Amenable Groups

Introduction

A group is amenable when it carries an invariant mean: a way of averaging bounded functions on the group that is unchanged by translation. For a discrete group the notion was isolated by von Neumann, who observed that the Banach–Tarski paradox of the free group on two generators is exactly the failure of such a mean, and that the class of groups admitting one is closed under the constructions of group theory — subgroups, quotients, extensions and direct limits — and contains all abelian and finite groups. The Følner condition turns the averaging property into a finite combinatorial statement: for every finite set $F$ and every $\varepsilon > 0$ there is a finite non-empty set $A$ with $|FA\smallsetminus A| \leq \varepsilon|A|$, so that $A$ is almost invariant under left translation. For a locally compact group the same statement is made with compact sets and the Haar measure in place of cardinality, and the averaging property is the existence of a mean on the space of bounded continuous functions.

Amenability is the exact opposite of paradoxicality: a group is amenable if and only if it has no paradoxical decomposition, so it behaves like a set on which a finitely additive invariant probability measure exists. The class is large — abelian, nilpotent, solvable, compact, and all groups of subexponential growth — and it is a quasi-isometry invariant of finitely generated groups. It is nevertheless a genuine restriction: the free groups of rank at least two are not amenable, no amenable group contains them, no infinite amenable group has property (T), and a finitely generated amenable linear group is virtually solvable by the Tits alternative. The Grigorchuk group of intermediate growth is amenable but not elementary amenable, showing that amenability is strictly larger than the class generated by the elementary constructions.

The article develops the definition by means, the explicit paradoxical decomposition of the free group, the Følner condition and its equivalence with the mean condition, Day's heredity theorem, the examples and the growth connection, the interaction with property (T) and with the ends of a group, and the Tits alternative for linear groups. The metric input is that of Geometric Group Theory, where the growth and the ends of a finitely generated group are introduced; the property (T) input is that of the companion article Property (T), which defines weak containment and the Kazhdan condition and where the failure of (T) for amenable groups is recorded; the Haar measure used in the locally compact form is that of Locally Compact Groups and Haar Measure in Part III. The mean, the Følner condition and paradoxical decomposition are defined in line, since no earlier article introduces them.

The boundary with Part III is the one fixed for this block. What is developed here is the combinatorial and group-theoretic content: means, Følner sets, paradoxical decompositions, heredity, examples and the Tits alternative. What is deferred is the analytic theory: the convolution algebra $L^1(G)$ and the approximate identity of Reiter's property, the weak containment of the trivial representation in the regular representation as a statement about $L^2$, the Banach–Tarski paradox itself, which is a theorem about measurable decompositions of the ball, and the ergodic-theoretic characterisations of amenability of actions — all of which belong to Analysis on Groups, where the measure and the limit are available. The tree-theoretic characterisation of amenable actions is not covered here. No physics is invoked.

Invariant Means and the Definition

Means on Bounded Functions

Definition. Let $G$ be a topological group and let $C_b(G)$ be the space of bounded continuous real-valued functions on $G$, with the left action of $G$ given by $(\lambda_g f)(x) = f(g^{-1}x)$. A mean on $G$ is a linear functional $m : C_b(G) \to \mathbb{R}$ such that

$$ m(f) \geq 0 \ \text{whenever } f \geq 0, \qquad m(1_G) = 1, \qquad m(\lambda_g f) = m(f) \ \text{for all } g\in G , $$

where $1_G$ denotes the constant function $1$. The group $G$ is amenable if such a mean exists. A mean is left-invariant by the third condition; replacing it by $m(f^\vee)$ with $f^\vee(x) = f(x^{-1})$ gives a right-invariant mean, and a group with a left-invariant mean has a two-sided invariant one, so the distinction is immaterial.

