On harmonic functions and the linear-growth case of Gromov's theorem

Abstract

We show that the space of harmonic functions on a finitely generated infinite group G is finite dimensional if, and only if, G has a finite-index subgroup isomorphic to the integers. A key tool is Wilkie and van den Dries's quantitative version of the linear-growth case of Gromov's theorem on groups of polynomial growth.

arXiv commentThis paper has been withdrawn by the author due to a crucial error in the proof of Lemma 3.3. This came to light after an observation made by an anonymous referee, for which the the author is grateful