suppose that there is a cut point of B (L). By de nition of a block, no vertex of any block Wi is a cut point of Wi and since L is a simple loop and two blocks may only intersect at a single vertex, the only possible cut points of B (L) are v1 ; : : : ; vn . However if we remove any of these vertices the loop L becomes a path joining the remaining blocks so the remainder of B (L) after removing v is connected. So B (L) is connected and nontrivial and has no cut points. Hence it is contained in some block, which contradicts maximality of the blocks Wi . We have shown that T (X ) has no loops and is hence a forest. Since X is connected, T (X ) is a tree. 2 Following [4] we shall call T (X ) the block-cutpoint tree of X . Note that if Q is a quasi-tree, T (Q) is of bounded valency, since Q is, and since each block

is nite and bounded in cardinality by lemma 3.3. See gures 6 and 7 for an example of a quasi-tree and its block-cutpoint tree.

5 A Splitting Theorem. If a group G has a Cayley graph which is quasi-isometric to a tree then G is virtually free. This follows from Stallings' ends theorem and can be found in [3]. In this section we describe a result allowing us to characterise a subclass of virtually free groups by the graphs on which they act freely. De nition 5.1 We say that a nitely generated group is quasi-free if it is isomorphic to a free product of free groups and nite groups. So if G is a quasi-free group then there exist integers n and m such that
? G = (ni=1 Z)  m j =1 Hj ;

where each Hj is nite. For example, the Cayley graph of the quasi-free group Z3  Z3 is shown in gure 8, with its block-cutpoint tree shown in broken lines. 9

Figure 6: A Quasi-Tree.

Figure 7: The Block-Cutpoint Tree of the Quasi-Tree in Figure 6.

Observe the dynamics of the action of Z3  Z3 on its Cayley graph. Left multiplication by a \rotates" the graph about the block containing 1,a and a2 (so does left multiplication by 1 or a2 ) and left multiplication by b \rotates" the graph about the block containing 1,b and b2 . Note that there is a well de ned induced action on the block-cutpoint tree with nite vertex stabilizers. Quasi-free groups are virtually free since they are isomorphic to the fundamen10

Figure 8: The Cayley graph of a quasi-free group.

b

a

b2

a e

b

a

b

a2 a

b

Figure 9: Graphs of Groups and Quasi-Free Groups. H1

1 1

H2 1

1 1

H3 1

Hm

1 1

1

1

n single edge loops

tal group of a graph of nite groups [2, IV.1.9]. The converse is however false. For example Z  Zn is virtually free but freely indecomposable. By the Kurosh Subgroup Theorem [2, I.7.8] quasi-freedom is a property which is closed under taking nitely generated subgroups. We now give a characterisation of quasi-free groups in terms of graphs of groups. Proposition 5.2 Let G be a nitely generated group. Then G is quasi-free if and only if G is isomorphic to the fundamental group of a graph of groups ? = (X = (V; E; L); G ; V ; E ; fe j e 2 E g; fe j e 2 E g) such that X is nite and bipartite with respect to some 2-colouring c : V ! f0; 1g, where for all v 2 c(0), V (v) is the trivial group and for all v 2 c(1), V (v) is nite. Proof. Suppose that G is quasi-free, say G  = Fn  (m j =1 Hj ) where Fn is the free group of rank n and each Hi is nite. Then let ? be the graph of groups as

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Figure 10: Quasi-Trees Are Not Group-Invariant. a2

a2

a

a2

a2

a2

a2

a

a2

in gure 9 where all edge groups and black vertex groups are trivial and white vertex groups are labelled by the groups Hi for each i. We have G  = 1 (?). For the converse, suppose that G  = 1 (?), where ? is a graph of groups as in the statement of the proposition. Let e 2 E (X ). Then since X is bipartite, either c(i(e)) = 0 or c(t(e)) = 0. Thus either V (i(e)) or V (t(e)) is trivial. Since e : E (e) ! V (i(e)) and e : E (e) ! V (t(e)) are both monomorphisms, E (e) embeds into the trivial group and is trivial. Thus we have shown that X has trivial edge groups. Let T be a maximal tree in X and let n be the cardinality of E (X ) ? E (T ). Then 1 (?)  = (c(v)=1 V (v))  Fn , which is a quasi-free group. 2 If a group G is quasi-free then its Cayley graph with respect to the standard generating set is a quasi-tree, so G acts freely on a quasi-tree. The main result of this paper is the converse, giving us the following.

