## Abstract

We discuss a partial normalisation of a finite graph of finite groups \((\Gamma (-), X)\) which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include: (1) an important inequality for the number of edges in a Stallings decomposition \(\Gamma \cong \pi _1(\Gamma (-), X)\) of a finitely generated virtually free group, (2) the proof of equivalence of a number of conditions for such a group to be ‘large’, as well as (3) the classification up to isomorphism of virtually free groups of (free) rank 2. We also discuss some number-theoretic consequences of the last result.

### Keywords

Graphs of groups Fundamental group Amalgamation### Mathematics Subject Classification

Primary 20E06 Secondary 20E08## 1 Introduction

The purpose of this paper is to introduce, and demonstrate the usefulness of, a technique for partially normalising the presentation of a finitely generated virtually free group as the fundamental group of a finite graph of finite groups. Roughly speaking, our method avoids trivial amalgamations along a maximal tree of the connected graph underlying such a representation. This result, Lemma 1, in conjunction with an almost trivial graph-theoretic result (Lemma 2), provides us with a flexible and rather powerful tool in the study of such groups. We demonstrate the usefulness of our approach by describing various applications: (1) a short and elegant argument establishing the (well-known) classification of virtually infinite-cyclic groups due originally to Stallings and Wall, (2) the classification of virtually free groups of free rank 2 together with some number-theoretic consequences,^{1} and (3) the equivalence of a number of conditions on a finitely generated virtually free group \(\Gamma \) expressing, in one way or other, the fact that \(\Gamma \) is large; cf. Propositions 4, 7, and 11. In Sect. 5, we also show that, (4) for a normalised decomposition \((\Gamma (-), X)\) of a finitely generated virtually free group \(\Gamma \), the number of geometric edges of the graph *X* is bounded above by the free rank of \(\Gamma \); cf. Lemma 3. This important observation plays a role in the proof of Proposition 7 below, as well as in establishing certain finiteness results for the class of finitely generated virtually free groups with specified information concerning the number of free subgroups of finite index. For another, recent application of the normalisation provided by Lemma 1 see [9].

## 2 Some preliminaries on finitely generated virtually free groups

Our notation and terminology here follows Serre’s book [19]; in particular, the category of graphs used is described in [19, §2]. This category deviates slightly from the usual notions in graph theory. Specifically, a *graph**X* consists of two sets: *E*(*X*), the set of (directed) *edges*, and *V*(*X*), the set of *vertices*. The set *E*(*X*) is endowed with a fixed-point-free involution \({}^-: E(X) \rightarrow E(X)\) (*reversal of orientation*), and there are two functions \(o,t: E(X)\rightarrow V(X)\) assigning to an edge \(e\in E(X)\) its *origin**o*(*e*) and *terminus**t*(*e*), such that \(t(\bar{e}) = o(e)\). The reader should note that, according to the above definition, graphs may have loops (that is, edges *e* with \(o(e)=t(e)\)) and multiple edges (that is, several edges with the same origin and the same terminus). An *orientation*\(\mathcal {O}(X)\) consists of a choice of exactly one edge in each pair \(\{e, \bar{e}\}\) (this is indeed always a *pair*—even for loops—since, by definition, the involution \(^-\) is fixed-point-free). Such a pair is called a *geometric edge*.

*T*canonically associated with \(\Gamma \) in the sense of Bass–Serre theory; cf. [1, Chap. IX, Prop. 7.3] or [18, Prop. 14]. We remark that a finitely generated virtually free group \(\Gamma \) is largest among finitely generated groups in the sense of Pride’s preorder [15] (i.e., \(\Gamma \) has a subgroup of finite index, which can be mapped onto the free group of rank 2) if, and only if, \(\chi (\Gamma )<0\); see Proposition 11 in Sect. 8.

*n*for each positive integer

*n*if, and only if, \(\tau (\Gamma _1) = \tau (\Gamma _2)\); cf. [13, Theorem 2]. We have \(\zeta _\kappa (\Gamma )\ge 0\) for \(\kappa <m_\Gamma \) and \(\zeta _{m_\Gamma }(\Gamma )\ge -1\) with equality occurring in the latter inequality if, and only if, \(\Gamma \) is the fundamental group of a tree of groups; cf. [12, Prop. 1] or [13, Lemma 2]. We observe that, as a consequence of (2.2), the Euler characteristic of \(\Gamma \) can be expressed in terms of the type \(\tau (\Gamma )\) via

*n*for every

*n*, then their Euler characteristics must coincide.

*torsion-free*\(\Gamma \)

*-action*on a set \(\Omega \) to be a \(\Gamma \)-action on \(\Omega \) which is free when restricted to finite subgroups, and let

^{2}

*X*; cf. [13, Prop. 3].

*free rank*\(\mu (\Gamma )\) of a finitely generated virtually free group \(\Gamma \) to be the rank of a free subgroup of index \(m_\Gamma \) in \(\Gamma \) (existence of such a subgroup follows, for instance, from Lemmas 8 and 10 in [19]; it need not be unique, though). We note that, in view of (2.1), the quantity \(\mu (\Gamma )\) is connected with the Euler characteristic of \(\Gamma \) via

## 3 Normalising a finite graph of groups

It will be important to be able to represent a finitely generated virtually free group \(\Gamma \) by a graph of groups avoiding trivial amalgamations along a maximal tree. This is achieved via the following.