Definition (discrete form). For a discrete group $G$ a mean may be taken on the space $\ell^\infty(G)$ of all bounded functions, and by restriction to characteristic functions it is the same thing as a finitely additive left-invariant probability measure on the power set of $G$: a function $\mu : \mathcal{P}(G) \to [0,1]$ with $\mu(G) = 1$, $\mu(A\cup B) = \mu(A)+\mu(B)$ for disjoint $A,B$, and $\mu(gA) = \mu(A)$ for all $g \in G$. The passage between the two descriptions is by $m(\chi_A) = \mu(A)$ and linearity; the choice of $C_b(G)$, $\ell^\infty(G)$ or the characteristic functions does not change the class of amenable groups.

Proposition (basic closure). Let $G$ be amenable.

(a) Every closed subgroup $H$ of $G$ is amenable.

(b) Every quotient $G/N$ by a closed normal subgroup is amenable.

(c) If $N \trianglelefteq G$ is closed and both $N$ and $G/N$ are amenable, then $G$ is amenable, and more generally an extension of an amenable group by an amenable group is amenable.

Proof sketch. (a) A mean on $C_b(G)$ restricts to a mean on $C_b(H)$ after choosing a Bruhat-type extension of functions; since $H$ is closed, a bounded continuous function on $H$ extends to a bounded continuous function on $G$ with the same bounds (by the Tietze extension theorem applied locally and a partition argument), and the restriction of the mean is invariant because $H$ acts on the extended functions correctly. (b) A mean on $C_b(G)$ descends to a mean on $C_b(G/N)$, which is the subspace of $C_b(G)$ of functions constant on cosets, and invariance under $G$ gives invariance under $G/N$. (c) Averaging in two stages: for a bounded function $f$ on $G$, the function $x \mapsto m_N(f(x\cdot))$ on $G/N$ is bounded and continuous, and applying $m_{G/N}$ to it gives a mean on $G$; the invariance uses the invariance of $m_N$ and of $m_{G/N}$. The details are Day's and are quoted from the literature.

Remark. The invariance of the mean is the same property as the existence of a fixed point for the natural action of $G$ on the convex set of means: a group is amenable exactly when every compact convex $G$-space on which $G$ acts affinely and continuously has a fixed point. This is the fixed-point characterisation, and it is the form in which amenability is used in the rigidity theory; it is also the reason amenability contradicts property (T), since a group with (T) may fail to fix a point in a compact convex set, as in Property (T).

The Discrete Case and Tarski's Theorem

Definition. A paradoxical decomposition of a $G$-set $X$ consists of pairwise disjoint subsets $A_1,\dots,A_n, B_1,\dots,B_m$ of $X$ and elements $g_1,\dots,g_n, h_1,\dots,h_m \in G$ such that

$$ X = A_1\sqcup\cdots\sqcup A_n\sqcup B_1\sqcup\cdots\sqcup B_m, \qquad X = g_1A_1\sqcup\cdots\sqcup g_nA_n, \qquad X = h_1B_1\sqcup\cdots\sqcup h_mB_m , $$

all three unions being disjoint. The set $X$ is paradoxical if such a decomposition exists.

Theorem (Tarski). Let $G$ be a group. Then $G$ is amenable if and only if the $G$-set $G$ admits no paradoxical decomposition. Consequently a finitely additive left-invariant probability measure on the subsets of a group exists exactly when the group admits no paradoxical decomposition.

Proof sketch. If $G$ admits a paradoxical decomposition, a finitely additive invariant measure $\mu$ with $\mu(G)=1$ would give, using invariance and additivity on the two partitions with one member each translated,

$$ 1 = \mu(G) = \sum_i \mu(g_iA_i) = \sum_i \mu(A_i) , \qquad 1 = \mu(G) = \sum_j \mu(h_jB_j) = \sum_j \mu(B_j) , $$

whence $\sum_i\mu(A_i)+\sum_j\mu(B_j) = 2$, in contradiction with the disjoint partition $G = \bigsqcup_iA_i\sqcup\bigsqcup_jB_j$, which gives the same sum equal to $\mu(G) = 1$. Conversely, if no invariant mean exists one considers the family of finitely additive invariant set functions of total mass at most $1$ and shows, by a compactness argument on the finite subsets, that the failure of the mean is witnessed by finitely many sets; these produce the two families $A_i$ and $B_j$ of a paradoxical decomposition. The theorem is Tarski's and is quoted from the literature.

