A distal property of groups and the growth of connected locally compact groups

Abstract

If G is a topological group then we can think of G acting on itself by multiplying on the left. We would like to know when this action has the property that whenever g and h are distinct elements of G , then the element xg does not get arbitrarily close to xh as x varies in G . It is natural to say that this is the case if {( xg , xh ): x ∈ G } is separated from the diagonal of G × G by a uniform neighbourhood of the diagonal.