### Lemma 1

*(*connected

*)*graph of groups with fundamental group \(\Gamma ,\) and suppose that

*X*has only finitely many vertices. Then there exists a graph of groups \((\Delta (-), Y)\) with \(\vert V(Y)\vert < \infty \) and a spanning tree

*T*in

*Y*, such that \(\pi _1(\Delta (-),Y) \cong \Gamma \), and such that

^{3}

\((F_1)\)

*X*is a finite graph,

\((F_2)\) \(\Gamma (v)\) is finite for every vertex \(v\in V(X),\)

### Proof

*S*in

*X*, and call an edge \(e\in E(S)\)

*trivial*, if at least one of the associated embeddings \(e: \Gamma (e) \rightarrow \Gamma (t(e))\) and \(\bar{e}: \Gamma (e) \rightarrow \Gamma (o(e))\) is an isomorphism. If

*S*contains a trivial edge \(e_1\)—to fix ideas, say \(\Gamma (e_1)^{e_1} = \Gamma (t(e_1))\)—then we contract the edge \(e_1\) into the vertex \(o(e_1)\) and re-define incidence and embeddings where necessary, to obtain a new graph of groups \((\Gamma '(-), X')\) with spanning tree \(S'\) in \(X'\). More precisely, this means that we let

*S*is generated by the groups \(\Gamma (v)\) for \(v\in V(X)\) plus extra generators \(\gamma _e\) for \(e\in \mathcal {O}(X)-E(S)\), where \(\mathcal {O}(X)\) is any orientation of

*X*, subject to the relations

*S*induced by \(\mathcal {O}(X)\), with a corresponding presentation for \(\pi _1(\Gamma '(-),X',S')\); see §5.1 in [19, Chap. I]. The relations (3.3) corresponding to the geometric edge \(\{e_1, \bar{e}_1\}\) identify \(\Gamma (t(e_1))\) isomorphically with a subgroup of \(\Gamma (o(e_1))\); we can thus delete the generators \(\gamma \in \Gamma (t(e_1))\) against those relations by Tietze moves. This yields a presentation for \(\pi _1(\Gamma (-), X,S)\) with the same set of generators as \(\pi _1(\Gamma '(-), X',S')\). Moreover, those relations (3.3)–(3.4) coming from edges

*e*with \(t(e)=t(e_1)\) have to be re-expressed in terms of elements of \(\Gamma (o(e_1))\), which leads exactly to the corresponding relations of \(\pi _1(\Gamma '(-), X',S')\) obtained by extending the embedding \(e: \Gamma (e) \rightarrow \Gamma (t(e_1))\) in the natural way as given in (3.2). Hence, \(\pi _1(\Gamma (-), X,S) \cong \pi _1(\Gamma '(-), X', S')\). Since

*V*(

*X*) is finite, the tree

*S*is finite; thus, proceeding in the manner described, we obtain, after finitely many steps, a graph of groups \((\Delta (-), Y)\) with fundamental group \(\Gamma \) and a spanning tree

*T*in

*Y*without trivial edges, such that \((\Delta (-),Y)\) enjoys the finiteness properties \((F_1), (F_2)\) whenever \((\Gamma (-),X)\) does. \(\square \)

## 4 A graph-theoretic lemma

The following auxiliary result, which is of an entirely graph-theoretic nature, will be used frequently in the rest of the paper.

### Lemma 2

Let *T* be a tree, and let \(v_0\in V(T)\) be any vertex. Then there exists one, and only one, orientation \(\mathcal {O}(T)\) of *T*, such that the assignment \(e\mapsto t(e)\) defines a bijection \(\psi _{v_0}: \mathcal {O}(T) \rightarrow V(T){\setminus }\{v_0\}\). This orientation is obtained by orienting each geometric edge so as to point away from the root \(v_0;\) that is, travelling along an edge of \(\mathcal {O}(X),\) the distance from \(v_0\) in the path metric always increases.

Lemma 2 is easy to show, even in this generality. Moreover, for our present purposes, the trees considered will all be finite, in which case the assertion of Lemma 2 may be proved by a straightforward induction on \(\vert V(T)\vert \), which we sketch briefly: by our condition on the map \(\psi _{v_0}\), all (geometric) edges incident with \(v_0\) will have to be oriented away from the root \(v_0\). Delete \(v_0\) together with edges incident to \(v_0\). The result is a disjoint union of finitely many subtrees, in which we choose the (previous) neighbours of \(v_0\) as new roots. An application of the induction hypothesis to these rooted subtrees now finishes the proof.

In what follows, the orientation of a tree *T* with respect to a base point \(v_0\) described in Lemma 2 will be denoted by \(\mathcal {O}_{v_0}(T)\).

## 5 An inequality for the number of edges of a graph of groups

An important consequence of normalisation is the following.

