# Strong spectral gaps for compact quotients of products of $PSL(2,\mathbb{R})$)

Journal of the European Mathematical Society (2009)

- Volume: 011, Issue: 2, page 283-313
- ISSN: 1435-9855

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topKelmer, Dubi, and Sarnak, Peter. "Strong spectral gaps for compact quotients of products of $\operatorname{PSL}(2,\mathbb {R})$)." Journal of the European Mathematical Society 011.2 (2009): 283-313. <http://eudml.org/doc/277419>.

@article{Kelmer2009,

abstract = {The existence of a strong spectral gap for quotients $\Gamma \setminus G$ 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, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice $\Gamma $ in $G=\operatorname\{PSL\}(2,\mathbb \{R\})^d$ for $d\ge 2$, 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.},

author = {Kelmer, Dubi, Sarnak, Peter},

journal = {Journal of the European Mathematical Society},

keywords = {spectral gap; Selberg trace formula; product of hyperbolic planes; spectral gap; Selberg trace formula; product of hyperbolic planes},

language = {eng},

number = {2},

pages = {283-313},

publisher = {European Mathematical Society Publishing House},

title = {Strong spectral gaps for compact quotients of products of $\operatorname\{PSL\}(2,\mathbb \{R\})$)},

url = {http://eudml.org/doc/277419},

volume = {011},

year = {2009},

}

TY - JOUR

AU - Kelmer, Dubi

AU - Sarnak, Peter

TI - Strong spectral gaps for compact quotients of products of $\operatorname{PSL}(2,\mathbb {R})$)

JO - Journal of the European Mathematical Society

PY - 2009

PB - European Mathematical Society Publishing House

VL - 011

IS - 2

SP - 283

EP - 313

AB - The existence of a strong spectral gap for quotients $\Gamma \setminus G$ 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, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice $\Gamma $ in $G=\operatorname{PSL}(2,\mathbb {R})^d$ for $d\ge 2$, 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.

LA - eng

KW - spectral gap; Selberg trace formula; product of hyperbolic planes; spectral gap; Selberg trace formula; product of hyperbolic planes

UR - http://eudml.org/doc/277419

ER -

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