Separability properties of higher-rank GBS groups

TL;DR

This paper classifies higher-rank generalized Baumslag-Solitar groups based on their separability properties, such as residual finiteness and subgroup separability, providing a comprehensive understanding of their algebraic structure.

Contribution

It offers a complete classification of these groups in terms of various separability properties, extending the understanding of their algebraic and geometric features.

Findings

Determines conditions for residual finiteness.

Identifies when groups are subgroup separable.

Establishes criteria for cyclic subgroup separability.

Abstract

A rank $n$ generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. In this paper we classify these groups in terms of their separability properties. Specifically, we determine when they are residually finite, subgroup separable and cyclic subgroup separable.