### Lemma 3

*X*is bounded above in terms of the free rank of \(\Gamma \) via

### Proof

- (a)\(\vert V(X)\vert = 1\). Then \(m_\Gamma = \vert \Gamma (v)\vert \), where \(V(X) = \{v\}\), and the Euler characteristic of \(\Gamma \) becomeswhere \(\mathcal {O}(X)\) is an arbitrary orientation of$$\begin{aligned} \chi (\Gamma )&= \frac{1}{\vert \Gamma (v)\vert }\,-\, \sum _{e\in \mathcal {O}(X)} \frac{1}{\vert \Gamma (e)\vert }\\&= m_\Gamma ^{-1} \bigg (1 - \sum _{e\in \mathcal {O}(X)} \big (\Gamma (v) : \Gamma (e)^e\big )\bigg )\\&\le - m_\Gamma ^{-1} \big (\vert \mathcal {O}(X)\vert - 1\big ), \end{aligned}$$
*X*. It follows thatwhence our claim in this case.$$\begin{aligned} \mu (\Gamma ) = 1 - m_\Gamma \chi (\Gamma ) \ge \vert \mathcal {O}(X)\vert , \end{aligned}$$ - (b)\(\vert V(X)\vert \ge 2\). Then \(E(T) \ne \emptyset \), and we may choose some edge \(e_1\in E(T)\). Consider the tree
*T*as rooted with root \(v_1 = o(e_1)\) and associated orientation \(\mathcal {O}_{v_1}(T)\) in the sense of Lemma 2. Extending \(\mathcal {O}_{v_1}(T)\) to an orientation \(\mathcal {O}(X)\) of*X*, we writeSince the edge \(e_1\) is not trivial, we have \(2\vert \Gamma (e_1)\vert \le \vert \Gamma (o(e_1))\vert \) as well as \(2\vert \Gamma (e_1)\vert \le \vert \Gamma (t(e_1))\vert \), thus$$\begin{aligned} \chi (\Gamma )= & {} \Big (\frac{1}{\vert \Gamma (o(e_1))\vert } + \frac{1}{\vert \Gamma (t(e_1))\vert } - \frac{1}{\vert \Gamma (e_1)\vert }\Big )\\&\quad +\sum _{e\in \mathcal {O}_{v_1}(T){\setminus }\{e_1\}}\Big (\frac{1}{\vert \Gamma (t(e))\vert } - \frac{1}{\vert \Gamma (e)\vert }\Big ) - \sum _{e\in \mathcal {O}(X){\setminus }\mathcal {O}_{v_1}(T)} \frac{1}{\vert \Gamma (e)\vert }. \end{aligned}$$For the same reason, for \(e\in \mathcal {O}_{v_1}(T){\setminus }\{e_1\}\), we have$$\begin{aligned} \frac{1}{\vert \Gamma (o(e_1))\vert } + \frac{1}{\vert \Gamma (t(e_1))\vert }\, \le \, \frac{1}{\vert \Gamma (e_1)\vert }. \end{aligned}$$Putting together these observations, we find that$$\begin{aligned} \frac{1}{\vert \Gamma (t(e))\vert } - \frac{1}{\vert \Gamma (e)\vert } = \frac{1-(\Gamma (t(e)) : \Gamma (e)^e)}{\vert \Gamma (t(e))\vert } \,\le \, - \frac{1}{\vert \Gamma (t(e))\vert }\, \le \, -\frac{1}{m_\Gamma }. \end{aligned}$$from which our claim follows as before.\(\square \)$$\begin{aligned} \chi (\Gamma ) \le -m_\Gamma ^{-1}\big (\vert \mathcal {O}_{v_1}(T)\vert - 1\big ) - m_\Gamma ^{-1} \vert \mathcal {O}(X){\setminus }\mathcal {O}_{v_1}(T)\vert = -m_\Gamma ^{-1}\big (\vert \mathcal {O}(X)\vert - 1\big ), \end{aligned}$$

## 6 Classifying virtually infinite-cyclic groups

Virtually infinite-cyclic groups play a certain role in topology as they are precisely the finitely generated groups with two ends. Their structure is well-known; cf. [21, 5.1] or [22, Lemma 4.1]. In this section, we shall give a short proof of the corresponding result (Proposition 4) based on the tools developed in Sects. 3 and 4. As a consequence of this classification result, we find that the function \(f_\lambda (\Gamma )\) is constant for \(\mu (\Gamma ) = 1\); cf. Corollary 6.

### Proposition 4

- (i)
\(\Gamma \) has a finite normal subgroup with infinite-cyclic quotient.

- (ii)
\(\Gamma \) is a free product \(\Gamma = G_1 \underset{A}{*} G_2\) of two finite groups \(G_1\) and \(G_2\), with an amalgamated subgroup

*A*of index 2 in both factors.

### Proof

Let \((\Gamma (-),X)\) be a finite graph of finite groups with fundamental group \(\Gamma \) and spanning tree *T*, chosen according to Lemma 1. The reader should observe that the assumption that \(\Gamma \) is virtually infinite-cyclic in combination with (2.8) implies that \(\chi (\Gamma )=0\).

