Twisted conjugacy in $BS(n,1)$

Abstract

In this article, we solve the twisted conjugacy problem for solvable Baumslag--Solitar groups $BS(n,1)$, i.e., we propose an algorithm which, given two elements $u,v\in BS(n,1)$ and an automorphism $φ\in \Aut(BS(n,1))$, decides whether $v=(w\ddot{I}†{)}^{-1}uw$ for some $w\in BS(n,1)$. Also we prove that the automorphism group $\Aut(BS(n,1))$ is orbit decidable -- given two words on the generators $u,v\in F(X)$, decide whether the corresponding elements $u,v\in G$ can be mapped to each other by some automorphism in $\Aut(BS(n,1))$.