Abstract

Let G be a group generated by a finite symmetric set S. By analogy with harmonic functions on manifolds, one can define the space H(G) of harmonic functions on G with respect to S as consisting of those functions f : G → R for which f(x) is always equal to the average of the values of f(xs) with s in S.

I will describe some results and conjectures relating certain properties of H(G) to certain algebraic properties of G. In particular, I will present a proof that H(G) is finite dimensional if and only if G is virtually cyclic. The proof uses functional analysis, polynomials on groups, and random walks, amongst other things.

Questions of this type are to some extent motivated by Kleiner’s proof of Gromov’s polynomial growth theorem.

Some of the work I will discuss is joint with Meyerovitch, Perl and Yadin.