1. Introduction

The $\ell ^{2}$-Betti numbers were introduced by Atiyah as dimensions of heat kernels of certain operators on Riemannian manifolds. The modern formulation assigns $\ell ^{2}$-Betti numbers $b^{(2)}_i(G)$ to arbitrary groups $G$. We refer the reader to Lück's account where the history can be found in the introduction of [Reference Lück12]. Technical results about $\ell ^{2}$-Betti numbers that we need can be found in chapters 6 and 8 of loc. cit. The $\ell ^{2}$-Euler characteristic $\chi ^{(2)}(G)$ is defined to be the alternating sum of these Betti numbers when this series is absolutely convergent. Let $\mathfrak {C}$ denote the class of groups $F$ such that

• • $\sum \nolimits _{i\geq 0}b^{(2)}_i(G)$ is finite (this being the condition for absolute convergence),

• • $b^{(2)}_1(F)=b^{(2)}_2(F)=0$, and

• • either $\chi ^{(2)}(F)=0$ or $F$ is finite.

Note that $\mathfrak {C}$ contains all $\ell ^{2}$-acyclic groups (i.e. the groups for which $b_i^{(2)}=0$ for all $i>0$) and, in particular, it contains all amenable groups. Relevant background on $\ell ^{2}$-cohomology is included in § 2. In this note, we prove the following theorem.

Note that the hypotheses of this theorem guarantee that $N$ acts freely on the specified tree and, in particular, $N$ is necessarily a free group. Note also that, according to [Reference Bader, Furman and Sauer2, Corollary 1.4], if $b^{(2)}_1(G)>0$ then $G$ has no commensurated infinite amenable subgroup and according to [Reference Bekka and Valette3, Corollary 6] does not have property (T). If we also have $b_2^{(2)}(G)=0$, then $G$ is in the class $\mathcal {D}_\textrm {reg}$ by [Reference Thom15, Lemma 2.8]. We refer the reader to [Reference Bekka, de la Harpe and Valette4] for background on property (T) and to [Reference Thom15, Definition 2.6] for the definition of the class $\mathcal {D}_{\mathrm {reg}}$. Acylindrically hyperbolic groups form a large class of groups admitting coarsely proper actions on hyperbolic metric spaces. The class is a generalization of relative hyperbolicity including many Artin groups, mapping class groups, and $\mathrm {Out}(F_n)$. By the main result of Osin's paper [Reference Osin13], we have the following corollary.

The simplest way in which the indicability hypothesis may arise is through stable letters: Let $T$ denote the $F$-tree of Theorem 1.1. Let $K$ denote the (necessarily normal) subgroup generated by the vertex stabilizers. Then there is a subgroup $E\le F$ that complements $K$ and all such subgroups are free of uniquely determined rank. Such a subgroup may be referred to as a subgroup of stable letters of the action. The group $G$ has an infinite cyclic quotient when $N\cap E$ has infinite index in $E$, in other words, when there is a stable letter that is faithfully represented in $G$, and in this case, $G$ is indicable.

Recall that a group $G$ is $C^{\ast }$-simple if the reduced group $C^{\ast }$-algebra, denoted $C_r^{\ast }(G)$, has exactly two norm closed $2$-sided ideals, $0$, and the algebra $C_r^{\ast }(G)$ itself. By [Reference Breuillard, Kalantar, Kennedy and Ozawa5, Corollary 6.7], we obtain the following.

These two corollaries highlight the relation between the first $\ell ^{2}$-Betti number and other areas of geometric group theory. It is an interesting matter for further research to find out whether they lead to new examples.

Theorem 1.1 has some historical pedigree. It originally began life as a result about quotients of free groups due to Thomas (see Theorem 1.4(i)) and was proved using combinatorial methods [Reference Thomas16]. The result was generalized by Allcock to incorporate a bound on the rank of the abelianisation of the quotient group [Reference Allcock1]. The introduction of $\ell ^{2}$-cohomology came when Peterson–Thom [Reference Peterson and Thom14, Theorem 3.6] and Kar–Niblo [Reference Kar and Niblo11] independently linked the inequality of Thomas to the first $\ell ^{2}$-Betti number. These discoveries are summarized in the following result.

