Kazhdan Constants for SL_n(Z)

2003/11/26 by Martin Kassabov, Kassabov, Martin · 1 citation
Computer Science · Mathematics · #20E46 (Primary) 15A36 #22E40 #22E67 (Secondary) #Advanced Topics in Algebra #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #math.CO #math.GR #math.RT #msc:15A36 #msc:20E46 #msc:22E40 #msc:22E67

paper · pdf · doi:10.48550/arxiv.math/0311487

22 pages

arxiv created 2003/11/26 · openalex publication_date 2003/11/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we improve the known Kazhdan constant for SL_n(Z) with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by [42√(n)+860]^-1, which gives the exact asymptotic behavior of the Kazhdan constant, as n goes to infinity, since √(2/n) is an upper bound. We can use this bound to improve the bounds for the spectral gap of the Cayley graph of SL_n(F_p) and for the working time of the product replacement algorithm for abelian groups.

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