##### Department of Mathematics,

University of California San Diego

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### Math 211A - Algebra Seminar

## Professor Tsachik Gelander

#### Northwestern University

## Spectral gap for irreducible subgroups and a strong version of Margulis normal subgroup theorem.

##### Abstract:

Let \(\Gamma\) be a discrete group. A subgroup \(N\) is called confined if there is a finite set \(F\) in \(\Gamma\) which intersects every conjugate of \(N\) outside the trivial element. For example, a nontrivial normal subgroup is confined.

A discrete subgroup of a semisimple Lie group is confined if the corresponding locally symmetric orbifold has bounded injectivity radius. We proved a generalization of the celebrated NST: Let \(\Gamma\) be an irreducible lattice in a higher rank semisimple Lie group G. Let \(N<\Gamma\) be a confined subgroup. Then \(N\) is of finite index.

The case where \(G\) has Kazhdan's property (T) was established in my joint work with Mikolaj Frakzyc. As in the original NST, without property (T) the problem is considerably harder. The main part is to prove a spectral gap for \(L_2(G/N)\).

This is a joint work with Uri Bader and Arie Levit.

Host: Alireza Golsefidy

### April 29, 2024

### 3:00 PM

APM 7321

Research Areas

Algebra Ergodic Theory and Dynamical Systems Functional Analysis / Operator Theory Representation Theory****************************