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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On an analogue of Selberg's eigenvalue conjecture for $\text{SL}_{3}(\mathbf{Z})$


Authors: Sultan Catto, Jonathan Huntley, Jay Jorgenson and David Tepper
Journal: Proc. Amer. Math. Soc. 126 (1998), 3455-3459
MSC (1991): Primary 11F55; Secondary 22E40, 11F72
MathSciNet review: 1600116
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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.


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Additional Information

Sultan Catto
Affiliation: The Graduate School and Baruch College, The City University of New York, New York, New York 10010 and Department of Physics, The Rockefeller University, 1230 York Avenue, New York, New York 10021-6339

Jonathan Huntley
Affiliation: Department of Mathematics, Baruch College CUNY, 17 Lexington Avenue, New York, New York 10010

Jay Jorgenson
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: jjorgen@littlewood.math.okstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04831-X
PII: S 0002-9939(98)04831-X
Received by editor(s): January 28, 1997
Additional Notes: The first named author acknowledges support from DOE grants DE-AC-0276-ER3074 and 3075 and PSC-CUNY Research Award No. 9203393.
The second named author acknowledges support from several PSC-CUNY grants. The third named author acknowledges support from NSF grant DMS-93-07023 and from the Sloan Foundation.
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1998 American Mathematical Society