Relative cohomological dimension of a relatively hyperbolic pair

TL;DR

This paper investigates the properties of the relative cohomological dimension in relatively hyperbolic pairs, establishing finiteness and invariance under quasi-isometries for torsion-free groups, and computing this dimension in various cases.

Contribution

It proves the finiteness and invariance of the relative cohomological dimension for torsion-free relatively hyperbolic pairs and computes this dimension in specific scenarios.

Findings

Relative cohomological dimension is finite for torsion-free groups.

This dimension is preserved under quasi-isometries under certain conditions.

Explicit calculations of the cohomological dimension in various cases.

Abstract

We show that the relative cohomological dimension $\cd(G,H)$ of a relatively hyperbolic pair $(G,H)$ is always finite when $G$ is torsion-free. We also show that this dimension is preserved under quasi-isometries, provided that $G$ is torsion-free and the peripheral subgroup $H$ is unconstricted and of type $F_{\infty}$. As a corollary of our methods, we compute $\cd(G,H)$ in a range of cases.