KAZHDAN CONSTANTS FOR SLn(ℤ)

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 (𝔽 p ) and for the working time of the product replacement algorithm for abelian groups.