I will explain several results relating the volume and the injectivity radius of locally symmetric manifolds, focusing mainly on the recently proved conjecture of Margulis:

Theorem: Let M be an irreducible locally symmetric manifold of rank at least 2. If M has infinite volume, then it admits injected contractible balls of arbitrary large radius. 

This result implies the celebrated normal subgroup theorem of Margulis: Let L be a higher rank arithmetic group and N a nontrivial normal subgroup, then |L:N| is finite.In fact the theorem implies that any infinite index subgroup of L admits a sequence of conjugates that converges to the trivial group (in the topology of subgroups).

The case where the isometry group of the universal cover has property (T) obtained in a joint paper with M. Fraczyk (Annals 2023) and the general case in a recent paper with U. Bader and A. Levit. The general case required us to proved a  spectral gap theorem for irreducible actions of product groups.

Both works, as well as other related results that I may mention, relied on the theory of random subgroups which has been developed in the last 15 years and repeatedly proved to be a successful new approach to study discrete subgroups of Lie groups and locally symmetric manifolds.