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Kazhdan Constants for $SL_n(Z)$

Martin Kassabov
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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 $[42\sqrt{n}+860]^{-1}$, 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.

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Martin Kassabov. 2003-11-26. Kazhdan Constants for $SL_n(Z)$. https://arxiv.org/abs/math/0311487

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