Endo-Twisted Conjugacy and Outer Fixed Points in Solvable Baumslag--Solitar Groups

TL;DR

This work solves the endo-twisted conjugacy problem for solvable Baumslag--Solitar groups $BS(1,n)$ by classifying endomorphisms into two types and reducing the problem to computable algebraic checks. It develops a framework linking endo-twisted conjugacy to outer fixed points via specialized endomorphisms, and proves decidability through a sequence of reductions that culminate in solvable Diophantine-type conditions. A key contribution is the algorithmic bridge between fixed-point dynamics and twisted conjugacy, including the introduction of weakly outer fixed points and their role in the decision procedure. The results advance algorithmic understanding of twisted conjugacy in HNN-extensions and illuminate fixed-point phenomena in $BS(1,n)$ with potential implications for Nielsen fixed-point theory in this setting.

Abstract

In this article, we solve the twisted conjugacy problem with respect to endomorphisms for solvable Baumslag--Solitar groups $BS(1,n)$, i.e., we propose an algorithm which, given two elements $u,v \in BS(1,n)$ and an endomorphism $ψ\in End(BS(1,n))$, decides whether $v=(xψ)^{-1} u x$ for some $x\in BS(1,n)$. Also, we connect the outer fixed points of a given endomorphism $ψ$ with $\varphi$-twisted conjugacy problem for two words $u, v \in BS(1,n)$, where $\varphi \in End(BS(1,n))$ and $u, v$ depend on $ψ$. Furthermore, we define the weakly (outer) fixed points and discuss its interplay with the endo-twisted conjugacy problem in $BS(1, n)$.

Theorems & Definitions (20)

• Lemma 2.1
• Lemma 2.2
• Proposition 2.3: J. O'Neill, N
• Lemma 2.5
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Definition 4.1