2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69365In 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.22 pagesGroup TheoryCombinatoricsRepresentation Theory20E46 (Primary) 15A36, 22E40, 22E67 (Secondary)text