A note on fine graphs and homological isoperimetric inequalities (Q2788801)

A graph \(\Gamma \) is \textit{fine} if for every edge \(e\) and every integer \(L>0\), the number of circuits of length at most \(L\) that contain \(e\) is finite.NEWLINENEWLINEThe authors show that the 1-skeleton of a simply connected 2-complex with a linear isoperimetric inequality, a bound on the length of attaching maps of the 2-cells, and finitely many 2-cells adjacent to any edge is fine.NEWLINENEWLINELet \(G\) be a group and let \(S\) be a finite collection of subgroups. The author shows that \(G\) is hyperbolic relative to \(S\) if and only if \(G\) acts cocompactly with finite edge stabilizers on a connected 2-dimensional cell complex with a linear isoperimetric inequality and \(S\) is a collection of representatives of conjugacy classes of vertex stabilizers.