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AMENABILITY, FREE SUBGROUPS, AND HAAR NULL SETS IN NON-LOCALLY COMPACT GROUPS

Published online by Cambridge University Press:  13 October 2006

SŁAWOMIR SOLECKI
SŁAWOMIR SOLECKI
Affiliation:
Department of Mathematics, 1409 W. Green Street, University of Illinois, Urbana, IL 61801, USAssolecki@math.uiuc.edu
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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.

Keywords

22A1043A07 (primary)20F2428C10 (secondary)
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 93 , Issue 3 , November 2006 , pp. 693 - 722
Copyright
2006 London Mathematical Society

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Footnotes

This research was supported by NSF grant DMS-0400931.