Dimensions of l^p-cohomology groups

TL;DR

This paper investigates the l^p-cohomology and homology of infinite groups, establishing that these groups are either trivial or infinitely dimensional, and analyzing the possible sizes of the p-harmonic boundary.

Contribution

It proves that for groups of type FP-infinity, l^p-cohomology and homology are either zero or infinite dimensional, and classifies the cardinality of the p-harmonic boundary.

Findings

l^p-cohomology groups are either 0 or infinite dimensional

l^p-homology groups are either 0 or infinite dimensional

p-harmonic boundary cardinality is 0, 1, or infinity

Abstract

Let G be an infinite discrete group of type FP-infinity and let p>1 be a real number. We prove that the l^p-homology and cohomology groups of G are either 0 or infinite dimensional. We also show that the cardinality of the p-harmonic boundary of a finitely generated group is either 0, 1, or infinity.