Relative cohomological dimension of a relatively hyperbolic pair

TL;DR

The paper proves that for a torsion-free group G relatively hyperbolic to a subgroup H, the relative cohomological dimension cd_{\mathbb{Z}}(G,H) is finite by constructing a finite-type free resolution of the G-module Δ_{G/H}. It further shows that cd_{\mathbb{Z}}(G,H) is preserved under quasi-isometries when the peripheral subgroup H is unconstricted and of type F_{\infty}, by tying relative cohomology to the Bowditch boundary via Manning–Wang and Groff's boundary invariance results. The authors establish corollaries linking finiteness of cd_{\mathbb{Z}}(G,H) to that of H, and provide a formula cd_{\mathbb{Z}}(G,H) = sup{i : H^{i}(G,H;\mathbb{Z}G) ≠ 0}, with several explicit examples illustrating computations of cd in diverse relativized settings. The work also generalizes to multiple peripherals and demonstrates how these invariants can distinguish relatively hyperbolic groups up to quasi-isometry, connecting algebraic invariants to boundary topology and geometric group theory. The results offer a robust algebraic toolkit for understanding the interplay between relative cohomology, boundary geometry, and quasi-isometric classifications in relatively hyperbolic groups.

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.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Corollary 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Definition 2.1
• Definition 2.2