A note on fine graphs and homological isoperimetric inequalities

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.

Version accepted by the Canadian Mathematical Bulletin

Cited by in corpus (3)

• Dismantlable classifying space for the family of parabolic subgroups of a relatively hyperbolic group
• Bounded cohomology of classifying spaces for families of subgroups
• Finiteness of Homological Filling Functions