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

Cristina Costoya, Antonio Viruel

TL;DR

The paper addresses whether the isomorphism type of a broad class of groups can be recovered from their faithful actions on differential graded algebras over the rational numbers. It develops a graph-to-DGA construction that realizes automorphism groups of graphs as semidirect products $K \rtimes Aut(graph)$ with $K$ abelian and torsion-free, enabling a representation-theoretic fingerprint via DGAs. The main result shows that for groups $G,H$ in the class, $G \cong H$ if and only if their faithful actions on all DGAs coincide, i.e., $G$ acts faithfully on $(A,d)$ precisely when $H$ does. This work encompasses finite groups, Tarski groups, solvable Artinian groups, and certain mapping class groups, providing a new invariant at the crossroads of group theory and rational homotopy theory.

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.

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

TL;DR

The paper addresses whether the isomorphism type of a broad class of groups can be recovered from their faithful actions on differential graded algebras over the rational numbers. It develops a graph-to-DGA construction that realizes automorphism groups of graphs as semidirect products with abelian and torsion-free, enabling a representation-theoretic fingerprint via DGAs. The main result shows that for groups in the class, if and only if their faithful actions on all DGAs coincide, i.e., acts faithfully on precisely when does. This work encompasses finite groups, Tarski groups, solvable Artinian groups, and certain mapping class groups, providing a new invariant at the crossroads of group theory and rational homotopy theory.

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 -algebras on which these groups act faithfully.

Paper Structure

This paper contains 4 sections, 3 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. The class ${\frak G}$ of groups
  3. From automorphisms of graphs to automorphisms of DGA's
  4. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.2

For $G$ and $H$ groups in ${\frak G}$, the following statements are equivalent:

Theorems & Definitions (11)

  • Theorem 1.2
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4: Finite groups are in ${\frak G}$
  • Example 2.5: Tarski groups are in ${\frak G}$
  • Example 2.6: Solvable Artinian groups are in ${\frak G}$
  • Example 2.7: Mapping class groups of some compact surfaces are in ${\frak G}$
  • Theorem 3.1
  • proof
  • ...and 1 more