Faithful actions on Differential Graded Algebras determine the isomorphism type of a large class of groups

TL;DR

This paper shows that for many groups, including finite, Artinian, and some surface mapping class groups, their isomorphism type can be uniquely identified by the differential graded algebras they act faithfully upon.

Contribution

It establishes a new method to determine the isomorphism class of certain groups through their faithful actions on differential graded algebras.

Findings

Faithful actions on differential graded algebras determine group isomorphism types.

Applicable to finite, Artinian, and some surface mapping class groups.

Provides a new algebraic approach to classify these groups.

Abstract

We prove that the isomorphism type of a large class of groups (containing finite groups, countable Artinian groups and mapping class groups of certain surfaces, among others) is determined by the set of differential graded $\mathbb Q$-algebras on which these groups act faithfully.