AMENABILITY AND SUPERHARMONIC FUNCTIONS

Let G be a countable group and mu a symmetric and aperiodic probabilit y measure on G. We show that G is amenable if and only if every positi ve superharmonic function is nearly constant on certain arbitrarily la rge 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 t hat G is amenable if and only if each member of a certain family of G- spaces contains a fixed point.