Amenability, free subgroups, and Haar null sets in non-locally compact groups

Sławomir Solecki

Research output: Contribution to journalArticle

Abstract

The paper has two objectives. On the one hand, we study left Haar null sets, a measure-theoretic notion of smallness on Polish, not necessarily locally compact, groups. On the other hand, we introduce and investigate two classes of Polish groups which are closely related to this notion and to amenability. We show that left Haar null sets form a $\sigma$-ideal and have the Steinhaus property on Polish groups which are 'amenable at the identity', and that they lose these two properties in the presence of appropriately embedded free subgroups. As an application we prove an automatic continuity result for universally measurable homomorphisms from inverse limits of sequences of amenable, locally compact, second countable groups to second countable groups.

Original languageEnglish (US)
Pages (from-to)693-722
Number of pages30
JournalProceedings of the London Mathematical Society
Volume93
Issue number3
DOIs
StatePublished - Jan 1 2006

ASJC Scopus subject areas

  • Mathematics(all)

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