Amenable Groups

Abstract

This chapter is devoted to the class of amenable groups, a class of groups which contains all finite groups and all abelian groups and which is closed under several group operations, in particular taking subgroups, taking extensions, and taking direct limits.

Appendices

Notes

A detailed exposition of the theory of amenable groups may be found for example in [41, 85, 88], and [22, Chap. 4]. The notes by Tao [103] provide an especially nice introduction to amenability via Følner sequences.

Amenability theory has its roots in the difficulties raised at the beginning of the 20th century by both the definition of the Lebesgue integral and the Banach-Tarski paradox (see [27] for a historical survey). Amenable groups were introduced by von Neumann [113] in 1929. The original definition of von Neumann requires that the group admits an invariant finitely-additive probability measure defined on the set of all of its subsets. A fundamental observation of M. Day is that von Neumann’s definition is equivalent to the existence of an invariant mean on the space of bounded functions on the group. The introduction of means has the advantage of allowing the use of the powerful tools of functional analysis. Let G be a group. For $g \in G$, we denote by $L_g$ and $R_g$ the left and right multiplication by g, that is, the maps $L_g :G \rightarrow G$ and $R_g :G \rightarrow G$ defined by $L_g(h) = gh$ and $R_g(h) = hg$ for all $h \in G$. Consider the vector space $\ell ^\infty (G)$ consisting of all bounded real-valued maps $f :G \rightarrow {\mathbb {R}}$. A mean on G is a linear map $m :\ell ^\infty (G) \rightarrow {\mathbb {R}}$ such that $\inf _{g \in G} f(g) \le m(f) \le \sup _{g \in G} f(g)$ for all $f \in \ell ^\infty (G)$. One says that a mean m on G is left-invariant (resp.  right-invariant ) if it satisfies $m(f \circ L_g) = m(f)$ (resp. $m(f \circ R_g) = m(f)$) for all $f \in \ell ^\infty (G)$ and $g \in G$. A mean m on G is said to be bi-invariant if it is both left and right-invariant. The following conditions are all equivalent: (1) G is amenable; (2) G admits a left-invariant mean; (3) G admits a right-invariant mean; (4) G admits a bi-invariant mean. There is a natural one-to-one correspondence between means on a group G and finitely-additive probability measures on subsets of G which is given by $m \mapsto \mu _m$, where $\mu _m(A)$ is the value taken by m at the characteristic map $\chi _A$ of $A \subset G$. This correspondence between means and finitely-additive probability measures preserves left and right-invariance and explains the equivalence between von Neumann’s and Day’s definitions. In fact, many other equivalent definitions of amenability for groups may be found in the literature and a complete list, if it exists, would be certainly far too long to be given here. The interest of choosing one of these definitions rather than another depends on the context. It seems that the term “amenable” was used for the first time by M. Day in 1949. The German word originally used by von Neumann in 1929 was “messbar”. Note that “amenable” is an anagram of “meanable” and that the French word that is currently used for “amenable” is “moyennable”.

The Tits alternative [106] asserts that every finitely generated linear group either contains a non-abelian free subgroup or is virtually solvable (a group G is called linear if there exist a field K and an integer $n \ge 1$ such that G is isomorphic to a subgroup of ${{\mathrm{GL}}}_n(K)$). One deduces from the Tits alternative that if G is a linear group then the following conditions are equivalent: (1) G is amenable; (2) G is locally virtually solvable; (3) G contains no non-abelian free subgroups. The group ${{\mathrm{GL}}}_2(K)$, where K is the algebraic closure of a finite field, provides an example of a linear group that is locally virtually solvable (it is even locally finite, see Exercise 9.10) but not virtually solvable. However, in characteristic 0, every linear group that is locally virtually solvable is virtually solvable. There exist finitely generated amenable groups that are not virtually solvable. The groups of intermediate growth, e.g., Grigorchuk groups [42], are examples of such groups. The infinite finitely generated amenable simple groups exhibited by Juschenko and Monod in [52] provide further examples of finitely generated amenable groups that are not virtually solvable, since it is clear that an infinite simple group cannot be virtually solvable. On the other hand, there exist non-amenable groups that contain no non-abelian free subgroups among Tarski monsters [82] and free Burnside groups [2]. These examples illustrate the fact that both the class of amenable groups and the class of non-amenable groups are very hard to apprehend from an algebraic viewpoint.

