Amenability of Groups and G-Sets

Abstract

This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of Göttingen and theÉcole Normale Supérieure. The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collaboration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups.(This last property is often called finite additivity, as opposed to the σ -additivity property enjoyed by measures, in which countable unions are allowed).It easily follows from the definition that m( / 0) = 0; that m(A) ≤ m(B) if A ⊆ B; and that m(A 1 · · · A k ) = m(A 1 ) + · · · + m(A k ) for pairwise disjoint A 1 , . . . , A k .We denote by M (X) the set of means on X, with the usual topology on a set of functions; namely, a sequence m n ∈ M (X) converges to m precisely if for every ε > 0 and every finite collection A 1 , . . . , A k ⊆ X we have |m n (A i ) − m(A i )| < ε for all i ∈ {1, . . . , k} and all n large enough.Observe that M is a covariant functor: if f : X → Y , then we have a natural mapIn particular, if a group G acts on X, then it also acts on M (X). For a right action · : X × G → X, we have a right action on M (X) given by (m · g)(A) = m(A · g −1 ) for all A ⊆ X. [103]). Let G be a group and let X G be a set on which G acts. The G-set X is amenable if there is a G-fixed element in M (X). Definition 2.2 (von NeumannA group G is amenable if all non-empty right G-sets are amenable.In other words, the G-set X is amenable if M (X) G = / 0, namely if there exists a mean m on X such that m(Ag) = m(A) for all g ∈ G and all A ⊆ X.

Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

“…In [9], Bartholdi studied the amenability of Γ-set, here Γ is a group, which was induced by John Von Neumann in 1929. Fundamentally, the notion exhibited the following property of a group acting on a Γ-set X: the Γ-set X right is called amenable if there exists a Γ-invariant mean m on the power set for all A ⊂ X and g ∈ Γ.…”

“…In Section 3, we construct a one-to-one correspondence between the orbit space of the action of W(G, S 1 , T) on H 1 (S 1 , T) and H 1 (S 1 , G). In Section 4, we discuss some new developments in the field as well as its relations with amenability of groups [9][10][11][12]. For basic knowledge on compact Lie groups and twisted conjugate actions, one can refer [2,13,14]; for the nonabelian cohomology of Lie groups, one can refer [5,[15][16][17].…”

Non-triviality of the Poisson boundary of random walks on the group of Monod

“…Clearly they do not contain free subgroups. For more information and properties of amenability, see [5], [9], [17], [46].…”

Upper and lower bounds for rich lines in grids

“…The second question is an inverse problem for point-line incidences. The inverse problem for the Szemerédi-Trotter theorem is to show that if n points and n lines in R 2 have Ω(n 4/3 ) incidences, then the point set has some structure [10,Problem 5.7]; sharpness examples suggest that the point set might contain a large Cartesian product of arithmetic progressions. Even under the assumption that the point set is a Cartesian product, little is known about the inverse problem for Szemerédi-Trotter.…”

“…The upper bound in Theorem 3 applies when F = R, since we may consider points and lines defined over R to be contained in C 2 .…”