On an Analog of Selberg's Eigenvalue Conjecture for SL_3(Z)

Abstract

Let H be the homogeneous space associated to the group PGL_3(R). Let X=\Gamma/H where \Gamma=SL_3(Z) and consider the first non-trivial eigenvalue \lambda_1 of the Laplacian on L^2(X). Using geometric considerations, we prove the inequality \lambda_1<pi^2/10. Since the continuous spectrum is represented by the band [1,\infty), our bound on \lambda_1 can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.