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Amenable group

In mathematics, particularly within group theory, an amenable group is a discrete or locally compact group that admits a finitely additive, left-invariant probability measure on its power set, equivalently, one that possesses a left-invariant mean on the space of bounded continuous functions.^[1] This property captures groups that avoid certain paradoxical decompositions, such as those arising in the Banach-Tarski paradox, and serves as a foundational concept in areas like ergodic theory, operator algebras, and geometric group theory.^[2] The notion of amenability originated in John von Neumann's 1929 work on the measurement of sets in the context of the special unitary group, where he identified the existence of an invariant mean as a key obstruction to paradoxical behavior in non-abelian groups.^[2] Von Neumann demonstrated that finite groups and abelian groups satisfy this condition, and he showed that the free group on two generators does not, establishing it as a canonical example of a non-amenable group.^[1] In 1949, Mahlon M. Day formalized and popularized the terminology "amenable group," extending von Neumann's ideas to broader classes and introducing the subclass of elementary amenable groups, generated by finite and abelian groups under extensions and increasing unions.^[3] Prominent examples of amenable groups include all abelian groups, compact groups (via the Haar measure), nilpotent groups, and solvable groups, while non-amenable examples encompass free groups on two or more generators, the special linear group SL(3,ℤ), and many hyperbolic groups.^[4] The class of amenable groups is closed under taking subgroups, quotients, and extensions by amenable groups, but not under free products, which often yield non-amenable structures.^[1] Amenability has profound implications, such as the validity of certain ergodic theorems and the vanishing of bounded cohomology in positive degrees for amenable groups.^[5]

Definitions and Basic Concepts

Definition for locally compact groups

A locally compact group is a topological group equipped with a topology that is Hausdorff and locally compact, meaning every point has a compact neighborhood basis. Central to the study of such groups is the left Haar measure, a non-trivial Borel measure \mu on G that is left-invariant under the group operation—that is, \mu(gE) = \mu(E) for all g \in G and measurable sets E \subseteq G—finite on compact subsets, and positive on non-empty open sets. This measure normalizes the group structure by providing a canonical way to integrate functions over G, facilitating the definition of convolutions and other operations in harmonic analysis on groups. The primary definition of amenability for a locally compact group G relies on the existence of a left-invariant mean on the space L^\infty(G) of essentially bounded measurable functions on G with respect to a fixed left Haar measure \mu. Specifically, G is amenable if there exists a positive linear functional m: L^\infty(G) \to \mathbb{C} such that m(1) = 1 and m(f \circ \lambda_g) = m(f) for all g \in G and all f \in L^\infty(G), where \lambda_g: G \to G denotes left translation by g, defined by \lambda_g(h) = gh. Such a mean extends the intuitive notion of averaging over the group while preserving invariance under left translations. For functions f \in L^\infty(G) and test functions \phi \in L^1(G), the invariance property implies

m(f * \phi) = m(f) \int_G \phi \, d\mu,

where f * \phi is the convolution (f * \phi)(x) = \int_G f(y) \phi(y^{-1}x) \, d\mu(y). This formulation underscores the measure-theoretic perspective, where the mean behaves like integration against an invariant "probability" measure, though no such finitely additive extension may exist on all subsets. The concept of amenability via invariant means originated in the work of John von Neumann in 1929, who developed it to characterize groups admitting no paradoxical decompositions with respect to free group actions, thereby avoiding measure-theoretic paradoxes like the Banach-Tarski phenomenon.^[6] This foundational idea was later generalized and formalized for locally compact groups, establishing amenability as a key property linking group structure, measure theory, and ergodic behavior.

Definition for discrete groups

A discrete group G is amenable if there exists a finitely additive probability measure \mu on the power set \mathcal{P}(G) such that \mu(E) = \mu(gE) for all g \in G and subsets E \subseteq G, with \mu(G) = 1.^[7] This measure is left-invariant under the group action and extends the notion of a translation-invariant "size" function without requiring \sigma-additivity, capturing a combinatorial form of uniformity across the group elements.^[7] This condition is equivalent to the existence of a left-invariant mean m on \ell^\infty(G), the space of bounded functions on G equipped with the supremum norm.^[8] Specifically, m is a positive linear functional with m(1) = 1 satisfying m(T_g f) = m(f) for all g \in G and f \in \ell^\infty(G), where the left translation operator is defined by (T_g f)(h) = f(g^{-1} h). In terms of the measure, the mean can be expressed as