If \(\vert V(X)\vert = 1\), \(V(X)=\{v\}\) say, then the above observation together with Formula (2.2) shows that *X* has exactly one geometric edge \(\{e, \bar{e}\}\), and that the associated embeddings \(e, \bar{e}: \Gamma (e)\rightarrow \Gamma (v)\) are isomorphisms. Hence, \(\Gamma (v) \unlhd \Gamma \) and \(\Gamma /\Gamma (v) \cong C_\infty \), which gives the desired result in Case (i).

*X*, and let \(v_1=t(e_1)\). We then split the Euler characteristic of \(\Gamma \) as follows:

### Remark 5

- 1.
In Case (i) of Proposition 4, we have \(\zeta _\kappa =0\) for all \(\kappa \mid m_\Gamma \) whereas, in Case (ii), \(\zeta _{m_\Gamma }=-1\). Hence, groups occurring in Case (i) are not isomorphic to groups belonging to Case (ii).

- 2.
In Part (ii) of Proposition 4,

*A*is a finite normal subgroup of \(\Gamma \) with quotient \(C_2*C_2\), the infinite dihedral group.

### Corollary 6

If \(\Gamma \) is virtually infinite-cyclic, then the function \(f_\lambda (\Gamma )\) is constant. More precisely, we have \(f_\lambda (\Gamma )=m_\Gamma \) for \(\lambda \ge 1\) in Case *(i)* of Proposition 4, while in Case *(ii)* we have \(f_\lambda (\Gamma ) = \vert A\vert = m_\Gamma /2\).

### Proof

If \(\Gamma \) is as described in Case (i) of Proposition 4, then (2.7) shows that \(g_\lambda (\Gamma )=1\) for \(\lambda \ge 0\), leading to \(f_\lambda (\Gamma )=m_\Gamma \) for all \(\lambda \ge 1\) by (2.5) and an immediate induction on \(\lambda \).

## 7 The case where \(\mu (\Gamma )=2\)

### 7.1 The classification result

### Proposition 7

- (i)
\(\Gamma \) is an HNN-extension \(\Gamma = G\underset{A,\phi }{*}\) with finite base group

*G*, associated subgroups*A*and \(B=\phi (A),\) associated isomorphism \(\phi : A\rightarrow B,\) and \((G:A)=2\). - (ii)
\(\Gamma \) contains a finite normal subgroup

*G*with quotient \(\Gamma /G \cong F_2\) free of rank 2. - (iii)\(\Gamma \) is a free product \(\Gamma = G_1 \underset{S}{*} G_2\) of two finite groups \(G_i\) with an amalgamated subgroup
*S*, whose indices \((G_i:S)\) satisfy one of the conditions(iii)\(_1\)\(\{(G_1:S),\,(G_2:S)\} = \{2,3\},\)

(iii)\(_2\)\((G_1:S) = 3 = (G_2:S),\)

(iii)\(_3\)\(\{(G_1:S),\, (G_2:S)\} = \{2, 4\}\).

- (iv)
\(\Gamma \) is a free product \(\Gamma = G_1\underset{S}{*} \Gamma _2,\) where \(G_1\) is finite, \(\Gamma _2\) is a virtually infinite-cyclic group of type

*(i)**(*see Proposition 4*)*, and \((G_1:S) = 2 = (G_2:S),\) where \(G_2\) is the base group of the HNN-extension \(\Gamma _2\). - (v)
\(\Gamma \) is of the form \(\Gamma = (G_1 \underset{S_1}{*} G_2) \underset{S_2}{*} G_3\) with finite factors \(G_1, G_2, G_3\) and subgroups \(S_1, S_2\) satisfying \(\vert G_1\vert = \vert G_2\vert = \vert G_3\vert = 2\vert S_1\vert = 2\vert S_2\vert \).

### Proof

*T*be a spanning tree in

*X*satisfying the normalisation condition (3.1) of Lemma 1. By Lemma 3,

*X*has at most two geometric edges, while, by Eq. (2.8), we have \(\chi (\Gamma ) = - \frac{1}{m_\Gamma }\). There are five possibilities for the isomorphism type of the graph

*X*underlying the decomposition of \(\Gamma \), and the proof of the proposition (as well as its statement) breaks into cases accordingly.

