Several equivalent definitions of amenable groups

1. Existence of left-invariant mean

   Let G be a group. Denote by l^{\infty}(G, R) the (Banach) space of bounded functions from G to the set R of real numbers. A left-invariant mean on a group G is an R-linear map m: l^{\infty}(G, R) -> R satisfying the following properties: 
   (i) (Normalization) m(1) = 1 
  (ii) (Positivity) If f(g) >= 0 for all g in G, we have m(f) >= 0
 (iii) (Left-invariance) For all g in G and f in l^{\infty}(G, R), we have m(gf) = m(f). Here gf is the function f* satisfying f*(a) = f(ga) for all a in G.  
   The group is amenable if it admits a left-invariant mean. 

Example 1: If G is a finite group, G is amenable. Indeed, if we define m(f) to be the average of all the numbers f(g) with g in G, the action of any element doesn't change this average since this action sends G to itself. The two other properties are also obvious. 

2. Amenable graphs and Folner sequences

   The notion of amenability can also be described graph-theoretically as follows. First, in graph theory, one might be interested in the edge boundary of a subset A of the vertex set, which is defined to be the number of edges connecting a vertex in A and a vertex not in A. Informally speaking, the edge boundary is the set of "bridges" between A and the complement of A. If for every arbitrarily small e > 0, there exists a finite subset A of the vertex set such that the number of edges in the edge boundary of A is less than e|A|, the graph is said to be amenable. For example, if the graph is finite, it is obviously amenable since we might choose A to be the whole vertex set (in this case the edge boundary is empty). 

   Here is a second definition of amenability: a group G is amenable if its Cayley graph (with respect to a finite generating set) is amenable. 

Example 2: The direct sum of two copies of the additive group of integers Z, or (Z^2, +), is amenable. Consider the Cayley graph of this group with respect to the generating set {(1, 0), (0, 1)}.  For any positive integer n, the set V(n) = {(x, y) | 0 <= x, y <= n} has (n + 1)^2 vertices and its edge boundary has 4(n + 1) edges, thus the ratio is 4(n+1)/(n+1)^2 = 4/(n + 1), which can be arbitrarily small. 


The set V(5) and its edge boundary   

   The definition via Cayley graph is (almost) the same as the definition via Folner sequences (named after Erling Folner). The only difference is that Folner sequence considers the set of vertices whose distances to A are smaller than a given positive integer r, instead of the edge boundary of A (corresponding to r = 1). However, since the degree (or valence) of every vertex is 2s, where s is the number of elements of the generating set, changing from r = 1 to an arbitrary r doesn't matter. 

3. Paradoxical sets, paradoxical actions and paradoxical groups

   Let G be a group acting on a set X. A subset E of X is paradoxical if there exists two families of pairwise disjoint subsets {A(1), A(2), ..., A(n)}, {B(1), B(2), ..., B(m)} of E and g(1), g(2), ..., g(n), h(1), h(2), ..., h(m) in G such that the union of g(1)A(1), g(2)A(2), ..., g(n)A(n) is E and the union of h(1)B(1), h(2)B(2), ..., h(m)B(m) is also E. If the set X itself is paradoxical, the action of G on X is said to be paradoxical. A group G is paradoxical if the left multiplication (i.e. the natural action of G on itself) is paradoxical. There is a famous theorem of Tarski which states that a group is paradoxical if and only if it is non-amenable. 



Alfred Tarski (1901 - 1983), a Polish mathematician who was
best known for his contributions to mathematical logic

Example 3: A free group F of rank 2 is paradoxical. Indeed, let {a, b} be the basis of F. Define w(a), w(b), w(a^(-1)), w(b^(-1)) be respectively the set of words in F starting with a, b, a^(-1), b^(-1). Then, F is the union of w(a) and w(a^(-1)). Similarly, F is also the union of w(b) and w(b^(-1)). Therefore, F is paradoxical and non-amenable.

4. Other equivalent definitions

   The first definition, in terms of a finitely additive measure on subsets of a group, was introduced by John von Neumann in 1929 under the German name "messbar", which means "measurable" in English. In 1984, Jean-Paul Pier gave a list of tens of equivalent definitions for amenability. Some of them are Kesten's condition (named after Harry Kesten), Fixed-point property, etc. 

   

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