Property 𝑅∞ for some spherical and affine Artin–Tits groups

Abstract

Abstract Let n ⩾ 2 n\geqslant 2 . In this note, we give a short uniform proof of property R ∞ R_{\infty} for the Artin–Tits groups of spherical types A n A_{n} , B n B_{n} , D 4 D_{4} , I 2 ⁢ ( m ) I_{2}(m) ( m ⩾ 3 m\geqslant 3 ), their pure subgroups, and for the Artin–Tits groups of affine types A ~ n - 1 \tilde{A}_{n-1} and C ~ n \tilde{C}_{n} . In particular, we provide an alternative proof of a recent result of Dekimpe, Gonçalves and Ocampo, who established property R ∞ R_{\infty} for pure Artin braid groups.