A note on fine graphs and homological isoperimetric inequalities

TL;DR

This paper proves that certain 2-complexes with linear homological isoperimetric inequalities and bounded attaching maps are necessarily fine graphs, advancing the understanding of homological characterizations of relative hyperbolicity.

Contribution

It provides a positive answer to Groves and Manning's question and refines the homological criteria for relative hyperbolicity of groups.

Findings

Proves that such 2-complexes are fine graphs.

Characterizes relative hyperbolicity via group actions on 2-complexes.

Establishes equivalence between homological inequalities and group hyperbolicity.

Abstract

In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite edge stabilizers on an connected $2$-dimensional cell complex with a linear homological isoperimetric inequality and $\mathcal P$ is a collection of representatives of conjugacy classes of vertex stabilizers.