A Characterization of Relative Hyperbolicity via Morse and Contracting Boundaries

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let $G$ be a finitely generated group with a finite collection $\mathcal{H}$ of finitely generated subgroups, and let $G^h$ denote the associated cusped space. We prove that the pair $(G,\mathcal{H})$ is non-elementary relatively hyperbolic if and only if the Morse boundary ${\mathrm{\partial }}_M^{\mathcal{D}\mathcal{L}}{G}^h$ or the contracting boundary ${\mathrm{\partial }}_c^{\mathcal{F}\mathcal{Q}}{G}^h$ is non-empty and compact.