- (i)
*X**consists of a single loop**e**with*\(o(e)=t(e)=v\). Setting \(G:= \Gamma (v)\) and \(S:= \Gamma (e)\), we have \(m_\Gamma = \vert G\vert \) andimplying \((G:S^e)=2\). Thus, setting \(A:=S^e\), \(B:=S^{\bar{e}}\), and with the isomorphism \(\phi : A \rightarrow B\) given by \(x^e \mapsto x^{\bar{e}}\) (in keeping with the notation of [11, Chap. IV.2]), the definition of \(\pi _1(\Gamma (-), X)\) yields that$$\begin{aligned} \chi (\Gamma ) = \frac{1}{\vert G\vert } - \frac{1}{\vert S\vert } = \frac{1 - (G:S^e)}{m_\Gamma } = -\frac{1}{m_\Gamma }, \end{aligned}$$whence the result in that case.$$\begin{aligned} \Gamma \cong \big \langle G, t\,\big \vert \,tat^{-1} = \phi (a),\, a\in A\big \rangle , \end{aligned}$$ - (ii)
*X**consists of a single vertex**v*,*supporting two loops*\(e_i\), \(i=1,2\). Set \(G:= \Gamma (v)\) and \(S_i:= \Gamma (e_i)\). Then \(m_\Gamma = \vert G\vert \), andimplying$$\begin{aligned} \chi (\Gamma ) = \frac{1}{\vert G\vert } - \frac{1}{\vert S_1\vert } - \frac{1}{\vert S_2\vert } = \frac{1 - (G:S_1^{e_1}) - (G:S_2^{e_2})}{m_\Gamma } = -\frac{1}{m_\Gamma }, \end{aligned}$$Hence, the maps \(e_i: S_i \rightarrow G\) are isomorphisms, and we obtain the presentation$$\begin{aligned} (G: S_1^{e_1}) = 1 = (G: S_2^{e_2}). \end{aligned}$$It follows that the finite group$$\begin{aligned} \Gamma \cong \big \langle G, s_1, s_2\,\big \vert \,s_1 a_1^{e_1} s_1^{-1} = a_1^{\bar{e}_1}\, (a_1\in S_1),\, s_2 a_2^{e_2} s_2^{-1} = a_2^{\bar{e}_2}\, (a_2\in S_2)\big \rangle . \end{aligned}$$*G*is normal in \(\Gamma \) with quotient a free group of rank two, as claimed. - (iii)\(X=T\)
*is a segment**e**with vertices*\(v_1, v_2,\)*say*\(t(e)=v_2\). Set \(G_i:= \Gamma (v_i), \,i=1,2,\) and \(S:= \Gamma (e)\). Then \(\Gamma = G_1 \underset{S}{*} G_2\), with the canonical embeddings given by \(\bar{e}: S\rightarrow G_1\) and \(e: S\rightarrow G_2\). Moreover, let \(a_1:= (G_1:S^{\bar{e}})\) and \(a_2:= (G_2:S^e)\). By symmetry, we may suppose that \(a_1\le a_2\), we have \(a_1\ge 2\) by our assumption that \((\Gamma (-), X, T)\) is normalised, and the requirement that \(\mu (\Gamma )=2\) boils down to the (equivalent) equationSince \(\gcd (a_1, a_2) \le a_1\), Eq. (7.1) implies that$$\begin{aligned} a_1a_2 - a_1 - a_2 = \gcd (a_1, a_2). \end{aligned}$$(7.1)which in turn leads to \(a_1^2\le 3a_1\). Given our present constraints, the last inequality is satisfied only for \(a_1 = 2\) and \(a_1 = 3\). If \(a_1=2\), then we find from (7.2) that \(2\le a_2\le 4\), while, for \(a_1=3\), we get \(a_2=3\). Thus, the only possibilities are$$\begin{aligned} a_1 \le a_2 \le \frac{2a_1}{a_1-1}, \end{aligned}$$(7.2)and, inserting these into (7.1), the possible solution \(a_1=2=a_2\) is eliminated, while the remaining three pairs all solve (7.1), whence the result in that case.$$\begin{aligned} (a_1, a_2) = (2,2),\, (2,3),\, (2,4),\, (3,3), \end{aligned}$$ - (iv)
*X**consists of a segment*\(e_1\)*with vertices*\(v_1\)*and*\(v_2,\)*say*\(t(e_1) = v_2,\)*with a loop*\(e_2\)*attached at*\(v_2\). For \(i=1,2\), set \(G_i:= \Gamma (v_i)\), and let \(S_i:= \Gamma (e_i)\). Then \(\Gamma = G_1 \underset{S_1}{*} \Gamma _2\), where \(\Gamma _2\) is the fundamental group of the loop \(e_2\) with bounding vertex \(v_2\), and the canonical embeddings are given by the maps \(\bar{e}_1: S_1\rightarrow G_1\) and \(\tilde{e}_1: S_1\overset{e_1}{\rightarrow } G_2\rightarrow \Gamma _2\). Let \(a_1:= (G_1:S_1^{\bar{e}_1})\), \(a_2:= (G_2:S_1^{e_1})\), and \(a_2':= (G_2:S_2^{e_2})\). Thenand the condition that \(\mu (\Gamma )=2\) translates into the equation$$\begin{aligned} m_\Gamma = {\text {lcm}}\!\big (\vert G_1\vert , \vert G_2\vert \big ) = \vert S_1\vert \cdot {\text {lcm}}(a_1, a_2) = \vert S_2\vert \cdot {\text {lcm}}(a_1, a_2) a_2'/a_2, \end{aligned}$$Moreover, we have \(a_1, a_2\ge 2\) by our assumption that \((\Gamma (-), X, T)\) is normalised, where$$\begin{aligned} a_1a_2 + a_1a_2' - a_1 - a_2 = \gcd (a_1, a_2). \end{aligned}$$(7.3)*T*is the unique spanning tree of*X*. Suppose first that \(a_1 \le a_2\). Then (7.3) givesThis forces \(a_1=a_2=2\), and from (7.3) we deduce that \(a_2'=1\). Now suppose that \(a_1\ge a_2\). Then (7.3) yields$$\begin{aligned} 2 \le a_1 \,\le \, a_2 \le \frac{(2-a_2')a_1}{a_1-1} \,\le \,\frac{a_1}{a_1-1}\, \le \, 2. \end{aligned}$$which again leads to the solution \(a_1 = a_2 = 2\) and \(a_2'=1\). Assertion (iv) now follows.$$\begin{aligned} 2 \le a_2 \,\le \, a_1 \,\le \, \frac{2a_2}{a_2 + a_2'-1} \, \le \, 2, \end{aligned}$$ - (v)\(X=T\)
*is a path*\((v_1, e_1, v_2, e_2, v_3)\)*of length*2. For \(i = 1,2,3\), set \(G_i:=\Gamma (v_i)\), and let \(S_j:=\Gamma (e_j)\) for \(j=1,2\). Then \(\Gamma = \big (G_1 \underset{S_1}{*} G_2\big ) \underset{S_2}{*} G_3\). Since \(\mu (\Gamma )=2\), we haveAs \((\Gamma (-), X, T)\) is normalised, we have$$\begin{aligned} \frac{m_\Gamma }{\vert S_1\vert }\, + \,\frac{m_\Gamma }{\vert S_2\vert }\, -\, \frac{m_\Gamma }{\vert G_1\vert }\, - \,\frac{m_\Gamma }{\vert G_2\vert }\, -\, \frac{m_\Gamma }{\vert G_3\vert } = 1. \end{aligned}$$(7.4)so that (7.4) gives$$\begin{aligned} \frac{m_\Gamma }{\vert G_1\vert } \,\le \, \frac{m_\Gamma }{2\vert S_1\vert } \, \text{ and } \, \frac{m_\Gamma }{\vert G_2\vert }\, \le \, \frac{m_\Gamma }{2\vert S_2\vert }, \end{aligned}$$Again by normalisation,$$\begin{aligned} \frac{m_\Gamma }{2\vert S_1\vert } \,+\,\frac{m_\Gamma }{2\vert S_2\vert } - \frac{m_\Gamma }{\vert G_3\vert }\, \le \,1. \end{aligned}$$thus$$\begin{aligned} \frac{m_\Gamma }{\vert G_3\vert } \,\le \, \frac{m_\Gamma }{2\vert S_2\vert }, \end{aligned}$$implying$$\begin{aligned} 1\,\le \,\max \left\{ \frac{\vert G_1\vert }{2\vert S_1\vert },\, \frac{\vert G_2\vert }{2\vert S_1\vert }\right\} \,\le \, \frac{m_\Gamma }{2\vert S_1\vert } \,\le \, 1, \end{aligned}$$Using this information in (7.4), we now find that$$\begin{aligned} m_\Gamma = \vert G_1\vert = \vert G_2\vert = 2\vert S_1\vert . \end{aligned}$$implying first \(m_\Gamma = 2 \vert S_2\vert \) by normalisation, and then \(\vert G_3\vert = m_\Gamma \). \(\square \)$$\begin{aligned} m_\Gamma \left( \frac{1}{\vert S_2\vert } \,-\,\frac{1}{\vert G_3\vert }\right) = 1, \end{aligned}$$

