The first eigenvalue of the Laplacian for the compact quotient of a certain Riemannian symmetric space (Q1098433)

Let G be a non-compact semisimple Lie group with finite centre and let K be a maximal compact subgroup of G. Let \(M=G/K\) with the metric induced from the Killing form. Let \(\Gamma\subset G\) be a co-compact discrete subgroup acting freely on M so \(M(\Gamma)=\Gamma \setminus G/K\) is a smooth manifold. Let \(\lambda_ 1(\Gamma)\) be the first positive eigenvalue of the Laplacian on M(\(\Gamma)\). Let \(\rho\) be the sum of the positive roots. If G is complex, the author shows \[ \limsup_{vol(M(\Gamma))\mapsto \infty} \lambda_ 1(\Gamma)\leq | \rho |^ 2; \] this result was obtained for the unit disk with Poincaré metric by \textit{H. Huber} [Comment. Math. Helv. 49, 251-259 (1974; Zbl 0287.35075)] and generalized to rank 1 symmetric spaces by \textit{H. Urakawa} [Nagoya Math. J. 78, 137-152 (1980; Zbl 0436.53049)].