math.GR — Group Theory

Twisted conjugacy in spherical Artin groups and their crystallographic quotients

Contributed by Jason Davies

A group has property $R_\infty$ if every automorphism has infinitely many twisted conjugacy classes. Let $A$ be an irreducible spherical Artin group of Coxeter rank at least two, with pure subgroup $P$. We classify property $R_\infty$ for the crystallographic quotient $\Gamma=A/[P,P]$. If $A$ has Coxeter rank at least three, then $\Gamma$ has $R_\infty$. In type $I_2(m)$, this holds exactly when $4\mid m$, or when $m$ is odd and $3\mid m$. Together with the known dihedral Artin-group cases, the positive quotient results imply that every irreducible spherical Artin group of rank at least two has $R_\infty$. The positive proofs combine the rational representation theory of the holonomy action with integral lattice constraints and explicit fixed-vector witnesses. In the negative dihedral cases, we construct automorphisms compatible with the crystallographic extension. For type $I_2(5)$, we exhibit an automorphism with Reidemeister number $18$, showing concretely that the quotient construction does not preserve $R_\infty$. This example, including the identification with the presented Artin quotient and the exact Reidemeister count, is formalised in Lean. An alternative proof for seven Artin groups is formalised under explicitly stated external hypotheses.