On volumes of arithmetic quotients of S O ( 1 , n )

Mikhail Belolipetsky[1]

  • [1] Sobolev Institute of Mathematics Koptyuga 4 630090 Novosibirsk, Russia and Max Planck Institute of Mathematics Vivatsgasse 7 53111 Bonn, Germany

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 4, page 749-770
  • ISSN: 0391-173X

Abstract

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We apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of S O ( 1 , n ) . As a result we prove that for any even dimension  n there exists a unique compact arithmetic hyperbolic n -orbifold of the smallest volume. We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces. We also study hyperbolic 4 -manifolds defined arithmetically and obtain a number theoretical characterization of the smallest compact arithmetic 4 -manifold.

How to cite

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Belolipetsky, Mikhail. "On volumes of arithmetic quotients of $SO (1, n)$." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.4 (2004): 749-770. <http://eudml.org/doc/84548>.

@article{Belolipetsky2004,
abstract = {We apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of $SO(1,n)$. As a result we prove that for any even dimension $n$ there exists a unique compact arithmetic hyperbolic $n$-orbifold of the smallest volume. We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces. We also study hyperbolic $4$-manifolds defined arithmetically and obtain a number theoretical characterization of the smallest compact arithmetic $4$-manifold.},
affiliation = {Sobolev Institute of Mathematics Koptyuga 4 630090 Novosibirsk, Russia and Max Planck Institute of Mathematics Vivatsgasse 7 53111 Bonn, Germany},
author = {Belolipetsky, Mikhail},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {749-770},
publisher = {Scuola Normale Superiore, Pisa},
title = {On volumes of arithmetic quotients of $SO (1, n)$},
url = {http://eudml.org/doc/84548},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Belolipetsky, Mikhail
TI - On volumes of arithmetic quotients of $SO (1, n)$
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 4
SP - 749
EP - 770
AB - We apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of $SO(1,n)$. As a result we prove that for any even dimension $n$ there exists a unique compact arithmetic hyperbolic $n$-orbifold of the smallest volume. We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces. We also study hyperbolic $4$-manifolds defined arithmetically and obtain a number theoretical characterization of the smallest compact arithmetic $4$-manifold.
LA - eng
UR - http://eudml.org/doc/84548
ER -

References

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