### Remark 8

By considering the type and the number of conjugacy classes of maximal finite subgroups, one shows again that any two groups from different classes in Proposition 7 are not isomorphic.

### 7.2 Some consequences of Proposition 7

- (a)In Cases (i) and (iv),$$\begin{aligned} f_{\lambda +1}(\Gamma ) = \frac{2\lambda +3}{2} m_\Gamma f_\lambda (\Gamma ) \,+\,\sum _{\mu =1}^{\lambda -1} f_\mu (\Gamma ) f_{\lambda -\mu }(\Gamma ),\quad \lambda \ge 1. \end{aligned}$$(7.5)
- (b)In Case (ii),$$\begin{aligned} f_{\lambda +1}(\Gamma ) = (\lambda +2) m_\Gamma f_\lambda (\Gamma )\,+\,\sum _{\mu =1}^{\lambda -1} f_\mu (\Gamma ) f_{\lambda -\mu }(\Gamma ),\quad \lambda \ge 1. \end{aligned}$$(7.6)
- (c)In Cases (iii) and (v),$$\begin{aligned} f_{\lambda +1}(\Gamma ) = (\lambda +1) m_\Gamma f_\lambda (\Gamma )\,+\,\sum _{\mu =1}^{\lambda -1} f_\mu (\Gamma ) f_{\lambda -\mu }(\Gamma ),\quad \lambda \ge 1, \end{aligned}$$(7.7)

- (a)
\(f_1(\Gamma ) = m_\Gamma ^2/2\),

- (b)
\(f_1(\Gamma ) = m_\Gamma ^2\),

- (c)
\(f_1(\Gamma ) = {\left\{ \begin{array}{ll} (m_\Gamma -\vert S\vert )\vert S\vert ,&{} \text{ Case } \text{(iii) },\\ (m_\Gamma /2)^2,&{} \text{ Case } \text{(v). } \end{array}\right. }\)

