The p-harmonic boundary for finitely generated groups and the first reduced \ell_p-cohomology

TL;DR

This paper introduces the concept of the p-harmonic boundary for finitely generated groups and links it to the vanishing of the first reduced ll^p-cohomology, providing new insights into geometric group theory.

Contribution

It defines the p-harmonic boundary for groups and characterizes the vanishing of first reduced ll^p-cohomology via this boundary, also exploring invariance under rough isometries.

Findings

p-harmonic boundary characterized for finitely generated groups

Vanishing of ll^p-cohomology linked to boundary cardinality

Properties preserved under rough isometries

Abstract

Let $p$ be a real number greater than one and let $G$ be a finitely generated, infinite group. In this paper we introduce the $p$-harmonic boundary of $G$. We then characterize the vanishing of the first reduced $\ell^p$-cohomology of $G$ in terms of the cardinality of this boundary. Some properties of $p$-harmonic boundaries that are preserved under rough isometries are also given. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $G$, the $p$-harmonic boundary of $G$ and the first reduced $\ell^p$-cohomology of $G$.