Boundary values, random walks, and lp-cohomology in degree one

The vanishing of reduced l(2)-cohomology for amenable groups can be traced to the work of Cheeger andGromov in [10]. The subjectmatter here is reduced l(p)-cohomology for p is an element of]1, infinity[, particularly its vanishing. Results for the triviality of (l(p) H-1) under bar (G) are obtained, for example: when p is an element of]1, 2] and G is amenable; when p is an element of]1, infinity[ and G is Liouville (e.g. of intermediate growth). This is done by answering a question of Pansu in [34, 1.9] for graphs satisfying certain isoperimetric profile. Namely, the triviality of the reduced l(p)-cohomology is equivalent to the absence of non-constant harmonic functions with gradient in l(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 very restrictive growth condition).