no. 4
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.

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 
@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},
     pages = {749--770},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {4},
     year = {2004},
     mrnumber = {2124587},
     zbl = {1170.11307},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2004_5_3_4_749_0/}
}
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
SP  - 749
EP  - 770
VL  - 3
IS  - 4
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2004_5_3_4_749_0/
LA  - en
ID  - ASNSP_2004_5_3_4_749_0
ER  - 
%0 Journal Article
%A Belolipetsky, Mikhail
%T On volumes of arithmetic quotients of $SO (1, n)$
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2004
%P 749-770
%V 3
%N 4
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2004_5_3_4_749_0/
%G en
%F 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://archive.numdam.org/item/ASNSP_2004_5_3_4_749_0/

[BG] M. Belolipetsky - W. T. Gan, The mass of unimodular lattices, J. Number Theory, to appear. | MR | Zbl

[BP] A. Borel - G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 119-171; Addendum, ibid. 71 (1990), 173-177. | Numdam | MR

[B] N. Bourbaki, “Groups et Algèbres de Lie, chapitres IV, V et VI”, Paris, Hermann, 1968.

[BT] F. Bruhat - J. Tits, Groupes réductifs sur un corps local, I; II, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5-251; 60 (1984), 5-184. | Numdam | MR | Zbl

[BFPOD] J. Buchmann - D. Ford - M. Pohst - M. Oliver - F. Diaz Y Diaz, Tables of number fields of low degree, ftp://megrez.math.u-bordeaux.fr/pub/numberfields/.

[CR] V. I. Chernousov - A. A. Ryzhkov, On the classification of maximal arithmetic subgroups of simply connected groups, Sb. Math. 188 (1997), 1385-1413. | MR | Zbl

[CF] T. Chinburg - E. Friedman, The smallest arithmetic hyperbolic three-orbifold, Invent. Math. 86 (1986), 507-527. | MR | Zbl

[CFJR] T. Chinburg - E. Friedman - K. N. Jones - A. W. Reid, The arithmetic hyperbolic 3-manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 1-40. | Numdam | MR | Zbl

[D] M. Davis, A hyperbolic 4-manifold, Proc. Amer. Math. Soc. 93 (1985), 325-328. | MR | Zbl

[EM] B. Everitt - C. Maclachlan, Constructing hyperbolic manifolds, In: “Computational and geometric aspects of modern algebra (Edinburgh, 1998)”, London Math. Soc. Lecture Note Ser., No. 275, Cambridge Univ. Press, Cambridge, 2000, pp. 78-86. | MR | Zbl

[GHY] W. T. Gan - J. Hanke - J.-K. Yu, On an exact mass formula of Shimura, Duke Math. J. 107 (2001), 103-133. | MR | Zbl

[Gi] G. W. Gibbons, Real tunnelling geometries, Classical Quantum Gravity 15 (1998), 2605-2612. | MR | Zbl

[Gr] B. H. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), 287-313. | MR | Zbl

[H] G. Harder, Halbeinfache Gruppenschemata ber Dedekindringen, Invent. Math. 4 (1967), 165-191. | MR | Zbl

[K] R. Kottwitz, Sign changes in Harmonic Analysis on Reductive Groups, Trans. Amer. Math. Soc. 278 (1983), 289-297. | MR | Zbl

[Le] S. Levy (ed.), “The eightfold way. The beauty of Klein's quartic curve”, Math. Sci. Res. Inst. Publ. 35, Cambridge Univ. Press, Cambridge, 1999. | MR | Zbl

[Lu] A. Lubotzky, Lattice of minimal covolume in SL 2 : a nonarchimedean analogue of Siegel’s theorem μ≥π/21, J. Amer. Math. Soc. 3 (1990), 961-975. | MR | Zbl

[Od] A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. Théor. Nombres Bordeaux (2) 2 (1990), 119-141. | Numdam | MR | Zbl

[Ono] T. Ono, On algebraic groups and discontinuous groups, Nagoya Math. J. 27 (1966), 279-322. | MR | Zbl

[Pl] V. P. Platonov, On the maximality problem of arithmetic groups, Soviet Math. Dokl. 12 (1971), 1431-1435. | MR | Zbl

[P] G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91-117. | Numdam | MR | Zbl

[RT] J. Ratcliffe - S. Tschantz, Volumes of integral congruence hyperbolic manifolds, J. Reine Angew. Math. 488 (1997), 55-78. | MR | Zbl

[R] J. Rohlfs, Die maximalen arithmetisch definierten Untergruppen zerfallender einfacher Gruppen, Math. Ann. 244 (1979), 219-231. | MR | Zbl

[S] J.-P. Serre, Cohomologie des groupes discrets, In: “Prospects in mathematics”, Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, 1971, pp. 77-169. | MR | Zbl

[T] J. Tits, Reductive groups over local fields | MR | Zbl

[V] E. B. Vinberg, On groups of unit elements of certain quadratic forms, Math. USSR-Sb. 16 (1972), 17-35. | Zbl

[W] A. Weil, “Adèles and algebraic groups”, Birkhäuser, Boston, 1982. | MR | Zbl