Boundary values, random walks and $\ell^p$-cohomology in degree one

Abstract

The vanishing of reduced $\ell^2$-cohomology for amenable groups can be traced to the work of Cheeger & Gromov. The subject matter here is reduced $\ell^p$-cohomology for $p \in ]1,\infty[$, particularly its vanishing. Results showing its triviality are obtained, for example: when $p \in ]1,2]$ and $G$ is amenable; when $p \in ]1,\infty[$ and $G$ is Liouville (in particular, of intermediate growth). This is done by answering a question of Pansu assuming the graph satisfies an isoperimetric profile. Namely, the triviality of the reduced $\ell^p$-cohomology is equivalent to the absence of non-constant bounded (equivalently, not necessarily bounded) harmonic functions with gradient in $\ell^q$ ($q$ depends on the profile). In particular, one reduces questions of non-linear analysis ($p$-harmonic functions) to linear ones (harmonic functions with a restrictive growth condition).