Twisted conjugacy in dihedral Artin groups II: Baumslag Solitar groups $\mathrm{BS}(n,n)$

TL;DR

This work resolves the twisted conjugacy problem for even dihedral Artin groups by identifying $G(m)$ with the Baumslag–Solitar group $\mathrm{BS}(n,n)$ and exploiting a semidirect product structure $F_{n}\rtimes \mathbb{Z}$. The authors determine the full outer automorphism group $\mathrm{Out}(G(m)) \cong D_{\infty} \times C_{2}$ and develop an algorithm that handles all outer automorphisms via shifts, ultimately reducing TCP to a finite set of representatives and an orbit-decision step. They prove that every finitely generated subgroup of $\mathrm{Aut}(G(m))$ is orbit decidable, which, together with existing results, yields decidability of the conjugacy problem in extensions $G(m) \rtimes H$ under suitable hypotheses. Complexity analysis shows TCP is linear for odd $m$ and quadratic for even $m$, and the framework provides a method to decide conjugacy in extensions, expanding the landscape of groups with solvable twisted conjugacy and conjugacy problems.

Abstract

In this second paper we solve the twisted conjugacy problem for even dihedral Artin groups, that is, groups with presentation $G(m) = \langle a,b \mid {}_{m}(a,b) = {}_{m}(b,a) \rangle$, where $m \geq 2$ is even, and $_{m}(a,b)$ is the word $abab\dots$ of length $m$. Similar to odd dihedral Artin groups, we prove orbit decidability for all subgroups $A \leq \mathrm{Aut}(G(m))$, which then implies that the conjugacy problem is solvable in extensions of even dihedral Artin groups.

Figures (2)

• Figure 1: Connections between elements of $\mathcal{D}_{u}$
• Figure 2: Connections between elements of $\overline{\mathcal{D}_{u_{0}}}$

Theorems & Definitions (86)

• Theorem 1.1
• Theorem 1.1
• Theorem 1.1
• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Proposition 2.4
• proof
• Corollary 2.5