The Free Group is not Amenable

Theorem (von Neumann). The free group $F_2 = \langle a, b\rangle$ is not amenable.

Proof. Let $\mathcal{W}$ be the set of reduced words in $a^{\pm1}, b^{\pm1}$ and let

$$ X_1 = \{w : w \text{ begins with } a\}, \quad X_2 = \{w : w \text{ begins with } a^{-1}\}, \quad Y_1 = \{w : w \text{ begins with } b\}, \quad Y_2 = \{w : w \text{ begins with } b^{-1}\} . $$

Every reduced word other than the empty word begins with exactly one of $a, a^{-1}, b, b^{-1}$, so

$$ F_2 = X_1 \sqcup X_2 \sqcup Y_1 \sqcup Y_2 \sqcup \{e\} $$

is a partition. The map $w \mapsto aw$ is injective, and its image is the set of reduced words not beginning with $a$: indeed $w = a^{-1}u$ is reduced exactly when $u$ does not begin with $a$, and then $aw = u$. Hence

$$ F_2 = X_1 \sqcup aX_2 , $$

a disjoint union because $aX_2$ is the complement of $X_1$. By symmetry,

$$ F_2 = Y_1 \sqcup bY_2 . $$

Suppose that $m$ is a finitely additive left-invariant probability measure on the subsets of $F_2$. Invariance gives $m(aX_2) = m(X_2)$ and $m(bY_2) = m(Y_2)$, and additivity gives, from the two decompositions,

$$ m(X_1) + m(X_2) = 1, \qquad m(Y_1) + m(Y_2) = 1 . $$

Adding and using the partition of $F_2$,

$$ 2 = m(X_1)+m(X_2)+m(Y_1)+m(Y_2) = m(F_2 \smallsetminus \{e\}) = 1 - m(\{e\}) \leq 1 , $$

a contradiction. Hence no invariant mean exists and $F_2$ is not amenable.

Corollary. Every group containing a subgroup isomorphic to $F_2$ is not amenable, by the heredity under subgroups. In particular the free groups $F_n$ with $n \geq 2$, the group $SL_2(\mathbb{Z})$ and the group $GL_n(\mathbb{Z})$ for $n \geq 2$ are not amenable; a free product $G_1 * G_2$ of two nontrivial groups is not amenable except when $G_1 \cong G_2 \cong \mathbb{Z}/2$, in which case it is the infinite dihedral group, virtually $\mathbb{Z}$ and hence amenable.

Remark. The paradox above is the group-theoretic core of the classical Banach–Tarski paradox, in which the failure of amenability of the isometry group of $\mathbb{R}^3$ produced a decomposition of the ball; the analytic form of the paradox is a statement about measurable sets and is treated in Part III.

The Følner Condition

Følner Sets and the Equivalence

Definition. A Følner sequence in a finitely generated group $G$ with finite generating set $S$ is a sequence $A_1, A_2, \dots$ of finite non-empty subsets such that

$$ \frac{|S A_n \smallsetminus A_n|}{|A_n|} \longrightarrow 0 . $$

A group is Følner if it has such a sequence, or more generally if for every finite set $F \subseteq G$ and every $\varepsilon > 0$ there is a finite non-empty $A$ with $|FA \smallsetminus A| \leq \varepsilon|A|$.

Theorem (Følner). A discrete group $G$ is amenable if and only if it satisfies the Følner condition: for every finite $F\subseteq G$ and every $\varepsilon>0$ there is a finite non-empty $A\subseteq G$ with $|FA\smallsetminus A|\leq\varepsilon|A|$.

Proof sketch. If a Følner set $A$ exists, the normalised counting measure of $A$, viewed as a mean defined on finitely supported functions by $m(f) = |A|^{-1}\sum_{x\in A}f(x)$, is approximately invariant under the finitely many translations in $F$; a compactness argument — taking a limit point of the family of these means with respect to increasingly large Følner sets — produces a genuine invariant mean, the limit being taken in the weak sense of pointwise convergence on the bounded functions. Conversely, if $G$ is amenable and no Følner set existed for some finite $F$ and $\varepsilon$, then one shows that $G$ admits a paradoxical decomposition with respect to $F$, contradicting Tarski's theorem; the equivalence of the approximate invariance with amenability is a theorem of Følner and is quoted from the literature.

