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).

arXiv comment25 pages