Cohomological characterization of relative hyperbolicity and combination theorem

TL;DR

This paper provides a cohomological framework to characterize relative hyperbolicity in groups and proves a converse to the combination theorem for graphs of relatively hyperbolic groups, extending classical results.

Contribution

It introduces a cohomological characterization of relative hyperbolicity and establishes a converse to the combination theorem for such groups.

Findings

Cohomological criteria for relative hyperbolicity

Proof of the converse to the combination theorem

Extension of Gersten's ideas to relative hyperbolicity

Abstract

We give a cohomological characterization of Gromov relative hyperbolicity. As an application we prove a converse to the combination theorem for graphs of relatively hyperbolic groups given in a previous paper of the first author. We build upon, and follow the ideas of, the work of S. Gersten in ``Cohom. lower bounds for isoperim. funct. on groups'' (Topology 37, 1998) about the same topics in the classical Gromov hyperbolic setting.