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

TL;DR

We prove that property $R_ty$ holds for Artin--Tits groups of spherical types $A_n$, $B_n$, $D_4$, $I_2(m)$ with $m\ge3$ (and $m\neq\infty$), their pure subgroups, and for affine types $A$-tilde and $C$-tilde, via a uniform strategy based on central quotients. By passing to $ ext{Gamma}=G/Z(G)$ and realizing $ ext{Gamma}$ and $ ext{Aut}( ext{Gamma})$ as finite-index subgroups of the extended mapping class group of a punctured surface, we obtain non-elementary actions on the curve complex, and apply Delzant's lemma together with a coset argument to deduce infinitely many twisted conjugacy classes for every automorphism. This yields $R_ $infty$ for these groups, and in particular provides a short independent proof of $R_ $infty$ for pure Artin braid groups; the approach also furnishes new cases beyond previously known results. The work connects Reidemeister theory with geometric group actions, offering a uniform framework across spherical and affine types.

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, Gonçalves and Ocampo, who established property $R_{\infty}$ for pure Artin braid groups.

Figures (1)

• Figure 1: The defining Coxeter graphs for the Artin--Tits groups under consideration.

Theorems & Definitions (12)

• Theorem 1
• Proposition 2: See e.g. FGD10
• Lemma 3
• proof
• Proposition 4: cf. FGD10
• proof
• Lemma 5: FGD10,LevittLustig
• Theorem 6: Fel
• Proposition 7
• proof