# Superharmonic functions and amenability

Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$.

Assume that there is a set of non-negative superharmonic functions $X$, which satisfies

(1) $X$ is convex and invariant under $P$

(2) There exists a sequence $(x_n)$ in $G$ such that $f(g.x_n)/f(x_n)\rightarrow 1$ for all g in $G$.

What are the conditions on this set one can impose in order to conclude amenability of the group? If the set of all non-negative superharmonic functions satisfies (2) then the paper "Amenability and superharmonic functions" by Northshield [1] implies amenability. It seems one should be able to read more information provided extra conditions on this set. I would be interested to have any citation on related articles.

[1] Northshield, S., Amenability and superharmonic functions, Proc. Am. Math. Soc. 119, No.2, 561-566 (1993). ZBL0785.31006.

• I suspect you mean $G$ rather than $P$ in condition (1). – R W Aug 9 '17 at 18:28
• I mean $P$. It also seems that it should be the same to ask. – Kate Juschenko Aug 10 '17 at 8:40