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 -buildings in detail. For
-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
based on our experimental results. For
and
, 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
.
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 [CitationKawakami 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.