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

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 ${\mathrm{\ell }}^p$. The proof relies on results from ${\mathrm{\ell }}^p$-cohomology, a form of radial isoperimetry, transport patterns and revisiting some results of Følner.