A characterization of relatively hyperbolic groups via bounded cohomology

TL;DR

This paper provides a new proof of a key property of relatively hyperbolic groups related to bounded cohomology and establishes a converse result under weaker conditions, advancing the understanding of their algebraic structure.

Contribution

It offers an alternative proof of a known surjectivity result and proves the converse implication with weaker hypotheses for relatively hyperbolic groups.

Findings

New proof of the surjectivity of the comparison map for relatively hyperbolic pairs.

Establishment of the converse implication under weaker hypotheses.

Enhanced understanding of the cohomological characterization of relatively hyperbolic groups.

Abstract

It was proved by Mineyev and Yaman that, if $(\Gamma, \Gamma')$ is a relatively hyperbolic pair, the comparison map $$ H_b^k(\Gamma, \Gamma'; V) \to H^k(\Gamma, \Gamma'; V) $$ is surjective for every $k \ge 2$, and any bounded $\Gamma$--module $V$. By exploiting results of Groves and Manning, we give another proof of this result. Moreover, we prove the opposite implication under weaker hypotheses than the ones required by Mineyev and Yaman.