, an amenable group
is a locally compact topological group G
carrying a kind of averaging operation on bounded functions that is invariant
under left (or right) 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.
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 (which can be made precise) is that the support of the regular representation is the whole space of irreducible representations.
In discrete group theory, where has no topological structure, a simpler definition is used. In this setting, a group is amenable if one can say what percentage of any given subset takes up.
If a group has a Følner sequence then it is automatically amenable.
Locally compact definition
be a locally compact group
be the Banach space
of all essentially bounded functions
with respect to the Haar measure
A linear functional on is called a mean if it maps the constant function to 1 and non-negative functions to non-negative numbers.
Let be the left action of on ,
Then, a mean is said to be left-invariant if
for all and
Similarly, is said to be right-invariant if
where is the right action
A locally compact group is amenable if there is a left- (or right-)invariant mean on
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 is amenable if there is a measure—a function that assigns to each subset of a number from 0 to 1—such that
- The measure is a probability measure: the measure of the whole group is 1.
- The measure is finitely additive: given finitely many disjoint subsets of , the measure of the union of the sets is the sum of the measures.
- The measure is left-invariant: given a subset and an element of , the measure of equals the measure of . ( denotes the set of elements for each element in . That is, each element of is translated on the left by .)
This definition can be summarized thus: is amenable if it has a finitely-additive left-invariant probability measure. Given a subset of , the measure can be thought of as answering the question: what is the probability that a random element of is in ?
It is a fact that this definition is equivalent to the definition in terms of .
Having a measure on allows us to define integration of bounded functions on . Given a bounded function , 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 , the function is a right-invariant measure. Combining these two gives a bi-invariant measure:
Conditions for a discrete group
The following conditions are equivalent for a countable discrete group Γ:
- Γ is amenable.
- If Γ acts by isometries on a (separable) Banach space E, leaving a weakly closed convex subset C of the closed unit ball of E* invariant, then Γ has a fixed point in C.
- There is a left invariant norm-continuous functional μ on l∞(Γ) with μ(1) = 1 (this requires the axiom of choice).
- There is a left invariant state μ on any left invariant separable unital C* subalgebra of l∞(Γ).
- There is a set of probability measures μn on Γ such that ||g · μn - μn||1 tends to 0 for each g in Γ (M.M. Day).
- There are unit vectors xn in l2(Γ) such that ||g · xn - xn||2 tends to 0 for each g in Γ (J. Dixmier).
- There are subsets Sn of Γ such that | g · Sn Δ Sn | / |Sn| tends to 0 for each g in Γ (Følner).
- If μ is a symmetric probability measure on Γ with support generating Γ, then convolution by μ defines an operator of norm 1 on l2(Γ) (Kesten).
- If Γ acts by isometries on a (separable) Banach space E and f in l∞(Γ, E*) is a bounded 1-cocycle, i.e. f(gh) = f(g) + g·f(h), then f is a 1-coboundary, i.e. f(g) = g·φ - φ for some φ in E* (B.E. Johnson).
- The von Neumann group algebra of Γ is hyperfinite (A. Connes).
- Finite groups are amenable. Use the counting measure with the discrete definition.
- Subgroups of amenable groups are amenable.
- The direct product of two amenable groups is amenable, while the direct product of an infinite family of amenable groups need not be.
- The group of integers is amenable (they have a Følner sequence).
- A group is amenable if all its finitely generated subgroups are. That is, locally amenable groups are amenable.
- A group is amenable if it has an amenable normal subgroup such that the quotient is amenable. That is, extensions of amenable groups by amenable groups are amenable.
- It follows that a group is amenable if it has a finite index amenable subgroup. That is, virtually amenable groups are amenable.
- Furthermore, it follows that all solvable groups are amenable.
- Compact groups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).
- Finitely generated groups of subexponential growth are amenable.
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. In 2002, Sapir and Olshankii found finitely generated
counterexamples: non-amenable finitely presented groups
that have periodic normal subgroups of finite index.
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 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.
- F.P. Greenleaf, Invariant Means on Topological Groups and Their Applications, Van Nostrand Reinhold (1969).
- V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer (2002).
- M. Takesaki, Theory of Operator Algebras, Vol. 2 and 3, Springer.
- J. von Neumann, Zur allgemeinen Theorie des Maßes, Fund. Math. 13 (1929), 73−111.