2017/03/13 by Koji Fujiwara, Fujiwara, Koji, Yuichi Kabaya +1
Computer Science · Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Graph theory and applications #Group Theory (math.GR) #Operator Algebras (math.OA) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1703.04555
openalex publication_date 2017/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A2-buildings in detail. For A2-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 (√ 2-1)2 based on our experimental results. For SL(3,\Bbb Z) and 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 St3(\Bbb Z).