Boundary values, random walks and p-cohomology in degree one
Abstract
The vanishing of reduced 2-cohomology for amenable groups can be traced to the work of Cheeger & Gromov. The subject matter here is reduced p-cohomology for p ∈ ]1,∞[, particularly its vanishing. Results showing its triviality are obtained, for example: when p ∈ ]1,2] and G is amenable; when p ∈ ]1,∞[ 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 p-cohomology is equivalent to the absence of non-constant bounded (equivalently, not necessarily bounded) harmonic functions with gradient in 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).