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

Published:2003-06-01
Printed: Jun 2003
• Michael J. Puls
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## Abstract

Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of $G$ with coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.
 Keywords: group cohomology, $L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functional
 MSC Classifications: 43A15 - $L^p$-spaces and other function spaces on groups, semigroups, etc. 20F65 - Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F18 - Nilpotent groups [See also 20D15]

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