Margulis Lemma and RCD spaces

The classic Margulis Lemma states that in a hyperbolic manifold, the subgroup of the fundamental group generated by small loops around a certain point is virtually abelian.

I will present some generalizations of this result, includingrecent work (with Qin Deng, Jaime Santos, and Xinrui Zhao) where we include RCD spaces, a non-smooth analogue of manifolds with a lower bound on the Ricci curvature.