An l^{p}-Version of von-Neumann Dimension For Banach Space Representations of Sofic Groups

TL;DR

This paper introduces a new l^{p}-dimension concept for Banach space representations of sofic groups, extending previous work and exploring implications for l^{p}-Betti numbers of free groups.

Contribution

It provides an alternative l^{p}-dimension definition applicable to sofic groups and groups satisfying the Connes embedding conjecture, broadening the scope of Gournay's original framework.

Findings

Extended Gournay's results to broader classes of groups

Proposed new l^{p}-dimension definitions for various group types

Discussed implications for l^{p}-Betti numbers of free groups

Abstract

A. Gournay defined a notion of $l^{p}$-dimension for subspaces of the l^{q}-left-regular representation of an amenable discrete group. We give an alternative definition that works for sofic groups and a different notion for groups satisfying the Connes embedding conjecture, and for more general representations on Banach spaces. We extend certain results due to Gournay, as well as discuss l^{p}-Betti numbers of Free groups.