Computing Kazhdan Constants by Semidefinite Programming

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, G₁, …, G₄, 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 $\mathrm{SL}(3,\mathbb{Z})$ and $\mathrm{SL}(4,\mathbb{Z})$, 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 ${\mathrm{St}}_3\left(\mathbb{Z}\right)$.

KEYWORDS:

• Kazhdan's property (T)
• Kazhdan constant
• semidefinite programming
• Ã₂ -buildings

2000 AMS SUBJECT CLASSIFICATION:

• 22D10
• 90C22
• 16S34

Acknowledgments

We would like to thank Uri Bader, Pierre-Emmanuel Caprace, Mike Davis, Ian Leary, Pierre Pansu, Narutaka Ozawa, and Alain Valette. We are benefitted from Kawakami’s paper [Kawakami 15]. We are grateful to the referee, whose comments improved the presentation.

Funding

We are supported by Grant-in-Aid for Scientific Research (No. 15H05739). A part of the work was done while the first author was at MSRI during Fall 2016 semester supported by NSF Grant No. DMS-1440140.