l^p-cohomology for groups of type FP_n

TL;DR

This paper explores the duality between reduced l^p-homology and l^q-cohomology for groups of type FP_n, establishing conditions under which the cohomology vanishes, with implications for various classes of groups.

Contribution

It proves a duality theorem between reduced l^p-homology and cohomology for groups of type FP_n and identifies conditions leading to vanishing cohomology for specific group classes.

Findings

Reduced l^p-homology is dual to reduced l^q-cohomology for groups of type FP_n.

Vanishing of reduced l^p-cohomology for groups with infinite order central elements.

Extension of vanishing results to groups with infinite center, FCC groups, nilpotent groups, and polynomial growth groups.

Abstract

Let G be a group of type FP_n and let p>1. In this paper we show that the reduced l^p-homology of G is dual to the reduced l^q-cohomology for \frac{1}{p}+\frac{1}{q}=1. In our main theorem we show that for a group of type FP_n with a central element of infinite order the reduced l^p-cohomology vanishes. We generalize this fact for groups with infinitely many elements in the center of the group, for groups which are FCC, for groups with infinitely many finite conjugacy classes, for nilpotent groups, and for groups of polynomial growth.