Group Cohomology and L^p-Cohomology of Finitely Generated Groups

Abstract

Let G be a finitely generated, infinite group, let p > 1, and let Lp(G) denote the Banach space . In this paper we will study the first cohomology group of G with coefficients in Lp(G), and the first reduced Lp-cohomology space of G. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

The first Lp-cohomology of some finitely generated groups and p-harmonic functions

“…In this section we will describe the nonzero elements of H 1 (G, L p (G)) for groups that have polynomial growth of (precise) degree n > p. These results are a generalization of results from [14,Section 6], where H 1 (G, L 2 (G)) was discussed.…”

“…The next proposition is a generalization of[14, Proposition 6.2]. I would like to thank Thomas Schick for showing me this generalization.Proposition 5.2.…”

Mixing, malnormal subgroups and cohomology in degree one

“…It seems natural to introduce the space of p-Dirichlet functions associated to the representation. These function have been of great use to study the cohomology of the left-regular representation, see [25], Martin & Valette [41, §3], Puls [52,53] or §5 below. For S G some finite generating set of G, let…”

“…Henceforth, it will be referred to as the D p G-norm. See Martin & Valette [41, §3] or Puls [52,53] for more background.…”

Graphs of bounded degree and thep-harmonic boundary

“…This of course would resolve Gromov's conjecture. More information about the first reduced L p -cohomology (and the special case of L 2 -cohomology) can be found in [Pansu 1989;2008;Tessera 2009] for various manifolds, and in [Bekka and Valette 1997;Bourdon 2004;Bourdon et al 2005;Elek 1998;Martin and Valette 2007;Puls 2003;2006; for finitely generated groups. As implied earlier, there is a strong connection between the vanishing of the first reduced L p -cohomology and the nonexistence nonconstant p-harmonic functions; for a proof in the case of homogeneous Riemannian manifolds, see [Tessera 2009, Proposition 4.11].…”