A note on finiteness properties of vertex stabilisers

TL;DR

This work introduces a higher-dimensional criterion for the FP$_n$ finiteness properties of vertex stabilisers in $G$-CW-complexes, tying these algebraic properties to the finiteness of stabilisers of higher-dimensional cells and the topology of vertex links. It develops both algebraic (FP$_n$) and finite-presentability criteria via augmented cellular chain complexes and blow-up constructions, and applies them to produce uncountably many one-ended groups of type FP$_n$ that are not FP$_{n+1}$. The results generalize Haglund–Wise’s tree-case to higher dimensions and relate to relative hyperbolicity and other homological finiteness properties, with implications for quasi-isometry classifications. The paper thereby broadens methods for transferring finiteness from higher-dimensional cell stabilisers to vertex stabilisers and provides new families of groups with prescribed FP$_n$-type properties.

Abstract

We prove a criterion for the geometric and algebraic finiteness properties of vertex stabilisers of $G$-CW-complexes, given the finiteness properties of the group $G$ and of the stabilisers of positive dimensional cells. This generalises a result of Haglund--Wise for groups acting on trees to higher dimensions. As an application, for $n\ge 2$, we deduce the existence of uncountably many quasi-isometry classes of one-ended groups that are of type $\mathsf{FP}_n$ and not of type $\mathsf{FP}_{n+1}$.

Theorems & Definitions (22)

• Theorem 1.1: Brown87
• Theorem 1.2
• Corollary 1.3: Haglund-Wise21
• Lemma 2.1
• proof
• Lemma 2.2: Bieri81
• Lemma 2.3
• proof
• Proposition 2.4
• proof