Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1993-1164149-7
MathSciNet review: 1164149
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [A] David Aldous, personal communication.
  • [B] M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Math., vol. 175, Springer, New York, 1971. MR 0281940 (43:7654)
  • [DK] Joseph Dodziuk and Leon Karp, Spectral and function theory for combinatorial Laplacians, Geometry of Random Motion, Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 25-40. MR 954626 (89h:58220)
  • [K] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146-156. MR 0112053 (22:2911)
  • [KV] V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), 457-490. MR 704539 (85d:60024)
  • [LS] T. J. Lyons and D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), 299-323. MR 755228 (86b:58130)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 43A07, 31C05, 31C35

Retrieve articles in all journals with MSC: 43A07, 31C05, 31C35


Additional Information

DOI: https://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

American Mathematical Society