Diagonal complexes and the integral homology of the automorphism group of a free product

TL;DR

This paper introduces diagonal complexes and a geometric approach to compute the integral (co)homology of symmetric automorphism groups of free products, generalizing right-angled Artin groups through the concept of DRAAGs.

Contribution

It develops a new geometric interpretation of symmetric automorphisms using cactus products and introduces diagonal complexes, extending the theory of right-angled Artin groups.

Findings

Provides a method to compute integral (co)homology of automorphism groups

Introduces the concept of diagonal complexes and DRAAGs

Generalizes right-angled Artin groups

Abstract

The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of diagonal complexes is introduced. This offers a generalisation of the theory of right-angled Artin groups in that each diagonal complex defines what we call a diagonal right-angled Artin group (DRAAG).