Amenable group
Encyclopedia
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.
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 no topological
structure, a simpler definition is used. In this setting, a group is amenable if one can say what percentage of G any given subset takes up.
If a group has a Følner sequence
then it is automatically amenable.
be a locally compact hausdorff group
. Then it is well-known that it possesses a unique, up-to-scale left- (or right-) rotation invariant ring (borel regular in the case of
second countable) measure (left and right probability measure in the case of 
compact), the Haar measure
. Consider the banach space
of essentially-bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).
Definition 1.
An operator
is said to be a mean if 
has norm 1 and is non-negative (i.e. 
a.e. implies 
).
Definition 2.
A mean
is said to be a left-invariant (resp. right-invariant) if 
all 
with respect to the left (resp. right) shift action of 
(resp. 
).
Definition 3.
A locally compact hausdorff group is called amenable if it admits a left- (or right-)invariant mean.
, i.e. a group with no topological structure.
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
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.)
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:
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 group
s of 2-dimensional simplicial complex
es of non-positive curvature.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an amenable group is a locally compact topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
G carrying a kind of averaging operation on bounded functions that is invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
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
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
in 1929 under the German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
. 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
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, the definition is in terms of linear functionals. An intuitive way to understand this version is that the support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
of the regular representation
Regular representation
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 by translation....
is the whole space of irreducible representations.
In discrete group theory, where G has no topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
structure, a simpler definition is used. In this setting, a group is amenable if one can say what percentage of G any given subset takes up.
If a group has a Følner sequence
Følner sequence
In mathematics, a Følner sequence for a group is a sequence of sets satisfying a particular condition. If a group has a Følner sequence with respect to its action on itself, the group is amenable. A more general notion of Følner nets can be defined analogously, and is suited for the study of...
then it is automatically amenable.
Locally compact definition
Let be a locally compact hausdorff groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. Then it is well-known that it possesses a unique, up-to-scale left- (or right-) rotation invariant ring (borel regular in the case of
second countable) measure (left and right probability measure in the case of compact), the Haar measureHaar measure
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....
. Consider the banach space
of essentially-bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).Definition 1.
An operator
is said to be a mean if has norm 1 and is non-negative (i.e. a.e. implies ).Definition 2.
A mean
is said to be a left-invariant (resp. right-invariant) if all with respect to the left (resp. right) shift action of (resp. ).Definition 3.
A locally compact hausdorff group is called amenable if it admits a left- (or right-)invariant mean.
Discrete definition
The definition of amenability is quite a lot simpler in the case of a discrete groupDiscrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
, i.e. a group with no topological structure.
Definition. A discrete group G is amenable if there is a finitely additive measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
(also called a mean) —a function that assigns to each subset of G a number from 0 to 1—such that
- The measure is a probability measure: the measure of the whole group G is 1.
- 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.
- 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 L∞(G).
Having a measure
on G allows us to define integration of bounded functions on G. Given a bounded function , the integralis defined as in Lebesgue integration
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
. (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 finite 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 algebraVon Neumann algebraIn mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
of Γ is hyperfiniteHyperfiniteHyperfinite may refer to:*Hyperfinite set*von Neumann algebra...
(A. Connes).
Examples
- Finite groupFinite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s are amenable. Use the counting measureCounting measureIn 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 elements in the subset, if the subset is finite, and ∞ if the subset is infinite....
with the discrete definition. - SubgroupSubgroupIn group theory, given a group G under a binary operation *, a subset H of G is called 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 x H is a group operation on H...
s of amenable groups are amenable. - The direct productDirect product of groupsIn the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of two amenable groups is amenable, while the direct product of an infinite family of amenable groups need not be. - The group of integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is amenable (a sequence of intervals of length tending to infinity is a Følner sequenceFølner sequenceIn mathematics, a Følner sequence for a group is a sequence of sets satisfying a particular condition. If a group has a Følner sequence with respect to its action on itself, the group is amenable. A more general notion of Følner nets can be defined analogously, and is suited for the study of...
).The existence of a shift-invariant, finitely additive probability measure on the group Z also follows easily from the Hahn-Banach theorem this way. Let S be the shift operator on the sequence space ℓ∞(Z), which is defined by (Sx)i = xi+1 for all x ∈ ℓ∞(Z), and let u ∈ ℓ∞(Z) be the constant sequence ui =1 for all i ∈ Z. Any element y ∈ Y:=Ran(S-I) has a distance larger than or equal to 1 from u (otherwise yi = xi+1 - xi would be positive and bounded away from zero, whence xi could not be bounded). This implies that there is a well-defined norm-one linear form on the subspace Ru + Y taking tu + y to t. By the Hahn-Banach theorem the latter admits a norm-one linear extension on ℓ∞(Z), which is by construction a shift-invariant finitely additive probability measure on Z. - 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 groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s are amenable.
- By the fundamental theorem of finitely generated abelian groups, it follows that abelian group
- A group is amenable if it has an amenable normal subgroupNormal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
such that the quotientQuotient groupIn mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
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 indexIndex of a subgroupIn mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
amenable subgroup. That is, virtually amenable groups are amenable. - Furthermore, it follows that all solvable groupSolvable groupIn mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
s are amenable.
- It follows that a group is amenable if it has a finite index
- CompactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
groups are amenable. The Haar measure is an invariant mean (unique taking total measure 1). - Finitely generated groups of subexponential growthGrowth rate (group theory)In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length...
are amenable.
Non-examples
If a countable discrete group contains a (non-abelian) freeFree group
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 elements of S and their inverses...
subgroup on two generators, then it is not amenable. The converse to this statement is the so-called von Neumann conjecture
Von Neumann conjecture
In mathematics, the von Neumann conjecture 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. The conjecture was disproved in 1980....
, 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
Periodic group
In group theory, a periodic group or a torsion group is a group in which each element has finite order. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group, although all finite cyclic groups are periodic.The exponent of a periodic group...
, 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
Tits alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. It states that every such group is either virtually solvable In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about...
: 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
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
' proof used algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, 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 group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
s of 2-dimensional simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es of non-positive curvature.