Expansive actions on uniform spaces and surjunctive maps

TL;DR

This paper extends Gromov's results on surjunctivity to uniform spaces, explores applications, and proves the compactness of certain spaces of group algebras related to expansive group actions.

Contribution

It introduces a uniform framework for Gromov's surjunctivity results and establishes the compactness of spaces of group algebras with stable finiteness.

Findings

Uniform version of Gromov's surjunctivity results

Compactness of the space of groups with stably finite group algebras

Applications to expansive group actions

Abstract

We present a uniform version of a result of M. Gromov on the surjunctivity of maps commuting with expansive group actions and discuss several applications. We prove in particular that for any group $\Gamma$ and any field $\K$, the space of $\Gamma$-marked groups $G$ such that the group algebra $\K[G]$ is stably finite is compact.