m(f) = \sum_{g \in G} f(g) \, \mu(\{g\}),

though the summation is formal since \mu is only finitely additive.^[7]^[8] Unlike the definition for locally compact groups, which relies on integration with respect to a Haar measure over Borel sets, the discrete case uses the power set \mathcal{P}(G) and the counting measure as its underlying "Haar" measure, emphasizing finite additivity to handle the lack of topology.^[7] This combinatorial perspective originates from efforts to resolve Tarski's problem on the existence of invariant measures for group actions, particularly in avoiding paradoxical decompositions in discrete settings.^[7]

Equivalent Characterizations

Invariant means and Banach limits

A locally compact group G is amenable if and only if there exists a left-invariant mean m on L^\infty(G), that is, a positive linear functional m: L^\infty(G) \to \mathbb{R} such that m(1) = 1 and m(\lambda_g f) = m(f) for all g \in G and f \in L^\infty(G), where \lambda_g f(x) = f(g^{-1}x) denotes the left translation action.^[7] For discrete groups, this specializes to a mean on \ell^\infty(G) with the counting measure, providing a functional-analytic characterization of amenability.^[7] Invariant means on amenable groups can be constructed using free ultrafilters. Specifically, for an amenable group G, consider the Cesàro means along Følner sequences, and apply a free ultrafilter on the natural numbers to obtain limits that yield a translation-invariant functional on bounded functions.^[9] This ultrafilter construction ensures the resulting mean is positive and normalized, extending the standard limit on convergent sequences while preserving invariance under the group action.^[9] Banach limits provide a concrete realization of such invariant means, particularly for the integers \mathbb{Z}. A Banach limit L: \ell^\infty(\mathbb{N}) \to \mathbb{R} is a linear functional extending the usual limit on convergent sequences, satisfying L(x) \geq 0 if x_n \geq 0 for all n, L(1) = 1, and shift-invariance L(Sx) = L(x), where (Sx)_n = x_{n+1}.^[10] For \mathbb{Z}, extending this to bidirectional shifts on \ell^\infty(\mathbb{Z}) via the Hahn-Banach theorem yields a left-invariant mean, proving that \mathbb{Z} (and more generally abelian groups) is amenable.^[10] In general, for amenable groups, Banach limits generalize to group-invariant functionals on \ell^\infty(G) by analogous extensions that respect the left regular representation.^[10] Such means are not unique, as the space of invariant means is a convex set with extreme points corresponding to different ultrafilter limits; however, all invariant means agree on the subspace of invariant functions, those fixed by the group action.^[7]

Følner sequences and condition

A discrete group G is amenable if and only if there exists a Følner sequence (F_n)_{n \in \mathbb{N}} consisting of nonempty finite subsets of G such that for every g \in G,

\lim_{n \to \infty} \frac{|g F_n \Delta F_n|}{|F_n|} = 0,

where \Delta denotes the symmetric difference of sets.^[11] This condition provides a combinatorial characterization of amenability, emphasizing the existence of "large" finite sets that are almost invariant under left translation by any fixed group element. An equivalent formulation, particularly for finitely generated groups with a finite symmetric generating set S, involves sets with small boundary in the Cayley graph: G is amenable if and only if \inf \{ \frac{1}{|F|} \max_{s \in S} |s F \Delta F| : F \subseteq G, F \text{ finite}, |F| > 0 \} = 0 .^[7] This measures the average "leakage" under translations by generators, and the infimum condition ensures sets can be found with negligible relative boundary. The proof of equivalence between the Følner condition and the existence of a left-invariant mean on \ell^\infty(G) proceeds in two directions: if a Følner sequence exists, one constructs an approximate invariant mean by averaging functions over the sets F_n, yielding a sequence of functionals that converges weakly to a genuine invariant mean; conversely, starting from an invariant mean, one extracts a Følner sequence by selecting sets where the mean concentrates appropriately to achieve near-invariance.^[11] This link highlights how the geometric notion of Følner sequences approximates the functional-analytic property of invariant means. For locally compact groups, the Følner condition generalizes to sequences of relatively compact sets F_n with positive left Haar measure \mu(F_n) > 0, such that for every compact subset K \subseteq G and \varepsilon > 0, there exists n with \sup_{g \in K} \mu(g F_n \Delta F_n) / \mu(F_n) < \varepsilon, adjusted by the modular function \Delta(g) when considering right translates to ensure compatibility with right-invariant means.^[12] This adaptation accounts for the non-unimodular case, where \mu(F g) = \Delta(g) \mu(F). The Følner condition proves particularly useful for establishing amenability in polycyclic groups, where explicit constructions of Følner sequences can be derived from chains of normal subgroups and coset decompositions. In contrast, Day generalized the condition to amenable semigroups by replacing group inverses with suitable one-sided approximations to invariance, allowing the characterization to extend beyond invertible structures.^[13]

