Twisted conjugacy in $BS(n, 1)$

TL;DR

This paper presents an algorithm to solve the twisted conjugacy problem in solvable Baumslag--Solitar groups $BS(n,1)$ and proves that their automorphism groups are orbit decidable, advancing understanding of their algebraic structure.

Contribution

It introduces an explicit algorithm for the twisted conjugacy problem in $BS(n,1)$ and establishes orbit decidability of their automorphism groups, a novel result in this context.

Findings

Algorithm for twisted conjugacy problem in $BS(n,1)$

Proof of orbit decidability for automorphism groups

Enhanced understanding of automorphism structure in Baumslag--Solitar groups

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 $\varphi \in \Aut(BS(n,1))$, decides whether $v=(w\varphi)^{-1} u w$ 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 $\Aut(BS(n,1))$.