Negative curvature in graph braid groups

TL;DR

This paper explores the geometric properties of graph braid groups, identifying conditions under which they exhibit hyperbolic and acylindrical hyperbolic behaviors using special colorings.

Contribution

It provides a precise characterization of when graph braid groups are Gromov-hyperbolic, toral relatively hyperbolic, or acylindrically hyperbolic, advancing the geometric understanding of these groups.

Findings

Determined conditions for Gromov-hyperbolicity of graph braid groups.

Identified when graph braid groups are toral relatively hyperbolic.

Established criteria for acylindrical hyperbolicity in graph braid groups.

Abstract

In this article, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous article, we determine precisely when a graph braid group is Gromov-hyperbolic, toral relatively hyperbolic, and acylindrically hyperbolic.