Property $R_\infty$ for some spherical and affine Artin-Tits groups

TL;DR

This paper provides a concise, uniform proof that certain spherical and affine Artin-Tits groups, including pure subgroups and Artin braid groups, possess property R_infinity, confirming their complex algebraic structure.

Contribution

It offers a new, streamlined proof of property R_infinity for various spherical and affine Artin-Tits groups, including pure subgroups and braid groups, extending previous results.

Findings

Property R_infinity holds for specified Artin-Tits groups.

Alternative proof simplifies understanding of these groups.

Confirms complex algebraic behavior of pure Artin braid groups.

Abstract

Let $n\ge2$. In this note we give a short uniform proof of property $R_\infty$ for the Artin-Tits groups of spherical types $A_n$, $B_n$, $D_4$, $I_2(m)$ ($m\ge3$), their pure subgroups, and for the Artin-Tits groups of affine types $\widetilde A_{n-1}$ and $\widetilde C_n$. In particular, we provide an alternative proof of a recent result of Dekimpe, Gon\c{c}alves and Ocampo, who established property $R_{\infty}$ for pure Artin braid groups.