Kazhdan constants for Chevalley groups over the integers

TL;DR

This paper establishes explicit lower bounds for Kazhdan constants of Chevalley groups over integers, revealing new insights into their spectral properties and group structure, especially for types other than A_n.

Contribution

It introduces a novel method linking root system structures to the Laplace operator, providing the first sharp bounds for many Chevalley groups.

Findings

First explicit asymptotically sharp bounds for Chevalley groups (excluding type A_n)

New connection between root system gradings and Laplace operator behavior

Enhanced understanding of Kazhdan constants in algebraic groups

Abstract

We compute lower bounds for Kazhdan constants of Chevalley groups over the integers, endowed with the standard Steinberg generators. For types other than $\mathtt{A}_{n}$, these are the first explicit asymptotically sharp such bounds. The method relies on establishing a new connection between the structure of a root system grading a family of groups and the behaviour of the square of the Laplace operator in the family.