Kazhdan Constants for $SL_n(Z)$

TL;DR

This paper establishes a new lower bound for the Kazhdan constant of $SL_n(Z)$, revealing its asymptotic behavior and improving bounds for related spectral gaps and algorithmic performance.

Contribution

It provides the first explicit asymptotic lower bound for the Kazhdan constant of $SL_n(Z)$ with respect to elementary matrices.

Findings

Kazhdan constant bounded below by [42√n+860]^{-1}

Improved bounds for spectral gap of $SL_n(F_p)$

Enhanced estimates for product replacement algorithm performance

Abstract

In this article we improve the known Kazhdan constant for $SL_n(Z)$ with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by $[42\sqrt{n}+860]^{-1}$, which gives the exact asymptotic behavior of the Kazhdan constant, as $n$ goes to infinity, since $\sqrt{2/n}$ is an upper bound. We can use this bound to improve the bounds for the spectral gap of the Cayley graph of $SL_n(F_p)$ and for the working time of the product replacement algorithm for abelian groups.