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