On the twisted conjugacy problem for large-type Artin groups

TL;DR

The paper addresses the solvability of the $ ext{TCP}$ for large-type Artin groups under finite outer automorphism groups generated by graph automorphisms and the global inversion. It uses a geometric approach by extending the action to a thickening of the Cayley complex to obtain a systolic action for the semidirect product $A_ Gamma times ig<\varphi\big>$, and leverages a known equivalence between $CP$ of the semidirect product and $TCP_\varphi$ to deduce solvability of $TCP$ when $\varphi$ has finite order (as for graph automorphisms and global inversion). Under the stated hypotheses, since $Out(A_ Gamma)$ is generated by these automorphisms, $TCP(A_ Gamma)$ follows, with corollaries asserting solvable conjugacy in certain short exact sequences and biautomaticity via systolicity. The results extend the landscape of twisted conjugacy in Artin groups beyond previously known cases and connect automorphism structure to algorithmic solvability.

Abstract

We show that the twisted conjugacy problem is solvable for large-type Artin groups whose outer automorphism group is finite, generated by graph automorphisms and the global inversion. This includes XXXL Artin groups whose defining graph is connected, twistless, and not an even edge; and large-type Artin groups whose defining graph admits a twistless hierarchy terminating in twistless stars.

Figures (2)

• Figure 1: Subdivision of a precell with $m_{st}=7$.
• Figure 2: Adding a zigzag (black) between two intersecting precells (red and blue).

Theorems & Definitions (11)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: bmv2024homomorphisms,huang2024rigidity
• Theorem 2.6: soton69565
• proof : Proof of Theorem \ref{['thm:TCP']}