Internal Topological Phenomena

Abstract

The topological analogue of amenability for discrete groups is local finiteness. In contrast to the setting of C*-algebras, which we will turn to below, for discrete groups the topological notion of perturbation is trivial and thus, unlike the combinatorial measure-theoretic viewpoint, does not give us anything new beyond the merely group-theoretic. The group G is said to be locally finite if every finite subset of G generates a finite subgroup. Equivalently, G is the increasing union of finite subgroups. Obviously every finite group is locally finite. An example of a countably infinite locally finite group is the group of all permutations of N which fix all but finitely many elements. There are uncountably many pairwise nonisomorphic countable locally finite groups, and there is a countable locally finite group U which has the universal property that it contains a copy of every finite group and any two monomorphisms of a finite group into U are conjugate by an inner automorphism [47]. Note that every locally finite group is torsion. The converse is the general Burnside problem and is false, as was shown by Golod. In fact a torsion group need not even be amenable, which is the measure-theoretic analogue of local finiteness to be discussed below.