Theorem 5.3 A nitely generated group G acts freely on a quasi-tree Q if and only if G is quasi-free.

Note that the property of the Cayley graph being a quasi-tree is not invariant under change of generators.

Example 5.4 The Cayley graph of Z with respect to the standard generating set is a tree, so it is certainly a quasi-tree. However there are arbitrarily large simple loops in the Cayley graph of Z with the generators a,a?1 ,a2 and a?2 (see gure 10 ). Before the proof of theorem 5.3 we give two lemmas.

Lemma 5.5 If a group G acts freely on a set X and Y is a nite subset of X then Stab(Y ) = fg 2 GjgY = Y g is nite. 12

Lemma 5.6 If a group G acts on a bipartite tree T such that the action pre-

serves the colouring, the stabilizers of vertices of one colour are nite and the stabilizers vertices of the other colour are trivial then G is quasi-free.

Proof(of Lemma 5.6) Suppose that T is bipartite with respect to the 2-colouring c, and that vertices v of T with c(v) = 0 have trivial stabilizer and vertices v with c(v) = 1 have nite stabilizer. Now T=G is bipartite with respect to the quotient colouring c0 and the action of G on T is without inversions since it preserves the 2-colouring with respect to which T is bipartite. So by the BassSerre theorem G is isomorphic to the fundamental group of the graph of groups on T=G. Now if v 2 V (T=G) has c0 (v) = 0 then it is labelled by a trivial group and if it has c0 (v) = 1 then it is labelled by a nite group. Hence by proposition 5.2, G is quasi-free. 2 Proof(of Theorem 5.3.) Suppose that G is a nitely generated group which acts on a quasi-tree Q. Let T (Q) be the block-cutpoint tree of Q. Since G acts by automorphisms on Q it acts by isometries on the vertices of Q and hence by homeomorphisms of the geometric realisation of Q. Now cut points and blocks of Q are topological properties of its geometric realisation, so there is an induced action of G on the vertices of T (Q) which is clearly an action by automorphisms. We know that T (Q) is bipartite and the action of G on T (Q) clearly preserves the 2-colouring. Stabilizers of cut points of Q are trivial since G acts freely on Q and stabilizers of blocks are nite by lemmas 3.3 and 5.5. Thus by lemma 5.6, G is quasi-free. 2 Note. Suppose G is a nitely generated group acting on a quasi-tree with nite edge stabilizers. Then a similar proof shows that G is isomorphic to the fundamental group of a nite graph of nite groups. Moreover the orders of the vertex groups are bounded. G is hence virtually free by [2,IV.1.9]. Question. Suppose that G is a nitely presented group acting on a graph X which is quasi-isometric to a tree. Then is it true that 1 (X=G) is quasi-isometric to G? Is it easier if we assume that X is a quasi-tree? Bridson [1] has given a counterexample if we only assume that G is nitely generated, using wreath products of nite cyclic groups.

ACKNOWLEDGEMENTS I would like to thank my Ph.D. Supervisor David Epstein for helping me to clarify much of this work. I am also grateful to Martin Bridson, Derek Holt and 13

Paul Sanders for useful conversations. This work was supported by EPSRC studentship no. 93004628.

REFERENCES [1] Bridson, M. R. private communication. [2] Dicks, W. and Dunwoody, M. J. Groups acting on Graphs, Cambridge Studies in Advanced Mathematics 17 CUP (1989). [3] Ghys, E. and de la Harpe, P. Sur les Groupes Hyperboliques d'apres Mikhael Gromov, Birkhauser Progress in Mathematics Series (1990). [4] Harary, F. Graph Theory, Addison-Wesley (1969). [5] Serre, J. P. Trees, Springer-Verlag (1980)

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