### Corollary 9

### Proof

This follows from the above recurrence relations plus initial conditions by an immediate induction on \(\lambda \). \(\square \)

### Corollary 10

*(iii)*\(_1,\)

*(iii)*\(_3,\) and

*(v)*, we have, with \(\vert S\vert \equiv 1\ (\text {mod }2)\) respectively \(\vert S_1\vert \equiv 1\ (\text {mod }2),\)

### Proof

We focus on Case (iii)\(_1\) with \(\vert S\vert \equiv 1\ (\text {mod }2)\); the proof in Cases (iii)\(_3\) and (v) is completely analogous, while the fact that \(f_\lambda (\Gamma )\) is constant modulo 2 in all other cases is immediate.

- (1)In Case (i) of Proposition 7, we have \(2\mid m_\Gamma \) andIn particular, we obtain that \(\mu _2(\Gamma )>0\), so Case (III)\(_2\) of [8, Theorem 1] applies, asserting that \(f_\lambda (\Gamma )\) is ultimately periodic modulo 2 in this case. Indeed, by Corollary 10, \(f_\lambda (\Gamma )\) is constant modulo 2.$$\begin{aligned} \mu _2(\Gamma ) = {\left\{ \begin{array}{ll} 1,&{}\vert A\vert \equiv 1\ (\text {mod }2),\\ 2,&{} \vert A\vert \equiv 0\ (\text {mod }2).\end{array}\right. } \end{aligned}$$
- (2)
In Case (ii) of Proposition 7, we either have \(2\not \mid m_\Gamma \), or \(2\mid m_\Gamma \) and \(\mu _2(\Gamma )=2>0\), so \(f_\lambda (\Gamma )\) is ultimately periodic modulo 2 in this case according to Case (III)\(_1\) respectively (III)\(_2\) of [8, Theorem 1]. Indeed, \(f_\lambda (\Gamma )\) is again constant modulo 2 by Corollary 10.

- (3)In Case (iii)\(_1\) of Proposition 7, we have \(2\mid m_\Gamma \) andso \(f_\lambda (\Gamma )\) is ultimately periodic modulo 2 according to [8, Theorem 1] if, and only if, \(\vert S\vert \equiv 0\ (\text {mod }2)\), which coincides with the corresponding assertion of Corollary 10.$$\begin{aligned} \mu _2(\Gamma ) = {\left\{ \begin{array}{ll} 0,&{}\vert S\vert \equiv 1\ (\text {mod }2),\\ 2, &{} \vert S\vert \equiv 0\ (\text {mod }2),\end{array}\right. } \end{aligned}$$
- (4)
In Case (iii)\(_2\) of Proposition 7, we either have \(\vert S\vert \equiv 1\ (\text {mod }2)\), and so \(2\not \mid m_\Gamma = 3\vert S\vert \), or \(\vert S\vert \equiv 0\ (\text {mod }2)\), in which case \(2\mid m_\Gamma \) and \(\mu _2(\Gamma ) =2>0\). Hence, ultimate periodicity of the function \(f_\lambda (\Gamma )\) modulo 2 follows again from Case (III)\(_1\) respectively Case (III)\(_2\) of [8, Theorem 1], while Corollary 10 asserts that \(f_\lambda (\Gamma )\) is constant modulo 2 in that case.

- (5)In Case (iii)\(_3\) of Proposition 7, we have \(2\mid m_\Gamma = 4\vert S\vert \) andHence, according to [8, Theorem 1], the function \(f_\lambda (\Gamma )\) is ultimately periodic modulo 2 if, and only if, \(\vert S\vert \equiv 0\ (\text {mod }2)\), which is in accordance with the corresponding assertion of Corollary 10.$$\begin{aligned} \mu _2(\Gamma ) = {\left\{ \begin{array}{ll} 0,&{} \vert S\vert \equiv 1\ (\text {mod }2),\\ 2,&{} \vert S\vert \equiv 0\ (\text {mod }2).\end{array}\right. } \end{aligned}$$
- (6)In Case (iv) of Proposition 7, we have \(2\mid m_\Gamma = 2\vert S\vert \) andIn particular, we obtain that \(\mu _2(\Gamma )>0\), so that ultimate periodicity of \(f_\lambda (\Gamma )\) follows from Case (III)\(_2\) of [8, Theorem 1], while Corollary 10 asserts that \(f_\lambda (\Gamma )\) is constant modulo 2 in that case.$$\begin{aligned} \mu _2(\Gamma ) = {\left\{ \begin{array}{ll} 1,&{} \vert S\vert \equiv 1\ (\text {mod }2),\\ 2, &{} \vert S\vert \equiv 0\ (\text {mod }2).\end{array}\right. } \end{aligned}$$
- (7)Finally, in Case (v) of Proposition 7, we have \(2\mid m_\Gamma = 2\vert S_1\vert \) and$$\begin{aligned} \mu _2(\Gamma ) = {\left\{ \begin{array}{ll} 0, &{} \vert S_1\vert \equiv 1\ (\text {mod }2),\\ 2,&{} \vert S_1\vert \equiv 0\ (\text {mod }2).\end{array}\right. } \end{aligned}$$

## 8 Some criteria for a virtually free group to be ‘large’

*G*and

*H*be groups. Then we write \(H\preceq G\), if there exist

- (a)
a subgroup \(G^0\) of finite index in

*G*; - (b)
a subgroup \(H^0\) of finite index in

*H*, and a finite normal subgroup \(N^0\) of \(H^0\); - (c)
a homomorphism from \(G^0\) onto \(H^0/N^0\).

