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 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".[a]
The critical step in the Banach–Tarski paradox construction is to find inside the rotation group SO(3) a free subgroup on two generators. Amenable groups cannot contain such groups, and do not allow this kind of paradoxical construction.
Amenability has many equivalent definitions. 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. For example, any subgroup of the group of integers ( Z , + ) {\displaystyle (\mathbb {Z} ,+)} is generated by some integer p ≥ ≥ --> 0 {\displaystyle p\geq 0} . If p = 0 {\displaystyle p=0} then the subgroup takes up 0 proportion. Otherwise, it takes up 1 / p {\displaystyle 1/p} of the whole group. Even though both the group and the subgroup has infinitely many elements, there is a well-defined sense of proportion.
If a group has a Følner sequence then it is automatically amenable.
Let G be a locally compact Hausdorff group. Then it is well known that it possesses a unique, up-to-scale left- (or right-) translation 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 (respectively right-invariant) if Λ(g·f) = Λ(f) for all g in G, and f in L∞(G) with respect to the left (respectively right) shift action of g·f(x) = f(g−1x) (respectively 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.
By identifying Hom(L∞(G), R) with the space of finitely-additive Borel measures which are absolutely continuous with respect to the Haar measure on G (a ba space), the terminology becomes more natural: a mean in Hom(L∞(G), R) induces a left-invariant, finitely additive Borel measure on G which gives the whole group weight 1.
As an example for compact groups, consider the circle group. The graph of a typical function f ≥ 0 looks like a jagged curve above a circle, which can be made by tearing off the end of a paper tube. The linear functional would then average the curve by snipping off some paper from one place and gluing it to another place, creating a flat top again. This is the invariant mean, i.e. the average value Λ Λ --> ( f ) = ∫ ∫ --> R / Z f d λ λ --> {\displaystyle \Lambda (f)=\int _{\mathbb {R} /\mathbb {Z} }f\ d\lambda } where λ λ --> {\displaystyle \lambda } is Lebesgue measure.
Left-invariance would mean that rotating the tube does not change the height of the flat top at the end. That is, only the shape of the tube matters. Combined with linearity, positivity, and norm-1, this is sufficient to prove that the invariant mean we have constructed is unique.
As an example for locally compact groups, consider the group of integers. A bounded function f is simply a bounded function of type f : Z → → --> R {\displaystyle f:\mathbb {Z} \to \mathbb {R} } , and its mean is the running average lim n 1 2 n + 1 ∑ ∑ --> k = − − --> n n f ( k ) {\displaystyle \lim _{n}{\frac {1}{2n+1}}\sum _{k=-n}^{n}f(k)} .
Pier (1984) contains a comprehensive account of the conditions on a second countable locally compact group G that are equivalent to amenability:[2]
The definition of amenability is simpler in the case of a discrete group,[4] i.e. a group equipped with the discrete topology.[5]
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:[2]
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.[6]
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.
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.[12]
For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative:[13] 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.[14] 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.[15]
This article incorporates material from Amenable group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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