Theorem 1.4 (Thomas [Reference Thomas16], Allcock [Reference Allcock1], Peterson–Thom [Reference Peterson and Thom14], Kar–Niblo [Reference Kar and Niblo11])

Let $G$ be a group with a presentation

\[ \langle x_{1},\dots,x_{n}|\ r_{1}^{k_{1}},\dots,r_{m}^{k_{m}}\rangle \]

in which the elements $r_{i}$ have order $k_{i}$ when interpreted in $G$.

1. (i) If $n-\sum \nolimits _{i=1}^{m}\frac 1{k_{i}}\ge 1$ then $G$ is infinite.

2. (ii) If $G$ is finite then $|G|\ge \frac 1{1-n+\sum \nolimits _{i=1}^{m}k_{i}}$.

3. (iii) If $n-\sum \nolimits _{i=1}^{m}\frac 1{k_{i}} > 1$ then $G$ is non-amenable.

Deduction of Theorem 1.4 from Theorem 1.1 Let $G$ be a group with a presentation

\[ G=\langle x_{1},\dots,x_{n}|\ r_{1}^{k_{1}},\dots,r_{m}^{k_{m}}\rangle. \]

Adding $m$ fresh generators $y_{1},\,\dots,\,y_{m}$, we can give the following alternative presentation of the same group:

\[ G\cong \langle x_{1},\dots,x_{n},y_{1},\dots,y_{m}|\ y_{1}^{k_{1}},\dots,y_{m}^{k_{m}}, r_{1}y_{1}^{{-}1},\dots,r_{m}y_{m}^{{-}1}\rangle. \]

Let $F$ be the group with presentation

\[ F=\langle x_{1},\dots,x_{n},y_{1},\dots,y_{m}|\ y_{1}^{k_{1}},\dots,y_{m}^{k_{m}}\rangle \]

and let $N$ be the subgroup of $F$ normally generated by $r_{1}y_{1}^{-1},\,\dots,\,r_{m}y_{m}^{-1}$. Then $F$ is a free product of cyclic groups which has a natural action on a simplicial tree $T$ (the Bass–Serre tree). In particular, it is virtually free and has Euler characteristic $\chi (F)=\sum \nolimits _{i=1}^{m}\frac 1{k_{i}}-n-m+1$ (as explained in [Reference Brown6, Chapter IX.7]). This is equal to $\chi ^{(2)}(F)$ by [Reference Lück12, Remark 6.81]. The condition that the $r_{i}$ have order $k_{i}$ in the original presentation ensures that $N$ does not meet any of the finite subgroups of $F$, so is torsion-free and acts freely on $T$. Applying Theorem 1.1 with these choices of $F$, $N$, $G$ and $T$ yields Theorem 1.4.

Throughout this paper, for a group or subgroup $G$, we will adopt the convention that $\frac {1}{|G|}$ be interpreted as zero if $G$ is infinite.

Finally, we compute the first $\ell ^{2}$-Betti number for certain groups acting on trees leading to Theorem 1.5 below, which is proved in § 4. This generalizes a result of Lück [Reference Brown, Dykema and Jung7], which covers the case of an amalgamated free product, and a result of Tsouvalas [Reference Tsouvalas17, Corollary 3.7]. Tsouvalas assumes the vertex stabilisers are either residually finite or virtually torsion-free and the edge stabilisers are finite. Here we replace both of these assumptions with Lück's less restrictive assumption that the first $\ell ^{2}$-Betti numbers of the edge stabilisers vanish. So, for example, the theorem applies to fundamental groups of graphs of $\mathfrak {C}$-groups.

Theorem 1.5 Let $F$ be a group acting simplicially on a simplicial tree and let $V$ and $E$ denote sets of representatives of $F$-orbits of vertices and edges. Assume for each $e\in E$ that $b^{(2)}_1(F_e)=0,$ then we have

\[ b^{(2)}_1(F)=\sum_{v\in V}\left( b^{(2)}_1(F_v)- \frac{1}{|F_v|}\right)+\sum_{e\in E} \frac{1}{|F_e|} + \frac{1}{|F|}. \]