Strong Spectral Gaps for Compact Quotients of Products of $PSL2,bR$ - Mathematics > Number TheoryReport as inadecuate




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Abstract: The existence of a strong spectral gap for quotients $\Gamma\bs G$ ofnoncompact connected semisimple Lie groups is crucial in many applications. Forcongruence lattices there are uniform and very good bounds for the spectral gapcoming from the known bounds towards the Ramanujan-Selberg Conjectures. If $G$has no compact factors then for general lattices a strong spectral gap canstill be established, however, there is no uniformity and no effective boundsare known. This note is concerned with the strong spectral gap for anirreducible co-compact lattice $\Gamma$ in $G=\PSL2,\bbR^d$ for $d\geq 2$which is the simplest and most basic case where the congruence subgroupproperty is not known. The method used here gives effective bounds for thespectral gap in this setting.



Author: Dubi Kelmer, Peter Sarnak

Source: https://arxiv.org/



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