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

TL;DR

This paper presents an algorithm to solve the twisted conjugacy problem in solvable Baumslag--Solitar groups and explores the relationship between outer fixed points and twisted conjugacy, advancing understanding of endomorphisms in these groups.

Contribution

It introduces a novel algorithm for the twisted conjugacy problem in $BS(1,n)$ and links outer fixed points with twisted conjugacy, providing new insights into endomorphism dynamics.

Findings

Algorithm successfully decides twisted conjugacy in $BS(1,n)$

Establishes connection between outer fixed points and twisted conjugacy

Defines and discusses weakly fixed points in the context of $BS(1,n)$

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 $\psi \in End(BS(1,n))$, decides whether $v=(x\psi)^{-1} u x$ for some $x\in BS(1,n)$. Also, we connect the outer fixed points of a given endomorphism $\psi$ 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 $\psi$. Furthermore, we define the weakly (outer) fixed points and discuss its interplay with the endo-twisted conjugacy problem in $BS(1, n)$.