Geometry and combinatorics via right-angled Artin groups

TL;DR

This survey explores the deep connections between graph combinatorics, geometry, and the algebraic properties of right-angled Artin groups, highlighting their significance in geometric and complexity theory.

Contribution

It provides a comprehensive overview of how the combinatorial and geometric aspects of graphs relate to the algebraic structure of right-angled Artin groups, emphasizing the role of defining and extension graphs.

Findings

Relationships between graph combinatorics and group geometry elucidated

Connections to geometric group theory and complexity theory discussed

Frameworks for analyzing right-angled Artin groups established

Abstract

We survey the relationship between the combinatorics and geometry of graphs and the algebraic structure of right-angled Artin groups. We concentrate on the defining graph of the right-angled Artin group and on the extension graph associated to the right-angled Artin group. Additionally, we discuss connections to geometric group theory and complexity theory. The final version of this survey will appear in "In the tradition of Thurston, vol.~II", ed.~K.~Ohshika and A.~Papadopoulos.