A net in a set X is a family $(x_i)_{i \in I}$ of elements of X indexed by a directed set I (recall that a directed set is a partially ordered set $(I,\le )$ satisfying the following condition: for all $i_1,i_2 \in I$, there exists $j \in I$ such that $i_1 \le j$ and $i_2\le j$). In a topological space, the notion of a limit can be extended to nets of points. For example, one says that a net $(x_i)_{i \in I}$ of real numbers converges to 0, and one writes $\lim _i x_i = 0$, if, for every $\varepsilon >0$, there exists $i_0 \in I$ such that $|x_i| \le \varepsilon$ for all $i_0 \le i$. If G is a group, a Følner net for G is a net $(F_i)_{i \in I}$ of non-empty finite subsets of G such that $\lim _i |F_i {\setminus } g F_i| = 0$ for all $g \in G$. By adapting the proof of Lemma 9.2.1, one easily shows that a (possibly uncountable) group is amenable if and only if it admits a Følner net. On the other hand, given a Følner net $(F_i)_{i \in I}$, there is an associated net of means $(m_i)_{i \in I}$, where $m_i(f)$ is the average of $f \in \ell ^\infty (G)$ on $F_i$. Then, by the compactness of the closed unit ball of the dual space of $\ell ^\infty (G)$ for the weak-$\star$ topology (which follows from the Banach-Alaoglu theorem), there exists a subnet of $(m_i)_{i \in I}$ that converges to a mean m. From the fact that $(F_i)_{i \in I}$ is a Følner net, one easily deduces that the limit mean m is left-invariant. The converse implication, namely that the existence of a left-invariant mean implies the existence of a Følner net, is more delicate. The proof given by Følner [37] for this converse implication was subsequently simplified by Namioka [81].

The definition of amenability via the existence of invariant means may be extended to semigroups, i.e., sets equipped with a binary operation that is only assumed to be associative, but theoretical complications appear in this more general setting (see [25, 26, 81, 85]). For instance, when considering semigroups, one must distinguish between left-amenability and right-amenability. Also, no equivalent definition of amenability based on Følner-type conditions is available in this setting. However, for semigroups, there is a Følner-type condition that is implied by amenability and a stronger Følner-type condition that implies amenability.

The notion of amenability has been generalized in several other directions (group actions, groupoids, associative algebras, orbit equivalences, etc.) and plays now an important role in many branches of mathematics such as combinatorial and geometric group theory, ergodic theory, dynamical systems, geometry of manifolds, and operator algebras. This is due to the fact that amenable objects are easier to manipulate because they are close to finite and commutative ones.

The Baumslag-Solitar groups BS(m, n) were introduced in [12]. Given non-zero integers m, n , the Baumslag-Solitar group BS(m, n) is the group given by the presentation $\langle a,b : b a^m b^{-1} = a^n \rangle$. This means that BS(m, n) is the quotient of the free group F on two generators a and b by the smallest normal subgroup N of F such that $b a^m b^{-1} a^{-n} \in N$. Baumslag-Solitar groups are often used as counterexamples in combinatorial and geometric group theory. For instance, the Baumslag-Solitar group BS(2, 3) is non-Hopfian (a group G is called Hopfian if every surjective endomorphism of G is injective). In contrast, it follows from results due to Malcev that every finitely generated linear group is residually finite and that every finitely generated residually finite group is Hopfian (cf. Exercise 10.8 for the definition of residual finiteness). For a nice survey on Baumslag-Solitar groups, see [1].

A proof of Theorem 9.4.1, under the additional hypothesis that h is non-decreasing, was given by Lindenstrauss and Weiss in [74, Theorem 6.1]. Their proof is based on the theory of quasi-tiles in amenable groups that was developed by Ornstein and Weiss in [83]. An alternative proof of Theorem 9.4.1 was sketched by Gromov [44] (see [61] for a detailed exposition of Gromov’s argument). A version of Theorem 9.4.1 for cancellative one-sided amenable semigroups was given in [23]. The proof of Theorem 9.4.1 presented in the present chapter relies on Gromov’s ideas and closely follows the exposition that may be found in [23]. The extension to uncountable amenable groups, for which Følner sequences are replaced by Følner nets, is straightforward (cf. [23]).

Exercises

 

• 9.1 Show that the additive group ${\mathbb {Q}}$ of rational numbers is countable but not finitely generated.

• 9.2 Show that the sequence $(F_n)_{n \ge 1}$, where

  $$ F_n := \left\{ \frac{k}{n!} \mid \; k \in {\mathbb {N}}\text { and } k \le (n + 1)! \right\} $$

  for all $n \ge 1$, is a Følner sequence for ${\mathbb {Q}}$.