Properties and Structural Implications

Hereditary and stability properties

Amenability exhibits strong hereditary properties with respect to subgroups and quotients. Specifically, if a group G is amenable, then every subgroup H \leq G is amenable, and every quotient group G/N (for N \trianglelefteq G) is amenable.^[7] To see this for subgroups, let \mu be a left-invariant mean on \ell^\infty(G). Let M be a right transversal for H in G. For f \in \ell^\infty(H), extend to \tilde{f} \in \ell^\infty(G) by \tilde{f}(hm) = f(h) for h \in H, m \in M, and \tilde{f} zero elsewhere if needed; then \nu(f) = \mu(\tilde{f}) defines a left-invariant mean on \ell^\infty(H). For quotients, given \mu on \ell^\infty(G), for \phi \in \ell^\infty(G/N) define \tilde{\phi}(g) = \phi(gN); then \lambda(\phi) = \mu(\tilde{\phi}) is a left-invariant mean on \ell^\infty(G/N). In the context of induced representations, the quotient construction corresponds to the module action where functions on the quotient act on functions constant on cosets.^[14]^[7] Amenability is also stable under group extensions. If N \trianglelefteq G is a normal amenable subgroup and the quotient G/N is amenable, then G itself is amenable; this result is known as Day's theorem. The proof constructs an invariant mean on G by combining invariant means on N and G/N: for f \in \ell^\infty(G), average first over N-orbits using the mean on N to descend to G/N, then apply the mean on G/N.^[14]^[7] Further stability holds under finite extensions, as finite groups are amenable (admitting the uniform mean), so adjoining a finite index amenable supergroup preserves amenability via the extension theorem.^[7] The converse properties do not hold: there exist non-amenable groups containing amenable subgroups. For instance, the free group on two generators F_2 is non-amenable but admits cyclic subgroups, which are amenable as abelian groups.

Relations to paradoxical decompositions and growth

A paradoxical decomposition of a group G consists of finite disjoint subsets P_1, \dots, P_m, Q_1, \dots, Q_n \subseteq G (with m, n \geq 1) and group elements g_1, \dots, g_m, h_1, \dots, h_n \in G such that G = \bigcup_{i=1}^m g_i P_i = \bigcup_{j=1}^n h_j Q_j.^[15] The Tarski number T(G) is the infimum of m + n over all such decompositions if they exist, and T(G) = \infty otherwise.^[16] A group G is amenable if and only if it admits no paradoxical decomposition, i.e., T(G) = \infty; this equivalence is a theorem of Tarski.^[7] Non-amenable groups like the free group F_2 admit paradoxical decompositions with T(F_2) = 4, as shown by partitioning F_2 using its generators and their powers.^[16] This characterization links amenability directly to the absence of Banach-Tarski-type paradoxes in group actions, such as the decomposition of the unit ball in \mathbb{R}^3 using rotations from the non-amenable group \mathrm{SO}(3), which contains a copy of F_2.^[7] For discrete groups, the existence of a paradoxical decomposition implies the failure of any left-invariant finitely additive probability measure on the power set of G, underscoring the measure-theoretic foundation of amenability.^[15] Regarding growth, finitely generated amenable groups exhibit diverse asymptotic behaviors in their word growth functions, measured by the cardinality of balls B_S(r) = \{g \in G : d_S(g, e) \leq r\} in the Cayley graph with respect to a finite generating set S. Groups with subexponential growth, where \lim_{r \to \infty} |B_S(r)|^{1/r} = 1, are necessarily amenable, as the growth allows construction of a Følner sequence satisfying the amenability condition.^[7] Conversely, amenable groups can have exponential growth, with growth rate \lim_{r \to \infty} |B_S(r)|^{1/r} > 1; for example, certain finitely generated solvable groups that are not virtually nilpotent have exponential growth while remaining amenable.^[17] The growth rate of an m-generated amenable group is at most $2m - 1, the maximum for any m-generated group, but amenable examples can approach this bound arbitrarily closely, as demonstrated by quotients of free groups by normal subgroups with long relations.^[18] This flexibility highlights that amenability imposes no strict upper bound on growth beyond the general combinatorial limit, distinguishing it from stronger conditions like polynomial growth, which imply nilpotency by Gromov's theorem.^[7]

