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

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