Amenable group

In mathematics, an amenable group is a locally compact topological group "G" carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of "G", was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".
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 is that the support of the regular representation is the whole space of irreducible representations.
In discrete group theory, where "G" has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one can say what proportion of "G" any given subset takes up.
If a group has a Følner sequence then it is automatically amenable.
Definition for locally compact groups.
Let "G" be a locally compact Hausdorff group. Then it is well known that it possesses a unique, up-to-scale left- (or right-) rotation invariant nontrivial ring measure, the Haar measure. (This is a Borel regular measure when "G" is second-countable; there are both left and right measures when "G" is compact.) Consider the Banach space "L"∞("G") of essentially bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).
Definition 1. A linear functional Λ in Hom("L"∞("G"), R) is said to be a mean if Λ has norm 1 and is non-negative, i.e. "f" ≥ 0 a.e. implies Λ("f") ≥ 0.
Definition 2. A mean Λ in Hom("L"∞("G"), R) is said to be left-invariant (resp. right-invariant) if Λ("g"·"f") = Λ("f") for all "g" in "G", and "f" in "L"∞("G") with respect to the left (resp. right) shift action of "g"·"f"(x) = "f"("g"−1"x")(resp. "f"·"g"(x) = "f"("xg"−1) ).
Definition 3. A locally compact Hausdorff group is called amenable if it admits a left- (or right-)invariant mean.
Equivalent conditions for amenability.
 contains a comprehensive account of the conditions on a second countable locally compact group "G" that are equivalent to amenability:
Case of discrete groups.
The definition of amenability is simpler in the case of a discrete group, i.e. a group equipped with the discrete topology.
Definition. A discrete group "G" is amenable if there is a finitely additive measure (also called a mean) —a function that assigns to each subset of "G" a number from 0 to 1—such that
This definition can be summarized thus: "G" is amenable if it has a finitely-additive left-invariant probability measure. Given a subset "A" of "G", the measure can be thought of as answering the question: what is the probability that a random element of "G" is in "A"?
It is a fact that this definition is equivalent to the definition in terms of "L"∞("G").
Having a measure μ on "G" allows us to define integration of bounded functions on "G". Given a bounded function "f" : "G" → R, the integral
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 μ, the function μ−("A") = μ("A"−1) is a right-invariant measure. Combining these two gives a bi-invariant measure:
The equivalent conditions for amenability also become simpler in the case of a countable discrete group Γ. For such a group the following conditions are equivalent:
Note that A. Connes also proved that the von Neumann group algebra of any connected locally compact group is hyperfinite, so the last condition no longer applies in the case of connected groups.
Amenability is related to spectral theory of certain operators. For instance, the fundamental group of a closed Riemannian manifold is amenable if and only if the bottom of the spectrum of the Laplacian on the L2-space of the universal cover of the manifold is 0.
Examples.
All examples above are elementary amenable. The first class of examples below can be used to exhibit non-elementary amenable examples thanks to the existence of groups of intermediate growth.
Nonexamples.
If a countable discrete 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 by Olshanskii in 1980 using his "Tarski monsters". Adyan subsequently showed that free Burnside groups are non-amenable: since they are periodic, they cannot contain the free group on two generators. These groups are finitely generated, but not finitely presented. However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely presented groups that have a periodic normal subgroup with quotient the integers.
For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative: every subgroup of GL("n","k") with "k" a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators. Although Tits' proof used algebraic geometry, Guivarc'h later found an analytic proof based on V. Oseledets' multiplicative ergodic theorem. Analogues of the Tits alternative have been proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes of non-positive curvature.