Elementary equivalence in Artin groups of finite type

TL;DR

This paper investigates the relationship between elementary equivalence and isomorphism in irreducible Artin groups of finite type, establishing conditions under which these groups are elementarily equivalent or isomorphic, with implications for braid and mapping class groups.

Contribution

It demonstrates that in most infinite families of Artin groups of finite type, elementary equivalence coincides with isomorphism, and shows the existence of infinitely many elementary equivalence classes.

Findings

Two groups in the same family are elementarily equivalent iff they are isomorphic.

Two braid groups are elementarily equivalent iff they are isomorphic.

Mapping class groups are elementarily equivalent iff they are isomorphic.

Abstract

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces - a geometric analogue of braid groups - are elementarily equivalent if and only if they are isomorphic.