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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Amenability and superharmonic functions

Author: S. Northshield
Journal: Proc. Amer. Math. Soc. 119 (1993), 561-566
MSC: Primary 43A07; Secondary 31C05, 31C35
MathSciNet review: 1164149
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Abstract: Let $ G$ be a countable group and $ \mu $ 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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Keywords: Amenable group, superharmonic function, Martin boundary, random walk
Article copyright: © Copyright 1993 American Mathematical Society

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