Convergence and limits of linear representations of finite groups

TL;DR

This paper explores the convergence and limits of linear representations of finite groups over finite fields, introducing infinite dimensional limit objects and extending hyperfiniteness characterization.

Contribution

It introduces a framework for understanding limits of linear group representations and extends Schramm's hyperfiniteness characterization to this setting.

Findings

Limit objects are infinite dimensional representations in continuous algebras.

Under integrality conditions, these algebras are skew fields.

Extended hyperfiniteness characterization to linear representations.

Abstract

Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in continuous algebras. We show that under a certain integrality condition, the algebras above are skew fields. Our main result is the extension of Schramm's characterization of hyperfiniteness to linear representations.