  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 $[42\sqrt{n}+860]^{-1}$, which gives
the exact asymptotic behavior of the Kazhdan constant, as $n$ goes to infinity,
since $\sqrt{2/n}$ 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.
