Isomorphism invariance of the girth of Artin groups

TL;DR

This paper demonstrates that the girth of Artin groups is an isomorphism invariant and introduces the weighted girth, providing new tools for understanding the algebraic and geometric properties of these groups.

Contribution

It establishes girth as an isomorphism invariant for all Artin groups and introduces the weighted girth, linking it to minimal subgroups and commutation graphs.

Findings

Girth is an algebraic isomorphism invariant for Artin groups.

Artin groups based on cycle graphs are isomorphically rigid.

Weighted girth is characterized for hyperbolic type Artin groups.

Abstract

For all Artin groups, we characterise the girth (i.e. the length of a shortest cycle) of the defining graph algebraically, showing that it is an isomorphism invariant. Using this result, we prove that the Artin groups based on a cycle graph are isomorphically rigid. Alongside the girth, we introduce a new graph invariant, the weighted girth, which takes into account the labels of the defining graph. Within the class of two-dimensional Artin groups of hyperbolic type, we characterise the weighted girth in terms of certain minimal right-angled Artin subgroups, showing that it is an isomorphism invariant. Finally, under the further hypothesis of leafless defining graph, we recover the weighted girth as the girth of the commutation graph introduced by Hagen-Martin-Sisto.