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Proceedings of the American Mathematical Society
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1164149-7
PII: S 0002-9939(1993)1164149-7
Keywords: Amenable group, superharmonic function, Martin boundary, random walk
Article copyright: © Copyright 1993 American Mathematical Society