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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.
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
- The measure is a probability measure: the measure of the whole group <math>G<math> is 1.
- 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.
- 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>?
In this setting, a left-invariant measure is automatically right-invariant.
It is a fact that this definition is equivalent to the definition in terms of <math>L^\infty(G)<math>.
Examples of amenable groups
All finite groups, all abelian groups, all finitely generated groups of subexponential growth are amenable.
Compact groups are amenable as 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.
Reference
The initial version of this page was based on http://planetmath.org/encyclopedia/AmenableGroup.html.
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