Spectral gap and character limits in arithmetic groups

TL;DR

This paper investigates the limits of characters in discrete groups, especially higher rank lattices, showing convergence of finite-dimensional representations to the regular representation using geometric analysis of trace simplices.

Contribution

It introduces new vanishing results for character limits in higher rank lattices and links these to convergence properties of group representations.

Findings

Finite-dimensional representations converge to the regular representation.

Vanishing results for limits of characters in higher rank lattices.

Analysis of the geometry of trace simplices in groups with property (T).

Abstract

We establish vanishing results for limits of characters in various discrete groups, most notably irreducible lattices in higher rank semisimple Lie groups. As an application, we show that any sequence of finite-dimensional representations converges to the regular representation in the Fell topology. We achieve this by studying the geometry of the simplex of traces of discrete groups having Kazhdan's property (T) or its relative generalizations.