Radial isoperimetry and absence of harmonic functions with ${\ell }^p$-gradient

Abstract

In this paper we show that groups for which the probability of return of a random walk is bounded below by $K_{1}exp(-K_2{n}^c)$ have no non-constant harmonic functions with gradient in ${\ell }^p$. The proof relies on results from ${\ell }^p$-cohomology, a form of radial isoperimetry, transport patterns and revisiting some results of Følner.

18 pages, extends results from a older version of arXiv:2206.13275