Computing Kazhdan constants by semidefinite programming

TL;DR

This paper introduces a semidefinite programming method to compute lower bounds for Kazhdan constants of discrete groups, providing new bounds and insights into property (T) and spectral gaps.

Contribution

The authors develop a computational approach using semidefinite programming to estimate Kazhdan constants, achieving bounds that match known values and conjecturing spectral gaps for certain groups.

Findings

Numerical bounds match known Kazhdan constants for $ ilde{A}_2$-groups.

Lower bounds for $ ext{SL}(3,Z)$ and $ ext{SL}(4,Z)$ are improved.

Conjecture of spectral gap $( oot 2 rom 1 - 1)^2$ for specific groups.

Abstract

Kazhdan constants of discrete groups are hard to compute and the actual constants are known only for several classes of groups. By solving a semidefinite programming problem by a computer, we obtain a lower bound of the Kazhdan constant of a discrete group. Positive lower bounds imply that the group has property (T). We study lattices on $\tilde{A}_2$-buildings in detail. For $\tilde{A}_2$-groups, our numerical bounds look identical to the known actual constants. That suggests that our approach is effective. For a family of groups, $G_1, \cdots, G_4$, that are studied by Ronan, Tits and others, we conjecture the spectral gap of the Laplacian is $(\sqrt 2-1)^2$ based on our experimental results. For $\mathrm{SL}(3,\Bbb Z)$ and $\mathrm{SL}(4,\Bbb Z)$ we obtain lower bounds of the Kazhdan constants, 0.2155 and 0.3285, respectively, which are better than any other known bounds. We also obtain 0.1710 as a lower bound of the Kazhdan constant of the Steinberg group $\mathrm{St}_3(\Bbb Z)$.