Existence of positive harmonic functions on groups and on covering manifolds (Q1347277)

The authors continue the study of bounded or positive harmonic functions on locally compact groups and on Riemannian manifolds. Main results: (1) Let \(G\) be a compactly generated group for which there is a continuous homomorphism \(f\) from \(G\) into an almost connected group \(L\) (i.e., \(L\) has a co-compact connected subgroup) such that the closure of \(f(G)\) has exponential growth. Then any centered adapted probability measure on \(G\) with a third moment has non-constant continuous positive harmonic functions. (2) Let \(M\) be a regular covering of a compact manifold such that the deck transformation group is a closed subgroup of an almost connected group. Then there exist non-constant positive harmonic functions on \(M\) if and only if \(M\) is of exponential growth. In particular, a centered probability measure on a linear group of exponential growth has non-constant positive harmonic functions. The same conclusion holds for the Laplace-Beltrami operator on a co-compact Riemannian covering with such a covering group.