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 [Formula: see text], which gives the exact asymptotic behavior of the Kazhdan constant, as n goes to infinity, since [Formula: see text] is an upper bound. We can use this bound to improve the bounds for the spectral gap of the Cayley graph of SL n(𝔽p) and for the working time of the product replacement algorithm for abelian groups.
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