The isomorphism problem for large-type Artin groups

TL;DR

This paper solves the isomorphism problem for large-type Artin groups by algebraically reconstructing associated Coxeter groups and explores rigidity properties of 2-dimensional Artin groups, revealing structural invariants and non-co-Hopfian cases.

Contribution

It provides a complete algebraic solution to the isomorphism problem for large-type Artin groups and classifies dihedral subgroups in 2-dimensional cases, establishing new rigidity results.

Findings

Reconstructed Coxeter groups algebraically for large-type Artin groups.

Most standard generators in 2-dimensional Artin groups are preserved under isomorphisms.

Artin groups with even-labelled leaves are not co-Hopfian.

Abstract

In this paper we solve the isomorphism problem for all large-type Artin groups. Our strategy involves reconstructing the Coxeter groups associated with large-type Artin groups in a purely algebraic way. This answers several questions raised by Charney. We also study 2-dimensional Artin groups in general. By classifying all their dihedral Artin subgroups, we are able to give strong results of rigidity for all 2-dimensional Artin groups. We prove that "most" standard generators in 2-dimensional Artin groups are preserved under isomorphisms (up to conjugation). We also show that an isomorphism between large-type Artin groups preserves the set of spherical parabolic subgroups if and only if the defining graphs do not have even-labelled leaves. Finally, we show that Artin groups whose defining graphs have even-labelled leaves are never co-Hopfian.