Amenable group

In mathematics, an amenable group is a topological group G carrying a kind of averaging operation, that is invariant under translations by group elements. In the case where G is not an abelian group, that means translation on a fixed side (left- or right-translation).

The amenability property has a large number of equivalent formulations. In the field of analysis, the definition is in terms of linear functionals. An intuitive way to understand this version, that can be made precise, is that the support of the regular representation is the whole space of irreducible representations.

In discrete group theory, on the other hand, a simpler definition is used, in which <math>G<math> has no topological structure. In this setting, a group is amenable if you can say what percentage of <math>G<math> any given subset takes up.

If a group has a Følner sequence then it is automatically amenable.

Contents

1 Amenability in general 2 Amenability of discrete groups 3 Examples of amenable groups 4 Examples of non-amenable groups

Amenability in general

Let <math>G<math> be a locally compact group and <math>L^\infty(G)<math> be the Banach space of all essentially bounded functions <math>G \to <math>R with respect to the Haar measure.

Definition 1. A linear functional on <math>L^\infty(G)<math> is called a mean if it maps the constant function <math>f(g) = 1<math> to 1 and non-negative functions to non-negative numbers.

Definition 2. Let <math>L_g<math> be the left action of <math>g \in G<math> on <math>f \in L^\infty(G)<math>, i.e. <math>(L_g f)(h) = f(gh)<math>. Then, a mean <math>\mu<math> is said to be left-invariant if <math>\mu(L_g f) = \mu(f)<math> for all <math>g \in G<math> and <math>f \in L^\infty(G).<math> Similarly, right-invariant if <math>\mu(R_g f) = \mu(f),<math> where <math>R_g<math> is the right action <math>(R_g f)(h) = f(hg).<math>

Definition 3. A locally compact group <math>G<math> is amenable if there is a left- (or right-)invariant mean on <math>L^\infty(G).<math>

Amenability of discrete groups

The definition of amenability is quite a lot simpler in the case of a discrete group, i.e. a group with no topological structure.

Definition. A discrete group <math>G<math> is amenable if there is a measure—a function that assigns to each subset of <math>G<math> a number from 0 to 1—such that

1. The measure is a probability measure: the measure of the whole group <math>G<math> is 1.
2. The measure is finitely additive: given finitely many disjoint subsets of <math>G<math>, the measure of the union of the sets is the sum of the measures.
3. The measure is left-invariant: given a subset <math>A<math> and an element <math>g<math> of <math>G<math>, the measure of <math>A<math> equals the measure of <math>gA<math>. (<math>gA<math> denotes the set of elements <math>ga<math> for each element <math>a<math> in <math>A<math>. That is, each element of <math>A<math> is translated on the left by <math>g<math>.)

This definition can be summarized thus: <math>G<math> is amenable if it has a finitely-additive left-invariant probability measure. Given a subset <math>A<math> of <math>G<math>, the measure can be thought of as answering the question: what is the probability that a random element of <math>G<math> is in <math>A<math>?

It is a fact that this definition is equivalent to the definition in terms of <math>L^\infty(G)<math>.

Having a measure <math>\mu<math> on <math>G<math> allows us to define integration of bounded functions on <math>G<math>. Given a bounded function <math>f:G\to\mathbf{R}<math>, the integral

<math>\int_G f\,d\mu<math>

is defined as in Lebesgue integration. (Note that some of the properties of the Lebesgue integral fail here, since our measure is only finitely-additive.)

If a group has a left-invariant measure, it automatically has a bi-invariant one. Given a left-invariant measure <math>\mu<math>, the function <math>\mu^-(A)=\mu(A^{-1})<math> is a right-invariant measure. Combining these two gives a bi-invariant measure:

<math>\nu(A)=\int_{g\in G}\mu(Ag^{-1})d\mu^-.<math>

Examples of amenable groups

• Finite groups are amenable. Use the counting measure with the discrete definition.
• Subgroups of amenable groups are amenable.
• The direct product of amenable groups is amenable.
  • In fact, a group is amenable if it has an amenable normal subgroup such that the quotient is amenable. That is, amenable by amenable is amenable.
    • It follows that a group is amenable if it has a finite index amenable subgroup. That is, virtually amenable groups are amenable.
• A group is amenable if all its finitely generated subgroups are. That is, locally amenable groups are amenable.
  • By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable.
• Finitely generated groups of subexponential growth are amenable.
• Solvable groups are amenable.
• Compact groups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).

Examples of non-amenable groups

If a group contains a (non-abelian) free subgroup on two generators, then it is not amenable. The converse to this statement is the so-called von Neumann conjecture, which was disproved in 1980.

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