Amenability and superharmonic functions

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Issue Date
1993
Authors
Northshield, Sam
Publisher
Proceedings of the American Mathematical Society
Keywords
Amenable group , superharmonic function , Martin boundary , random walk
Abstract
Let G be a countable group and u a symmetric and aperiodic probability measure on G . We show that G is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets of G. We use this to show that if G is amenable, then the Martin boundary of G contains a fixed point. More generally, we show that G is amenable if and only if each member of a certain family of G-spaces contains a fixed point.
Description
This article has been published in the October 1993 issue of Proceedings of the American Mathematical Society.
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