Examples of Amenable Groups

Abelian and solvable groups

All abelian locally compact groups are amenable, with the left-invariant Haar measure serving as the basis for constructing an invariant mean on the space of bounded continuous functions. For the specific case of the discrete group \mathbb{Z}^d, amenability follows from the existence of Følner sequences consisting of cubes [-n, n]^d, where the uniform probability measure on these sets converges to a left-invariant mean, analogous to the Lebesgue measure in the continuous setting.^[19] In general, for a locally compact abelian group G, an invariant mean can be obtained by averaging over the cosets of compact subgroups, leveraging the bi-invariance of the Haar measure to ensure left-invariance under group translations. A concrete illustration of this is the additive group \mathbb{Q} of rational numbers, which is amenable despite being non-finitely generated; as an abelian group, it admits a left-invariant finitely additive probability measure on all subsets, constructed via limits of Følner sequences from finite approximations.^[19] Solvable groups extend this amenability further. Virtually solvable groups—those containing a finite-index solvable subgroup—are amenable, as the class of amenable groups is closed under finite extensions and subgroups.^[19] For solvable groups themselves, amenability holds by induction on the derived length: the base case of derived length 1 (abelian groups) is established as above, and assuming amenability for derived length k, for a solvable group of derived length k+1, the derived subgroup G' has derived length at most k and is thus amenable, while the quotient G/G' is abelian (derived length 1) and amenable; since amenable groups are closed under extensions, G is amenable.^[19] An example is the discrete Heisenberg group over the integers, defined as triples (x, y, z) \in \mathbb{Z}^3 with multiplication (x_1, y_1, z_1) \cdot (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2 + x_1 y_2), which is nilpotent (hence solvable) and amenable; it possesses Følner sequences given by boxes [1, n]^3, where the boundary measure relative to the total size tends to zero as n \to \infty.^[20]

Nilpotent and virtually nilpotent groups

All finitely generated nilpotent groups are amenable.^[21] This follows from an inductive argument on the nilpotency class: abelian groups (class 1) are amenable as they admit translation-invariant means on \ell^\infty(G), and for higher class c, a nilpotent group G has amenable center Z(G) and amenable central quotient G/Z(G), allowing construction of an invariant mean on G via the short exact sequence $1 \to Z(G) \to G \to G/Z(G) \to 1. Moreover, such groups exhibit polynomial growth, where the ball of radius rin the word metric has [cardinality](/page/Cardinality) asymptotic tor^dwithdequal to the Hirsch lengthh(G)$, the sum of the ranks of the abelian factors in the lower central series.^[22] Virtually nilpotent groups, those containing a finite-index nilpotent subgroup H, are also amenable. If [G:H]=k < \infty, an invariant mean on G can be obtained by averaging a mean on H over the k left cosets of H in G.^[23] This preserves the polynomial growth property, with degree again bounded by h(H).^[24] A concrete example is the discrete Heisenberg group H_3(\mathbb{Z}), consisting of $3 \times 3 upper-triangular integer matrices with 1s on the diagonal, generated by

x = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad y = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix},

which is nilpotent of class 2 and Hirsch length 3. Følner sets for H_3(\mathbb{Z}) can be constructed as products of symmetric intervals in the coordinates, such as F_N = [-N,N] \times [-N,N] \times [-N^2, N^2], which satisfy the Følner condition with boundary size O(N^2) relative to |F_N| \sim N^4.^[25] For contrast, Baumslag-Solitar groups BS(1,n) = \langle a,t \mid t a t^{-1} = a^n \rangle are amenable as solvable groups of derived length 2, but they are virtually nilpotent only when n = \pm 1 (reducing to \mathbb{Z}^2), highlighting that amenability holds more broadly than virtual nilpotency.^[26]

