Property $R_\infty$ for new classes of Artin groups

TL;DR

This work extends the class of groups known to have property $R_{\infty}$ by establishing it for Artin groups of spherical type $D_n$ with $n\ge6$ and their central quotients, and for large-type hyperbolic-type free-of-infinity Artin groups, among other large-type families. The authors combine up-to-date automorphism descriptions with geometric actions on Gromov-hyperbolic spaces, embedding automorphism groups into extended mapping class groups, and applying Delzant's Lemma to produce infinitely many twisted conjugacy classes. A central technical contribution is a detailed proof of Delzant's Lemma, which underpins several previous and future results on $R_{\infty}$. The paper thus broadens the scope of Artin groups known to possess $R_{\infty}$ and provides a robust geometric framework for verifying this property across new classes.

Abstract

We establish property $R_\infty$ for Artin groups of spherical type $D_n$, $n\ge6$, their central quotients, and also for large hyperbolic-type free-of-infinity Artin groups and some other classes of large-type Artin groups. The key ingredients are recent descriptions of the automorphism groups for these Artin groups and their action on suitable Gromov-hyperbolic spaces. We also provide a detailed proof of Delzant's Lemma, an important technical tool used in our work and in several other papers on the $R_\infty$ property.

Figures (4)

• Figure 1: The Coxeter graph of type $D_n$, $n\geqslant4$.
• Figure 2: Surface $(\Sigma_n,\mathcal{P}_n)$ with $\mathcal{P}_n=\{p_1,p_2\}$ for $n$ odd and $\mathcal{P}_n=\{p_1,p_2,p_3\}$ for $n$ even. The image of the standard generator $\bar{t}_i$ of $\overline{A[D_n]}$ under the embedding of Proposition \ref{['prop:mcg']} is the Dehn twist about the circle $c_i$. The hyperelliptic involution $\iota$ rotates the surface by angle $\pi$ in the ambient space.
• Figure 3: Surface $\Sigma_n^\partial$ with the images of standard generators $t_i$ of $A[D_n]$ under the Perron--Vannier embedding. (The image of $t_i$ is the Dehn twist about the circle marked $c_i$.)
• Figure 4: The graph $\Gamma_1$ is a twistless star, and $\Gamma=\Gamma_1\cup\Gamma_2$ is a twistless hierarchy terminating in twistless stars (here $st(v_i)$ denotes the star of vertex $v_i$).

Theorems & Definitions (45)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4: See e.g. FeGoDa1
• Lemma 5: Calvez--Soroko CalSor1
• Lemma 6: Calvez--Soroko CalSor1
• Corollary 7
• proof
• Remark 8
• Theorem 9: Castel--Paris CasPar1