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 Free Access
Abstract | References | Similar Articles | Additional Information
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.
- [A] David Aldous, personal communication.
- [B] Marcel Brelot, On topologies and boundaries in potential theory, Enlarged edition of a course of lectures delivered in 1966. Lecture Notes in Mathematics, Vol. 175, Springer-Verlag, Berlin-New York, 1971. MR 0281940
- [DK] Jozef Dodziuk and Leon Karp, Spectral and function theory for combinatorial Laplacians, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 25–40. MR 954626, https://doi.org/10.1090/conm/073/954626
- [K] Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, https://doi.org/10.7146/math.scand.a-10568
- [KV] V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457–490. MR 704539
- [LS] Terry Lyons and Dennis Sullivan, Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), no. 2, 299–323. MR 755228
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