2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77764Let H be the homogeneous space associated to the group PGL_3(R). Let X=Γ/H where Γ=SL_3(Z) and consider the first non-trivial eigenvalue λ_1 of the Laplacian on L^2(X). Using geometric considerations, we prove the inequality λ_1<pi^2/10. Since the continuous spectrum is represented by the band [1,\infty), our bound on λ_1 can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.Spectral Theorytext