Example. A finite group is Følner with $A = G$. The group $\mathbb{Z}^n$ is Følner with $A_n$ the ball of radius $n$: the boundary has measure growing like $n^{n-1}$ against the volume $n^n$. A group of subexponential growth is Følner with $A_n$ a ball of suitable radius, since the ratio of the boundary to the volume tends to zero by subexponentiality. The free group $F_2$ is not Følner: in the $4$-regular Cayley tree the subgraph induced by a finite set $A$ is a forest and so has at most $|A|-1$ internal edges, so at least $2|A|+2$ edges leave $A$; passing from edges to the set $SA\smallsetminus A$ divides by at most $4$, so the boundary-to-size ratio is bounded below by $\tfrac{1}{2}$, which recovers the failure of amenability from the geometry.

The Locally Compact Case

Definition. A locally compact group $G$ is Følner if for every compact set $K \subseteq G$ and every $\varepsilon>0$ there is a compact set $A$ of positive Haar measure with

$$ \mu(KA\smallsetminus A) < \varepsilon\,\mu(A) , $$

where $\mu$ is the left Haar measure of Locally Compact Groups and Haar Measure; the set $A$ is a Følner set. The passage from the discrete definition is by replacing cardinality with Haar measure.

Theorem. A locally compact group $G$ is amenable if and only if it is Følner, with $K$ ranging over compact sets and $\mu$ the Haar measure. In particular a compact group is amenable with $A = G$, and the normalised Haar measure of a compact group is the unique invariant mean.

Proof sketch. The proof is the same as in the discrete case: the normalised Haar measures of the Følner sets give approximately invariant linear functionals on $C_b(G)$, and a weak limit point gives the mean; conversely the failure of the Følner condition produces a paradoxical decomposition in the sense of the measurable $G$-space $G$, contradicting the existence of a mean. The details are standard and are quoted from the literature.

Remark. The invariant mean of an amenable locally compact group is closely related to the Haar measure, and for a compact group it is exactly the normalised Haar measure. For a non-compact amenable group no invariant probability measure exists, and the mean is a purely finitely additive object; the analytic approximation of the mean by absolutely continuous objects is Reiter's property, whose formulation uses the convolution algebra $L^1(G)$ and therefore belongs to Part III.

Constructions and Heredity

Day's Theorem

Theorem (Day). The class of amenable groups is closed under:

(a) the passage to subgroups and to quotients;

(b) extensions: if $N$ is a closed normal subgroup of $G$ and both $N$ and $G/N$ are amenable, then $G$ is amenable;

(c) the passage to directed unions of open subgroups: if $G$ is the union of a directed family of amenable open subgroups, then $G$ is amenable;

(d) finite direct products and finite direct sums.

Proof sketch. (a) and (b) are the closure properties proved above. (c) A mean on $G$ is obtained as a limit of means on the subgroups of the directed family, using the compactness of the set of means in the topology of pointwise convergence; the family is directed and each function is supported on some subgroup, so the limit is well defined and invariant. (d) follows from (b) by induction. The theorem is Day's and is quoted from the literature.

Corollary (examples). Every abelian group is amenable; every nilpotent group is amenable, by iterating the extension property along the lower central series with abelian quotients; every solvable group is amenable, by iterating along the derived series; every compact group is amenable, by the Haar measure; every finite group is amenable; and every group of subexponential growth is amenable.

Proof. Abelian groups: for a finitely generated abelian group the Følner sets are balls in the lattice, and the general case is a directed union of finitely generated subgroups. Nilpotent and solvable: the quotients in the central and derived series are abelian, so the extension property applies inductively. Compact: the normalised Haar measure is an invariant mean. Finite: a special case of compact. Subexponential growth: the Følner condition is satisfied by balls, as above.

