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 be a countable group and a symmetric and aperiodic probability measure on . We show that is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets of . We use this to show that if is amenable, then the Martin boundary of contains a fixed point. More generally, we show that is amenable if and only if each member of a certain family of -spaces contains a fixed point.

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1993-1164149-7

Keywords:
Amenable group,
superharmonic function,
Martin boundary,
random walk

Article copyright:
© Copyright 1993
American Mathematical Society