On an analogue of Selberg’s eigenvalue conjecture for $\text {SL}_3(\mathbf {Z})$

Abstract:

Let $\mathcal {H}$ be the homogeneous space associated to the group $\text {PGL}_{3}(\mathbf {R})$. Let $X = \Gamma {\backslash \mathcal {H}}$ where $\Gamma = \text {SL}_{3}(\mathbf {Z})$ and consider the first nontrivial eigenvalue $\lambda _{1}$ of the Laplacian on $L^{2}(X)$. Using geometric considerations, we prove the inequality $\lambda _{1} > 3\pi ^{2}/10$. 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.