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# A Note on Fine Graphs and Homological Isoperimetric Inequalities

Published:2015-11-30
Printed: Mar 2016
• Eduardo Martínez-Pedroza,
Memorial University, St. John's, Newfoundland A1C 5S7
 Format: LaTeX MathJax PDF

## 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.
 Keywords: isoperimetric functions, Dehn functions, hyperbolic groups
 MSC Classifications: 20F67 - Hyperbolic groups and nonpositively curved groups 05C10 - Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 20J05 - Homological methods in group theory 57M60 - Group actions in low dimensions

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