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

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 $φ$-twisted conjugacy problem for two words $u, v \in BS(1,n)$, where $φ\in End(BS(1,n))$ 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 $BS(1, n)$.

💡 Research Summary

The paper addresses the endo‑twisted conjugacy problem (E‑TCP) for the solvable Baumslag–Solitar groups (BS(1,n)=\langle a,t\mid a=t^{-1}a^{,n}t\rangle) with (|n|\ge 2). Given an endomorphism (\psi\in\operatorname{End}(BS(1,n))) and two group elements (u,v), the task is to decide whether there exists an element (x) such that (v=(x\psi)^{-1}ux). The authors first develop a convenient normal form for elements of (BS(1,n)): every element can be uniquely written as (a^{\alpha}t^{c}) where (\alpha\in\mathbb Z