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). Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

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. Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself. ... In mathematics, the term irreducible is used in several ways. ...

In discrete group theory, on the other hand, a simpler definition is used, in which G has no topological structure. In this setting, a group is amenable if you can say what percentage of G any given subset takes up. In mathematics, a discrete group is a group G equipped with the discrete topology. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...

If a group has a Følner sequence then it is automatically amenable. In mathematics, a Følner sequence for a group is a sequence of sets satisfying a particular condition. ...

Amenability in general

Let G be a locally compact group and be the Banach space of all essentially bounded functions with respect to the Haar measure. In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ...

Definition 1. A linear functional on is called a mean if it maps the constant function f(g) = 1 to 1 and non-negative functions to non-negative numbers.

Definition 2. Let L_g be the left action of on , i.e. (L_gf)(h) = f(gh). Then, a mean μ is said to be left-invariant if μ(L_gf) = μ(f) for all and Similarly, μ is said to be right-invariant if μ(R_gf) = μ(f), where R_g is the right action (R_gf)(h) = f(hg).

Definition 3. A locally compact group G is amenable if there is a left- (or right-)invariant mean on

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. In mathematics, a discrete group is a group G equipped with the discrete topology. ...

Definition. A discrete group G is amenable if there is a measure—a function that assigns to each subset of G a number from 0 to 1—such that In mathematics, a measure is a function that assigns a number, e. ...

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

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 .

Having a measure μ on G allows us to define integration of bounded functions on G. 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.) The integral can be interpreted as the area under a curve. ...

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:

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).

In mathematics, a finite group is a group which has finitely many elements. ... In mathematics, the counting measure is an intuitive way to put a measure on any set: the size of a subset is taken to be the number of the subsets elements if this is finite, and ∞ if the subset is infinite. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In group theory, the growth rate of a group with respect to a symmetric generating set is a notion that describes how fast a group grows. ... In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...

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. The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... In mathematics, the von Neumann conjecture, disproved in recent years, stated that a topological group G is not amenable if and only if G contains a subgroup that is a free group on two generators. ...

This article incorporates material from Amenable group on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...