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Amenability and superharmonic functions

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Amenable group
 superharmonic function
 Martin boundary
 random walk
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1993
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SUNY Plattsburgh Mathematics Faculty Work
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http://hdl.handle.net/20.500.12648/1125
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.
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Northshield, S. (1993). Amenability and superharmonic functions. Proceedings of the American Mathematical Society, 119(2). http://doi.org/10.1090/S0002-9939-1993-1164149-7
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This article has been published in the October 1993 issue of Proceedings of the American Mathematical Society.
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