Elementary Amenable Groups

Definition. The class of elementary amenable groups, written $EG$, is the smallest class of groups containing all finite groups and all abelian groups and closed under the passage to subgroups, to quotients, to extensions and to directed unions. By Day's theorem every elementary amenable group is amenable; the question of the converse was open for two decades and was answered in the negative by Grigorchuk.

Theorem (Grigorchuk). The Grigorchuk group of intermediate growth is amenable but not elementary amenable. Consequently the class of amenable groups is strictly larger than the elementary class, and amenability is not generated by the elementary constructions from the finite and abelian groups.

Proof sketch. Amenability of the Grigorchuk group follows from its intermediate growth, since a group of subexponential growth satisfies the Følner condition. The failure of elementary amenability is proved by a careful analysis of the self-similar action: an elementary amenable group has a finite series with elementary amenable factors satisfying a growth restriction, and the growth of the Grigorchuk group is too close to exponential to admit such a series. The result is Grigorchuk's and is quoted from the literature.

Examples and the Growth Connection

The Standard Examples

Example (abelian, nilpotent, solvable). The groups $\mathbb{Z}^n$, the finitely generated nilpotent groups such as the Heisenberg group, and the finitely generated solvable groups are amenable. The lamplighter group $\mathbb{Z} \wr \mathbb{Z} = \bigoplus_{\mathbb{Z}}\mathbb{Z}/2\mathbb{Z} \rtimes \mathbb{Z}$ is solvable, hence amenable, and has exponential growth: the Cayley graph contains a binary tree of exponential size, so amenability does not imply subexponential growth. This is the standard example separating the two notions.

Example (compact and discrete). Every compact group is amenable, in particular every finite group; the mean is the normalised Haar measure. Every discrete group of subexponential growth is amenable; the converse fails by the lamplighter group. Every virtually abelian group is amenable. The free group $F_n$, $n \geq 2$, and every group containing it are not amenable. The fundamental group of a closed surface of genus $g \geq 2$ is not amenable, being a non-elementary hyperbolic group and so containing $F_2$; this illustrates the coincidence, among finitely generated groups, of non-amenability with the presence of a free subgroup in a large class of cases, but not in general: there exist non-amenable groups without a non-abelian free subgroup, for instance the Tarski monsters of Ol'shanskii.

Growth and the Ends

Theorem. Let $G$ be a finitely generated group.

(a) If $G$ has subexponential growth then $G$ is amenable. Hence a group of intermediate growth is amenable, and the Grigorchuk group is an amenable group of intermediate growth.

(b) If $G$ is amenable then $G$ has at most two ends; a finitely generated amenable group with two ends is virtually $\mathbb{Z}$.

(c) A finitely generated group with infinitely many ends is not amenable.

Proof sketch. (a) Balls of radius $n$ in the Cayley graph satisfy $|S A_n \smallsetminus A_n|/|A_n| \to 0$ exactly when the growth is subexponential, because the boundary of a ball is contained in the difference of the balls of radius $n+1$ and $n-1$; the Følner condition follows. (b) A group with infinitely many ends splits over a finite subgroup by the Stallings theorem, and a nontrivial splitting over a finite subgroup with an infinite vertex group produces a free subgroup of rank two, which is not amenable; hence only $0$, $1$ or $2$ ends are possible, and the two-ended case is virtually $\mathbb{Z}$ by the theorem of Hopf. (c) is the contrapositive of (b).

Quasi-Isometry Invariance

Theorem (Rosenblatt). Amenability of finitely generated groups is a quasi-isometry invariant: if $G$ and $H$ are quasi-isometric finitely generated groups, then $G$ is amenable if and only if $H$ is.

Proof sketch. The Følner condition is a statement about the existence of finite sets with small relative boundary in the Cayley graph, and a quasi-isometry carries a ball of radius $n$ to a set at finite Hausdorff distance from a ball of radius $\lambda n + C$, so the Følner property is preserved; the converse uses the quasi-inverse. The invariance is the theorem of Rosenblatt and is quoted from the literature; it also follows from the characterisation of amenability by the existence of a mean invariant under the action on the boundary at infinity, framed in Geometric Group Theory.

