Higher-rank GBS groups: non-positive curvature and biautomaticity

TL;DR

The paper generalizes known relations between non-positive curvature and algorithmic properties to rank-$n$ GBS groups by analyzing the modular homomorphism $\Delta_{\mathcal{G}}$ into $\mathrm{GL}_n(\mathbb{Q})$. It establishes precise equivalences: biautomaticity (and virtual/biautomatic-subgroup status) corresponds to finite $\Delta_{\mathcal{G}}$-image, while CAT(0) is equivalent to the image being conjugate into $\mathrm{O}(n)$; when the image is finite, the group is biautomatic, and in particular any biautomatic rank-$n$ GBS group is CAT(0). The approach extends Leary–Minasyan’s results via embeddings into $\mathrm{AGL}_n(\mathbb{Q})$ and leverages commensurators and Bass–Serre theory to relate geometric and algorithmic properties. These criteria clarify the landscape of rank-$n$ GBS groups with respect to non-positive curvature and biautomaticity, and connect to broader questions about residually finite CAT(0) groups.

Abstract

We characterise when a rank $n$ generalised Baumslag-Solitar group is CAT(0) and when it is biautomatic.

Theorems & Definitions (9)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm:main']}\ref{['it:main-biaut']}
• Remark 3.2
• Theorem 4.1
• proof : Proof of Theorem \ref{['thm:main']}\ref{['it:main-CAT0']}