Separability properties of higher-rank GBS groups

TL;DR

The paper classifies separability properties for rank $n$ generalized Baumslag–Solitar groups. Using Bass–Serre theory and a modular homomorphism into $\mathrm{GL}(n,\mathbb{Q})$, it reduces questions to the structure of ascending HNN extensions and to virtually $\mathbb{Z}^n$-by-free quotients. It proves that residual finiteness and subgroup separability occur precisely in the virtually $\mathbb{Z}^n$-by-free case, while cyclic subgroup separability for ascending HNN extensions is governed by a polynomial factorization criterion on the characteristic polynomial of the edge automorphism $\varphi$, with explicit eigenvalue obstructions. These results connect algebraic invariants of the monomorphism $\varphi$ to decomposition properties and algorithmic decidability, enriching the understanding of higher-rank GBS groups.

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.

Figures (5)

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Theorems & Definitions (67)

• Theorem 1.1: Theorem \ref{['thm:css']}
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Theorem 3.1
• Proposition 3.2
• Lemma 3.3