Geodesic
Strong spectral gaps for compact quotients of products of PSL(2,ℝ)
Journal of the European Mathematical Society, Tome 11 (2009) no. 2, pp. 283-313 Cet article a éte moissonné depuis la source EMS Press

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The existence of a strong spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan–Selberg Conjectures. If G has no compact factors then for general lattices a spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice Γ in G=PSL(2,R)d for d≥2 which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting.
DOI : 10.4171/jems/151
Classification : 22-XX, 00-XX
Keywords: Spectral gap, Selberg trace formula, product of hyperbolic planes 
@article{JEMS_2009_11_2_a3,
     author = {Dubi Kelmer and Peter Sarnak},
     title = {Strong spectral gaps for compact quotients of products of {PSL(2,\ensuremath{\mathbb{R}})}},
     journal = {Journal of the European Mathematical Society},
     pages = {283--313},
     year = {2009},
     volume = {11},
     number = {2},
     doi = {10.4171/jems/151},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/151/}
}
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Dubi Kelmer; Peter Sarnak. Strong spectral gaps for compact quotients of products of PSL(2,ℝ). Journal of the European Mathematical Society, Tome 11 (2009) no. 2, pp. 283-313. doi: 10.4171/jems/151

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