Actions of random quotients on hyperbolic CAT(0) cube complexes

Combinatorial nonpositive curvature of CAT(0) cube complexes plays a surprising role both in topological characterization of hyperbolic 3-manifolds and also in studying algebraic properties of random groups.
With Einstein, Krishna MS, Montee, and Steenbock, we introduce a new model for random quotients of free products that generalizes Gromov’s destiny model. I will discuss challenges that arise in this new setting, connections to work of Futer-Wise and Martin-Steenbock on cubulating quotients, as well as applications to residual finiteness using recent work of Einstein and Groves on relative cubulation.