Aspects of negative curvature preserved under random quotients

Quotients are a powerful tool used for constructing exotic embeddings in groups that act on negatively curved metric spaces.  Models for random quotients originate in work of Gromov, Arzhantseva and Ol’shanskii where relations are sampled from spheres in free groups to study genericity of properties like hyperbolicity.  I will introduce a new model for random quotients of groups that instead samples relations using random walks and describe how this model is well-adapted to studying quotients of groups with more flexible actions on hyperbolic spaces and discuss geometric tools used to establish when these more general forms of negative curvature are preserved in random quotients. These techniques also provide new examples of groups that are quasi-isometrically rigid and exotic common quotients.  This talk will be based on joint work with Abbott, Berlyne, Mangioni, and Rasmussen.