Groups acting on quasi-median graphs. An introduction

TL;DR

This paper introduces the theory of groups acting on quasi-median graphs, a generalization of median graphs and CAT(0) cube complexes, highlighting their potential in geometric group theory.

Contribution

It provides an accessible introduction to the concept of groups acting on quasi-median graphs, expanding the toolkit for geometric group theory research.

Findings

Quasi-median graphs generalize median graphs and CAT(0) cube complexes.

Groups acting on quasi-median graphs can be studied using new geometric methods.

The paper offers foundational insights for future research in this area.

Abstract

Quasi-median graphs have been introduced by Mulder in 1980 as a generalisation of median graphs, known in geometric group theory to naturally coincide with the class of CAT(0) cube complexes. In his PhD thesis, the author showed that quasi-median graphs may be useful to study groups as well. In the present paper, we propose a gentle introduction to the theory of groups acting on quasi-median graphs.