The Steinhaus-Weil property: its converse, Solecki amenability and subcontinuity

Abstract

The Steinhaus-Weil theorem that concerns us here is the `interior points' property -- that in a topological group a non-negligible set S has the identity as an interior point of $SS^{-1}$. There are various converses; the one that mainly concerns us is due to Simmons and Mospan. Here the group is locally compact, so we have a Haar measure. The Simmons-Mospan theorem states that a (regular Borel) measure has such a Steinhaus-Weil property if and only if it is absolutely continuous with respect to the Haar measure. In Part I (Propositions 1-9, Theorems 1-3) we develop a number of relatives of the Simmons-Mospan theorem, drawing also on Solecki's amenability at 1 (and using Fuller's notion of subcontinuity). In Part II (Theorems 4, 5) we link this with topologies of Weil type.