ON AN ANALOG OF SELBERGS EIGENVALUE CONJECTURE FOR SL3(Z)

Let H be the homogeneous space associated to the group PGL(3)(R). Let X = Gamma\H where Gamma = SL3(Z) and consider the first nontrivial eig envalue lambda(1) of the Laplacian on L-2(X). Using geometric consider ations, we prove the inequality lambda(1) > 3 pi(2)/10. Since the cont inuous spectrum is represented by the band [1,infinity), our bound on lambda(1) can be viewed as an analogue of Selberg's eigenvalue conject ure for quotients of the hyperbolic half space.