• 9.3 The symmetric difference of two sets A and B is the set $A \bigtriangleup B$ consisting of all elements that are either in A or in B but not in both. Thus one has $A \bigtriangleup B = A\cup B {\setminus } A \cap B$.

  • (a) Show that one has $A \bigtriangleup B = (A {\setminus } B) \cup (B {\setminus } A)$.

  • (b) Show that one has $A \bigtriangleup B = \varnothing$ if and only if $A = B$.

  • (c) Show that if A and B are finite sets with the same cardinality then one has $|A \bigtriangleup B| = 2|A {\setminus } B| = 2|B {\setminus } A|$.

  • (d) Let G be a group and let $(F_n)_{n \ge 1}$ be a sequence of non-empty finite subsets of G. Show that $(F_n)_{n \ge 1}$ is a Følner sequence for G if and only if one has $\lim _{n \rightarrow \infty } \dfrac{|F_n \bigtriangleup g F_n|}{|F_n|} = 0$ for all $g \in G$.

• 9.4 Let G be a countable amenable group. Let $(F_n)_{n \ge 1}$ be a Følner sequence for G and let $(g_n)_{n \ge 1}$ be a sequence of elements of G. Show that $(F_ng_n)_{n \ge 1}$ is a Følner sequence for G.

• 9.5 Let G be a group and let A be a finite subset of G. Show that if $(F_n)_{n \ge 1}$ is a Følner sequence for G then $(F_n \cup A)_{n \ge 1}$ is also a Følner sequence for G.

• 9.6 One says that a Følner sequence $(F_n)_{n \ge 1}$ for a group G is a Følner exhaustion if it satisfies $F_n \subset F_{n + 1}$ for all $n \ge 1$ and $G = \bigcup _{n \ge 1} F_n$. Show that every countable amenable group G admits a Følner exhaustion.

• 9.7 Show that any group G satisfies the following condition: for every $s \in G$ and every $\varepsilon > 0$, there exists a non-empty finite subset $F \subset G$ such that $|F {\setminus } sF| \le \varepsilon |F|$.

• 9.8 Deduce from Lemma 9.2.12 that a group G is amenable if and only if it satisfies the following condition: for every finite subset S of G and every $\varepsilon > 0$, there exists a non-empty finite subset $F \subset G$ such that $| SF| \le (1 + \varepsilon )|F|$. Hint: observe that $SF \subset F \cup SF$ and $SF {\setminus } F= (S \cup \{1_G\})F {\setminus } F$ for all $F,S \subset G$.

• 9.9 Let G be a group.

  • (a) Show that the following conditions are equivalent: (1) G is countable and locally finite; (2) there exists a non-decreasing sequence $(H_n)_{n \ge 1}$ of finite subgroups of G such that $G = \bigcup _{n \ge 1} H_n$.

  • (b) Suppose that G is countable and locally finite. Let $(H_n)_{n \ge 1}$ be a non-decreasing sequence of finite subgroups of G such that $G = \bigcup _{n \ge 1} H_n$. Show that the sequence $(H_n)_{n \ge 1}$ is a Følner exhaustion of G (cf. Exercise 9.6).

• 9.10 Let K be the algebraic closure of a finite field and let n be a positive integer. Show that the group ${{\mathrm{GL}}}_n(K)$ is locally finite. Hint: Observe that K is the union of an increasing sequence of finite subfields.

• 9.11 Let G be a group. Suppose that G contains a normal subgroup H such that H is solvable of solvability degree m and G / H is solvable of solvability degree n. Show that G is solvable of solvability degree at most $m + n$.

• 9.12 Let G be a group and let $(D^n(G))_{n \ge 0}$ denote its derived series. Show that $D^n(G)$ is normal in G for every $n \ge 0$.

• 9.13 Show that every virtually solvable group that is a torsion group is locally finite. Hint: reduce to the case of a solvable group and then use induction on the solvability degree.

• 9.14 Let G be a finitely generated group. Suppose that $A \subset G$ is a finite generating subset of G. Show that the following conditions are all equivalent: (1) G is amenable; (2) for every $\varepsilon > 0$, there exists a non-empty finite subset $F \subset G$ such that $|F {\setminus } aF| \le \varepsilon |F|$ for all $a \in A$; (3) for every $\varepsilon > 0$, there exists a non-empty finite subset $F \subset G$ such that $|AF {\setminus } F| \le \varepsilon |F|$; (4) for every $\varepsilon > 0$, there exists a non-empty finite subset $F \subset G$ such that $|AF| \le (1 + \varepsilon ) |F|$.

