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Proceedings of the London Mathematical Society 2006 93(3):693-722; doi:10.1017/S002461150601584X
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© London Mathematical Society

Amenability, Free Subgroups, and Haar Null Sets in Non-Locally Compact Groups

Slawomir Solecki

Department of Mathematics 1409 W. Green Street, University of Illinois, Urbana, IL 61801, USA ssolecki{at}math.uiuc.edu

Received 7 April 2005. Revision received 4 November 2005.

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. 2000 Mathematics Subject Classification 22A10, 43A07 (primary); 20F24, 28C10 (secondary).

This research was supported by NSF grant DMS-0400931.


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