Volume vs injectivity radius of locally symmetric manifolds
Volume vs injectivity radius of locally symmetric manifolds
听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.听
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 听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.