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.[1]

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 ring measure, the Haar measure. (This is 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−1x)(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

Pier (1984) contains a comprehensive account of the conditions on a second countable locally compact group G that are equivalent to amenability:[2]

Case of discrete groups

The definition of amenability is simpler in the case of a discrete group,[3] i.e. a group equipped with the discrete topology.[4]

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 1such that

  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 L(G).

Having a measure μ on G allows us to define integration of bounded functions on G. Given a bounded function f : GR, the integral

\int_G f\,d\mu

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:

\nu(A) = \int_{g\in G}\mu \left (Ag^{-1} \right ) \, d\mu^-.

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:[5]

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 the spectral problem of Laplacians. For instance, the fundamental group of a closed Riemannian manifold is amenable if and only if the bottom of the spectrum of the Laplacian is 0 (R. Brooks, T. Sunada).

Properties

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.

Counterexamples

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.[11]

For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative:[12] 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.[13] 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.[14]

See also

Notes

  1. Day's first published use of the word is in his abstract for an AMS summer meeting in 1949, Means on semigroups and groups, Bull. A.M.S. 55 (1949) 1054–1055. Many text books on amenabilty, such as Volker Runde's, suggest that Day chose the word as a pun.
  2. Pier 1984
  3. See:
  4. Weisstein, Eric W., "Discrete Group", MathWorld.
  5. Pier 1984
  6. Ornstein, D.; Weiss, B. (1987). "Entropy and isomorphism theorems for actions of amenable groups". J. Analyse Math. 48: 1–141.
  7. Lewis Bowen (2011), "Every countably infinite group is almost Ornstein", ArXiv abs/1103.4424
  8. Leptin 1968
  9. See:
  10. Juschenko, Kate; Monod, Nicolas (2013), "Cantor systems, piecewise translations and simple amenable groups", Annals of Mathematics 178 (2): 775–787, doi:10.4007/annals.2013.178.2.7
  11. Olshanskii, Alexander Yu.; Sapir, Mark V. (2002), "Non-amenable finitely presented torsion-by-cyclic groups", Publ. Math. Inst. Hautes Études Sci. 96: 43–169, doi:10.1007/s10240-002-0006-7
  12. Tits, J. (1972), "Free subgroups in linear groups", J. Algebra 20 (2): 250–270, doi:10.1016/0021-8693(72)90058-0
  13. Guivarc'h, Yves (1990), "Produits de matrices aléatoires et applications aux propriétés géometriques des sous-groupes du groupes linéaire", Ergod. Th. & Dynam. Sys. 10 (3): 483–512, doi:10.1017/S0143385700005708
  14. Ballmann, Werner; Brin, Michael (1995), "Orbihedra of nonpositive curvature", Inst. Hautes Études Sci. Publ. Math. 82: 169–209, doi:10.1007/BF02698640

References

This article incorporates material from Amenable group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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