# Groups, Geometry, and Dynamics

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**Volume 9, Issue 4, 2015, pp. 1153–1184**

**DOI: 10.4171/GGD/337**

Published online: 2015-11-16

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

Antoine Gournay^{[1]}(1) Université de Neuchâtel, Switzerland

The vanishing of reduced $\ell^2$-cohomology for amenable groups can be traced to the work of Cheeger and Gromov in [10]. The subject matter here is reduced $\ell^p$-cohomology for $p \in ]1,\infty[$, particularly its vanishing. Results for the triviality of $\underline {\ell^pH}^1(G)$ are obtained, for example: when $p \in ]1,2]$ and $G$ is amenable; when $p \in ]1,\infty[$ 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 $\ell^p$-cohomology is equivalent to the absence of non-constant 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 very restrictive growth condition).

*Keywords: *Group cohomohology, $L^p$-cohomology, harmonic functions, Poisson boundary

Gournay Antoine: Boundary values, random walks, and $\ell^p$-cohomology in degree one. *Groups Geom. Dyn.* 9 (2015), 1153-1184. doi: 10.4171/GGD/337