Amenability, Property (T) and Rigidity

Amenable Groups with Property (T)

Theorem. Let $G$ be a locally compact group that is amenable and has property (T). Then $G$ is compact. Consequently an infinite discrete amenable group never has property (T), and a lattice in a higher-rank semisimple Lie group is not amenable.

Proof sketch. By the Delorme–Guichardet form of (T), an amenable group with (T) has the trivial representation isolated but also weakly contained in the regular representation, by the Følner condition; a representation that is simultaneously weakly contained in and isolated from the trivial representation contains an invariant vector, and the resulting invariant vector for the regular representation forces $G$ to be compact. The result is standard and is quoted from the literature; it is recorded in Property (T) from the other side.

Corollary. The class of (T) groups and the class of amenable groups meet exactly in the compact groups; an infinite discrete group is amenable only if it does not have (T), and a group with (T) and no compact factor is not amenable. In particular $SL_n(\mathbb{Z})$ for $n \geq 3$ is not amenable, although its subgroups that are virtually solvable are.

The Boundary with the Ends

Remark. For a finitely presented group the amenable groups have an action-theoretic characterisation: an amenable group acting on a tree fixes a point either in the tree or in the boundary, and this is the group-theoretic shadow of the fixed-point characterisation of amenability. The tree on which the group acts and the splitting it induces are the subject, and the action of an amenable group on a hyperbolic space and its relation to the boundary at infinity is the subject of Hyperbolic Groups.

Linear Groups and the Tits Alternative

Theorem (Tits alternative). Let $\Gamma$ be a finitely generated subgroup of $GL_n(F)$ for a field $F$. Then either $\Gamma$ is virtually solvable, or $\Gamma$ contains a free subgroup of rank two.

Proof sketch. The theorem is Tits's and is proved by considering the action of $\Gamma$ on the projective space over the algebraic closure of $F$: if the action is "far from compact" in the appropriate sense, the ping-pong lemma on the projective line, or on a suitable tree when the field is non-archimedean, produces two elements generating a free subgroup; otherwise the group preserves an invariant structure and is virtually solvable. The theorem is stated in Infinite Groups in Part I, and the details are quoted from the literature.

Corollary. Let $\Gamma$ be a finitely generated linear group over a field. Then $\Gamma$ is amenable if and only if $\Gamma$ is virtually solvable. In particular a finitely generated amenable group that is linear is virtually solvable, and no finitely generated linear group is a counterexample to the equivalence of amenability with virtual solvability.

Proof. A virtually solvable group is amenable by Day's theorem and the heredity for finite-index subgroups. Conversely, if $\Gamma$ is amenable it cannot contain $F_2$, by the non-amenability of the free group and heredity; the Tits alternative then forces $\Gamma$ to be virtually solvable.

The Boundary with Analysis

The means, the Følner condition and the paradoxes are combinatorial objects; the analysis built on them is Part III.

The convolution algebra $L^1(G)$, the approximate identity of Reiter's property $P_1$, and the characterisation of amenability by the existence of functions $f_n$ with $\|g\cdot f_n - f_n\|_1 \to 0$ are Analysis on Groups in Part III, where the $L^p$ spaces are available.
The statement that the trivial representation is weakly contained in the regular representation if and only if $G$ is amenable is stated here as an equivalence; the weak containment itself is defined in Property (T), and the $L^2$ form of the statement and the theory of the group von Neumann algebra belong to Part III and to Von Neumann Algebras in Topology on Linear Algebras.
The Banach–Tarski paradox, the Hausdorff paradox and the measurable forms of the paradoxical decompositions are theorems about measures, hence Part III; the group-theoretic decomposition of the free group is proved here.
The ergodic-theoretic characterisations of amenability of a measure-preserving action, the Følner conditions for actions and the Ornstein–Weiss theory are Part III's.
What is not deferred: the definition by invariant means; the equivalence of the mean and Følner conditions; paradoxical decompositions and the free-group paradox; Day's closure properties and the elementary amenable class; the examples, the growth and ends theorems, and the quasi-isometry invariance; the incompatibility with property (T); and the Tits alternative for linear groups.

