Abstract commensurators of solvable Baumslag - Solitar groups

TL;DR

This paper characterizes the abstract commensurator groups of Baumslag-Solitar groups and explores the embeddings of automorphism groups into commensurators and quasi-isometry groups for groups with the unique root property.

Contribution

It provides an explicit description of the commensurator groups of BS(1,n) and establishes embedding results for automorphism groups in broader classes of groups.

Findings

The commensurator of BS(1,n) is isomorphic to a group of 2x2 upper triangular matrices over rationals.

Automorphisms of groups with the unique root property embed into their commensurators and quasi-isometry groups.

The results extend understanding of the structure of automorphism and commensurator groups for specific classes of groups.

Abstract

We prove that for any natural n>1, the abstract commensurator group of the Baumslag - Solitar group BS(1,n) is isomorphic to the group of 2 by 2 upper triangular matrices A over rational numbers with A_{11}=1. We also prove that for any finitely generated group G with the unique root property the natural homomorphisms Aut(G)--> Comm(G)--> QI(G) are embeddings.