On fixed points and stabilizers in solvable Baumslag--Solitar groups

Abstract

In this article, we study the fixed-point subgroups of the solvable Baumslag-Solitar groups $\BS(1,n)= \langle a, t \mid t a t^{-1} = a^{n} \rangle$, $n>1$ of automorphisms and endomorphisms. We also investigate the stabilizers of subgroups of $\BS(1,n)$, considered as subgroups of the group of automorphisms and submonoids of the monoid of endomorphisms of $\BS(1,n)$. We show that the fixed-point subgroups of automorphisms are either infinite cyclic (in which case, a generator is computable), or they are equal to $\mathbb{Z}\left[\tfrac{1}{n}\right]$, an infinitely generated abelian group. We further prove that the stabilizer subgroup of an element in $\BS(1,n)$ is either a finitely generated abelian group whose rank equals the number of distinct prime divisors of $n$ (and in this case, a finite generating set is computable), or it is $\mathbb{Z}\left[\tfrac{1}{n}\right]$. As a corollary, we show that for all $k \in \mathbb{N}$, every element of $\BS(1,n)$ has a unique $k$-th root. We then proceed to examine the behaviour of fixed-point subgroups and stabilizers under endomorphisms and find similar results. We prove that the fixed point subgroups of endomorphisms are again infinite cyclic or $\mathbb{Z}\left[\tfrac{1}{n}\right]$, but the stabilizer submonoids are always infinitely generated.