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, p. 749-770
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
@article{ASNSP_2004_5_3_4_749_0,
     author = {Belolipetsky, Mikhail},
     title = {On volumes of arithmetic quotients of $SO (1, n)$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {4},
     year = {2004},
     pages = {749-770},
     zbl = {1170.11307},
     mrnumber = {2124587},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2004_5_3_4_749_0}
}
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://www.numdam.org/item/ASNSP_2004_5_3_4_749_0/

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