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

TL;DR

This paper investigates the conditions under which reduced ll^p-cohomology vanishes for certain groups, linking harmonic functions with gradient conditions to cohomological triviality, thus connecting nonlinear and linear analysis.

Contribution

It establishes new criteria for the vanishing of reduced ll^p-cohomology based on isoperimetric profiles and harmonic functions, extending previous results beyond ll^2-cohomology.

Findings

Vanishing of ll^p-cohomology for amenable groups when p 

Triviality of ll^p-cohomology for Liouville groups

Equivalence between cohomology triviality and absence of certain harmonic functions

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).