• 9.15 Show that a group G is amenable if and only if it satisfies the following condition: for every finite subset $K \subset G$ and every real number $\varepsilon >0$, there exists a non-empty finite subset $F \subset G$ such that $\alpha (F,K)\le \varepsilon$.

• 9.16 (Groups with subexponential growth). Let G be a finitely generated group. Let $A \subset G$ be a finite subset which generates G as a semigroup (i.e., every element of G can be written as a finite product of elements of A). For $n \ge 1$, let $B_n = B_n(G,A)$ denote the set consisting of all $g \in G$ that can be written in the form $g = a_1 a_2 \dots a_k$ with $0 \le k \le n$ and $a_i \in A$ for all $1 \le i \le k$.

  • (a) Show that the limit

    $$ \lambda = \lambda (G,A) :=\lim _{n \rightarrow \infty } \frac{\log |B_n|}{n} $$

    exists and that $0 \le \lambda < \infty$. Hint: see Exercise 6.4.

  • (b) One says that G has subexponential growth if $\lambda = 0$ and that G has exponential growth otherwise. Show that this definition does not depend on the choice of the finite subset $A \subset G$ which generates G as a semigroup. Hint: Observe that if $A'$ is another finite subset of G which generates G as a semigroup and $B_n' := B_n(G,A')$, then there exists a positive integer $C = C(G,A,A')$ such that $B_n \subset B_{Cn}'$ for all $n \ge 1$.

  • (c) Show that if G has subexponential growth then

    $$ \liminf _{n \rightarrow \infty } \frac{|B_{n + 1}|}{|B_n|} = 1. $$

  • (d) Show that if G has subexponential growth then G is amenable. Hint: Consider a finite subset $S \subset G$ and $\varepsilon > 0$. Choose a finite subset $A \subset G$ that generates G as a semigroup with $S \subset A$. Observe that the subsets $B_n := B_n(G,A)$ satisfy $S B_n \subset B_{n + 1}$ for all $n \ge 1$ and deduce from the result of the previous question that if G has subexponential growth then there exists $n \ge 1$ such that $|B_{n + 1} {\setminus } B_n| \le \varepsilon |B_n|$. Conclude by using the characterization of amenability in Lemma 9.2.12.

• 9.17 Let G be a finitely generated group and H a subgroup of G.

  • (a) Show that if H is finitely generated and G has subexponential growth, then H has subexponential growth.

  • (b) Suppose that H is of finite index in G. Show that H is finitely generated and that G has subexponential growth if and only if H has subexponential growth.

• 9.18 Show that every finitely generated abelian group has subexponential growth.

• 9.19 Show that the integral Heisenberg group $H_{\mathbb {Z}}$ described in Example 9.2.19 is finitely generated and has subexponential growth. Hint: Observe that the matrices $X := M(1,0,0)$, $Y := M(0,1,0)$, and $Z := M(0,0,1)$ generate the group $H_{\mathbb {Z}}$ and that they satisfy $[X,Y] = Z$ and $[X,Z] = [Y,Z] = 1$.

• 9.20 (An example of a finitely generated amenable group with exponential growth). Consider the Baumslag-Solitar group $G := BS(1,2)$, i.e., the group of affine transformations of the real line generated by the translation $t :x \mapsto x + 1$ and the homothety $h :x \mapsto 2 x$ (cf. Example 9.2.18). We have seen in Example 9.2.18 that G is metabelian. Therefore G is amenable by Corollary 9.2.16. The goal of this exercise is to show that G has exponential growth. Let a and b be the elements of G respectively defined by $a := h^{-1}$ and $b := a t$.

  • (a) Show that if $u_1,\dots ,u_n$ and $v_1,\dots ,v_m$ are two finite sequences of elements of $\{a,b\}$ such that $u_1 \dots u_n = v_1 \dots v_m$ then $n = m$ and $u_i = v_i$ for all $1 \le i \le n$. Hint: use a ping-pong-type argument after observing that a sends the open interval (0, 1) in (0, 1 / 2) and that b sends (0, 1) in (1 / 2, 1).

  • (b) Deduce from the result of the previous question that G has exponential growth.

  • (c) Find a similar argument to prove that BS(1, n) has exponential growth for every $n \ge 2$.