*G*] the equivalence class of the group

*G*under \(\sim \). By abuse of notation, we also denote by \(\preceq \) the preorder induced on the class of equivalence classes of groups. The finitely generated groups which are ‘largest’ in Pride’s sense are the ones having a subgroup of finite index which can be mapped homomorphically onto the free group \(F_2\) of rank 2.

### Proposition 11

- (i)
\(\chi (\Gamma ) < 0\).

- (ii)
\(\mu (\Gamma ) \ge 2\).

- (iii)
\(\Gamma \) has infinitely many ends.

- (iv)
The function \(f_\lambda (\Gamma )\) is strictly increasing.

- (v)
\(\Gamma \sim F_2\) in the sense of Pride’s preorder \(\preceq \) on groups, where \(F_2\) denotes the free group of rank 2.

- (vi)
\(\Gamma \) has fast subgroup growth in the sense that the inequality \(s_{nj}(\Gamma ) \ge c\cdot n!\) holds for some fixed positive integer

*j*, some constant \(c>0,\) and all \(n\ge 1\). Here \(s_m(\Gamma )\) denotes the number of subgroups of index*m*in \(\Gamma \). - (vii)
If

*X*has only one vertex*v*, then either*X*has more than one geometric edge, or \(E(X) = \{e_1, \bar{e}_1\}\) and \((\Gamma (v): \Gamma (e_1)^{e_1}) \ge 2;\) if \(\vert V(X)\vert \ge 2,\) then*X*is not a tree, or*X*is a tree with more than one geometric edge, or \(E(X) = \{e_1, \bar{e}_1\}\) and \(\chi (\Gamma _0)<0,\) where \(\Gamma _0:= \Gamma _{o(e_1)} \underset{\Gamma (e_1)}{*} \Gamma _{t(e_1)}\).

### Proof

(i) \(\Leftrightarrow \) (ii). This is immediate from Formula (2.8) plus the fact that \(\mu (\Gamma )\) is integral.

(ii) \(\Leftrightarrow \) (iii). This follows from [3, Prop. 2.1] (i.e., the fact that the number of ends is invariant when passing to a subgroup of finite index) and Examples 1 and 2 in [3] computing the number of ends of a free product, respectively of \(C_\infty \).

(ii) \(\Leftrightarrow \) (iv). This follows from [13, Theorem 4] in conjunction with Corollary 6.

(ii) \(\Rightarrow \) (v). If \(\mu (\Gamma )\ge 2\), then \(\Gamma \) contains a free group

*F*of rank at least 2, with \((\Gamma :F) = m_\Gamma <\infty \); in particular, \(F_2 \preceq \Gamma \). Since \([F_2]\) is largest with respect to the preorder \(\preceq \) among all equivalence classes of finitely generated groups, we also have \(\Gamma \preceq F_2\), so \(\Gamma \sim F_2\), as claimed.- (v) \(\Rightarrow \) (vi). Suppose that \(\Gamma \sim F_2\). Then there exists a subgroup \(\Delta \le \Gamma \) of index \((\Gamma :\Delta )=j<\infty \) and a surjective homomorphism \(\varphi : \Delta \rightarrow F_2\). From this plus Newman’s asymptotic estimate [14, Theorem 2]it follows that$$\begin{aligned} s_n(F_r) \sim n (n!)^{r-1} \text{ as } n\rightarrow \infty ,\quad r\ge 2, \end{aligned}$$for \(n\ge 1\) and some constant \(c>0\), whence (vi).$$\begin{aligned} s_{jn}(\Gamma ) \ge s_n(\Delta ) \ge s_n(F_2) \ge c\cdot n\cdot n! \ge c\cdot n! \end{aligned}$$
- (vi) \(\Rightarrow \) (ii). If \(\mu (\Gamma )\le 1\), then either \(\Gamma \) is finite, so \(s_n(\Gamma )=0\) for sufficiently large
*n*, or \(\Gamma \) is virtually infinite-cyclic, implyingfor some constant \(\alpha \), by [10, Cor. 1.4.3]; see also [17]. In both cases, Condition (vi) does not hold.$$\begin{aligned} s_n(\Gamma ) \le n^\alpha ,\quad n\ge 1, \end{aligned}$$ (ii) \(\Leftrightarrow \) (vii). This follows by splitting the Euler characteristic \(\chi (\Gamma )\) as in the proof of Proposition 4, making use of Lemmas 1 and 2.\(\square \)

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