Given a metric space X, we call a discrete subgroup G of Isom(X) confined if the quotient X/G has uniformly bounded injectivity radius. When X is a higher-rank symmetric space, Fraczyk and Gelander showed that all confined subgroups are lattices. When X is a rank-1 symmetric space, Gekhtman and Levit showed that all confined subgroups are as “big” as normal subgroups of lattices. In this talk, I will discuss an analogous result about confined subgroups acting on Gromov hyperbolic spaces and Teichmüller space. Joint work with Ilya Gekhtman, Wenyuan Yang and Tianyi Zheng.