Non-Amenable Groups and Counterexamples

Free groups and surface groups

Non-abelian free groups F_r on r \geq 2 generators are non-amenable, as established by von Neumann in his foundational work on the subject.^[27] These groups admit paradoxical decompositions, which can be constructed using the ping-pong lemma applied to their natural action on the Cayley tree. For the free group F_2 = \langle a, b \rangle, an explicit paradoxical decomposition arises from partitioning the group elements based on their reduced word representations. Define A as the set of non-identity elements whose reduced words begin with a or a^{-1}, and B as the set whose reduced words begin with b or b^{-1}. Then A \cup B = F_2 \setminus \{e\}, with A and B disjoint, and both A \cong F_2 and B \cong F_2 via bijections preserving the group structure (mapping generators appropriately to account for the initial letters). This partition facilitates a decomposition showing that F_2 can be equidecomposable to two copies of itself using left translations by group elements, confirming non-amenability. The fundamental groups \pi_1(\Sigma_g) of closed orientable surfaces \Sigma_g of genus g \geq 2 are likewise non-amenable, as they contain non-abelian free subgroups of rank 2. For instance, in the genus-2 case with presentation \langle a,b,c,d \mid [a,b][c,d] = 1 \rangle, the subgroup generated by a and c is free on two generators, embedding F_2 non-trivially. More generally, the hyperbolic geometry of \Sigma_g (via its uniformization as a quotient of the hyperbolic plane) ensures such free embeddings, as the fundamental group acts properly discontinuously on \mathbb{H}^2 with ping-pong dynamics yielding free subgroups. This embedding implies non-amenability, as subgroups of amenable groups are amenable. In these groups, no Følner sequence exists because boundary ratios in the Cayley graph remain bounded away from zero: for any finite set F, the relative boundary size |\partial F| / |F| satisfies \inf |\partial F| / |F| \geq \delta > 0, violating the Følner condition. This uniform non-amenability holds for non-elementary hyperbolic groups, including free and surface groups. Additionally, \mathrm{SL}(3,\mathbb{Z}) contains a free subgroup of rank 2 (via explicit matrix pairs satisfying ping-pong in the projective space), rendering it non-amenable.^[28]

Groups with rapid growth or paradoxical actions

Hyperbolic groups provide a key class of examples of non-amenable groups characterized by rapid growth. According to Gromov's foundational work, finitely generated hyperbolic groups are either virtually cyclic or have exponential growth.^[29] Non-elementary hyperbolic groups, which are those that are infinite and not virtually cyclic, exhibit uniform exponential growth with rate \lambda > 1.^[30] Such groups are non-amenable, as their rapid growth precludes the existence of Følner sequences; specifically, the uniform exponential growth rate \lambda > 1 ensures that no sequence of finite sets satisfies the Følner condition, since the boundary of balls grows too quickly relative to their volume.^[30] Lattices in non-amenable semisimple Lie groups offer another important family of non-amenable discrete groups. For instance, irreducible lattices in \mathrm{SL}(n, \mathbb{R}) for n \geq 3, such as \mathrm{SL}(n, \mathbb{Z}), are non-amenable because they contain non-abelian free subgroups, as established by the ping-pong lemma applied to suitable elements. Paradoxical actions also characterize non-amenability, particularly for groups acting on trees. Groups admitting a free action on a tree without global fixed points exhibit paradoxical decompositions, rendering them non-amenable by Tarski's theorem linking such actions to the failure of invariant means. A prominent example is the outer automorphism group \mathrm{Out}(F_n) for n \geq 2. For n=2, \mathrm{Out}(F_2) \cong \mathrm{GL}(2, \mathbb{Z}) contains free subgroups and is non-amenable. For n \geq 3, it is non-amenable as it lacks inner amenability.^[31] The growth criterion extends more broadly: any finitely generated group with uniform exponential growth rate \lambda > 1 fails the Følner condition and is hence non-amenable, providing a geometric obstruction to amenability beyond specific classes like hyperbolic groups.^[30] Von Neumann conjectured that a group is non-amenable if and only if it contains a subgroup isomorphic to the free group on two generators. This was disproved in 1980 by Ol'shanskii, who constructed finitely presented non-amenable groups without non-abelian free subgroups.^[32] Up to 2025, no significant revisions to these foundational results have emerged, though connections to other rigidity properties persist; notably, infinite discrete groups with Kazhdan's property (T) are non-amenable, as property (T) precludes nontrivial invariant means on \ell^\infty(G).^[7]

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