Summary

A topological group $G$ is amenable if there is a linear functional $m$ on the bounded continuous functions that is positive, unital and left-invariant; for discrete groups this is the same as a finitely additive left-invariant probability measure on the subsets. Tarski's theorem says that amenability is the absence of a paradoxical decomposition, and von Neumann's proof for $F_2 = \langle a,b\rangle$ uses the partition of the group into reduced words beginning with $a, a^{-1}, b, b^{-1}$ and the identity together with the disjoint decompositions $F_2 = X_1 \sqcup aX_2$ and $F_2 = Y_1 \sqcup bY_2$ to derive $2 \leq 1$. Hence $F_2$ is not amenable, and neither is any group containing it.

The Følner condition, $|FA\smallsetminus A|\leq\varepsilon|A|$ for a finite non-empty $A$, is equivalent to amenability for discrete groups and, with Haar measure in place of cardinality and compact sets in place of finite ones, for locally compact groups. The class of amenable groups is closed under subgroups, quotients, extensions, directed unions and finite products, so all abelian, nilpotent, solvable, compact and subexponentially growing groups are amenable; the elementary amenable groups are the smallest such class, and the Grigorchuk group of intermediate growth is amenable but not elementary amenable. The lamplighter group shows that an amenable group may have exponential growth.

Amenability is a quasi-isometry invariant of finitely generated groups; an amenable finitely generated group has at most two ends, and two ends mean virtually $\mathbb{Z}$; an amenable group with property (T) is compact, so the infinite amenable groups have no (T). A finitely generated linear group is amenable exactly when it is virtually solvable, by the Tits alternative. The analytic forms of the theory — Reiter's property, weak containment in the regular representation as an $L^2$ statement, the Banach–Tarski paradox, and the ergodic characterisations — belong to Part III.

Summary of Notation

Symbol	Meaning
$C_b(G)$	Bounded continuous real-valued functions on $G$
$\lambda_g f$	Left translate, $(\lambda_g f)(x) = f(g^{-1}x)$
mean $m$	Positive unital linear functional with $m(\lambda_gf) = m(f)$
amenable	Admits an invariant mean
$\mu$, finitely additive measure	Discrete form of a mean on $\mathcal{P}(G)$
Følner condition	For all finite $F$ and $\varepsilon$, a finite non-empty $A$ with $\vert FA\smallsetminus A\vert \leq\varepsilon\vert A\vert$
Følner set / sequence	Set or sequence satisfying the condition
paradoxical decomposition	$X = A_1\sqcup\cdots\sqcup A_n = g_1A_1\sqcup\cdots\sqcup g_nA_n$
Tarski's theorem	Amenable $\Leftrightarrow$ no paradoxical decomposition
$F_2 = \langle a,b\rangle$	The free group of rank two, non-amenable
$X_1,X_2,Y_1,Y_2$	Reduced words beginning with $a,a^{-1},b,b^{-1}$; $F_2 = X_1\sqcup X_2\sqcup Y_1\sqcup Y_2\sqcup\{e\}$
$EG$	Elementary amenable groups (Day closure of finite and abelian groups)
$\mathbb{Z}\wr\mathbb{Z}$	Lamplighter; amenable, exponential growth
Grigorchuk group	Amenable, intermediate growth, not elementary amenable
Ends	Amenable f.g. groups have $\leq 2$ ends (Hopf; see Geometric Group Theory)
$1_G \prec \lambda_G$	Trivial representation weakly contained in the regular one (amenability criterion; see Property (T))
$P_1$, Reiter's property	Analytic form of the Følner condition (Part III)
Tits alternative	F.g. linear group: virtually solvable or contains $F_2$
$\mu(A)$, Haar measure	Used in the locally compact Følner condition (see Locally Compact Groups and Haar Measure)