On Kazhdan constants of finite index subgroups in $SL_n(\mathbb{Z})$

TL;DR

This paper investigates the Kazhdan constants of finite index subgroups in $SL_n( ext{Z})$, establishing bounds and existence results that reveal the structure and limitations of these constants in relation to subgroup properties.

Contribution

It provides new bounds and existence results for Kazhdan constants in finite index subgroups of $SL_n( ext{Z})$, including asymptotically optimal bounds for principal congruence subgroups.

Findings

Existence of subgroups with uniformly bounded Kazhdan constants

Construction of subgroups with arbitrarily small Kazhdan constants

Kazhdan constant of principal congruence subgroup $ ext{Gamma}_n(m)$ is greater than $c/m$

Abstract

We prove that for any finite index subgroup $\Ga$ in $SL_n(\mathbb{Z})$, there exists $k=k(n)\in\mathbb{N}$, $\ep=\ep(\Ga)>0$, and an infinite family of finite index subgroups in $\Ga$ with a Kazhdan constant greater than $\ep$ with respect to a generating set of order $k$. On the other hand, we prove that for any finite index subgroup $\Ga$ of $SL_n(\mathbb{Z})$, and for any $\ep>0$ and $k\in \mathbb{N}$, there exists a finite index subgroup $\Ga'\leq \Ga$ such that the Kazhdan constant of any finite index subgroup in $\Ga'$ is less than $\ep$, with respect to any generating set of order $k$. In addition, we prove that the Kazhdan constant of the principal congruence subgroup $\Gamma_n(m)$, with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than $\frac{c}{m}$, where $c>0$ depends only on $n$. For a fixed $n$, this bound is asymptotically best possible.