Geodesic
On volumes of arithmetic quotients of SO(1,n)
Belolipetsky, Mikhail1
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, Série 5, Tome 3 (2004) no. 4, pp. 749-770

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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 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.

Classification : 11F06, 22E40, 20G30, 51M25

Belolipetsky, Mikhail 1

1 Sobolev Institute of Mathematics Koptyuga 4 630090 Novosibirsk, Russia and Max Planck Institute of Mathematics Vivatsgasse 7 53111 Bonn, Germany 
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     title = {On volumes of arithmetic quotients of $SO (1, n)$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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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, Série 5, Tome 3 (2004) no. 4, pp. 749-770. http://geodesic.mathdoc.fr/item/ASNSP_2004_5_3_4_749_0/