Cusped spaces for hierarchically hyperbolic groups, and applications to Dehn filling quotients

We introduce a construction that simultaneously yields cusped spaces of relatively hyperbolic groups, and spaces quasi-isometric to Teichmueller metrics. We use this to study Dehn-filling-like quotients of various groups, among which mapping class groups of punctured spheres. In particular, we show that the mapping class group of a five-holed sphere (resp. the braid group on four strands) has infinite hyperbolic quotients (strongly) not isomorphic to hyperbolic quotients of any other given sphere mapping class group (resp. any other braid group). These quotients are obtained by modding out suitable large powers of Dehn twists, and we further argue that the corresponding quotients of the extended mapping class group have trivial outer automorphism groups. We obtain these results by studying torsion elements in the relevant quotients.