Strong spectral gaps for compact quotients of products of $PSL(2,ℝ)$)

Kelmer, 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 -