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 -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 .

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