## upper bound on first non-zero eigenvalue of the Laplacian

I'm looking for an upper bound on the first non-zero eigenvalue of the Laplace-Beltrami operator on compact manifolds of dimension greater than four that have constant negative curvature. In particular, I would like to know whether or not there is an upper bound that is less than $\frac{1}{2}n^2 - 2$, where $n$ is the dimension, for the case when the sectional curvature is $-1$. I can find quite a few papers on lower bounds, but the few I could find on upper bounds required information that is hard to obtain. Any help would be greatly appreciated.

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Do you want an upper bound that depends on the dimension only? And is $\frac{1}{2}n^2-2$ known to be an upper bound? – Deane Yang Dec 16 at 1:20
I don't know an explicit upper bound, but one can be obtained by combining the Margulis constant, which gives a lower bound on the maximal injectivity radius, and use an estimate of the Dirichlet eigenvalues on the ball of that radius to get an upper bound of the eigenvalue on the manifold. You might have a look at the book by Gallot-Hulin-Lafontaine which gives an explicit estimate of the eigenvalue in terms of the injectivity radius (I don't have a copy now to give the chapter). – Agol Dec 16 at 6:09
What about Buser inequality? (this is a "converse inequality" to the Cheeger inequality). – Asaf Dec 16 at 7:25