logo AFST
Maximal radius of quaternionic hyperbolic manifolds
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 5, pp. 875-896.

Nous donnons une borne inférieure explicite sur le rayon d’une boule plongée dans une variété hyperbolique quaternionique (le rayon maximal). Nous en déduisons une minoration du volume de telles variétés. Les deux bornes exhibées décroissent avec la dimension, et il n’est pas clair que l’on doive s’attendre au même comportement pour le rayon maximal ou pour le volume minimal. En lien avec cette question, nous remarquons cependant que la constante de Margulis de l’espace hyperbolique quaternionique de dimension n est inférieure à C/n, et décroit donc quand la dimension augmente.

We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold (the maximal radius). We then deduce a lower bound on the volume of a quaternionic hyperbolic manifold. Both those bounds decrease with the dimension, when it is not clear that it should be the behaviour of the maximal radius or of the minimal volume. Related to that question, we note however that the Margulis constant of the quaternionic hyperbolic space of dimension n is smaller than C/n, so is decreasing as the dimension grows.

Reçu le : 2015-09-21
Accepté le : 2015-10-14
Publié le : 2019-01-21
DOI : https://doi.org/10.5802/afst.1586
@article{AFST_2018_6_27_5_875_0,
     author = {Zo\'e Philippe},
     title = {Maximal radius of quaternionic hyperbolic manifolds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {5},
     year = {2018},
     pages = {875-896},
     doi = {10.5802/afst.1586},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2018_6_27_5_875_0/}
}
Zoé Philippe. Maximal radius of quaternionic hyperbolic manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 5, pp. 875-896. doi : 10.5802/afst.1586. https://afst.centre-mersenne.org/item/AFST_2018_6_27_5_875_0/

[1] Ilesanmi Adeboye Lower bounds for the volume of hyperbolic n-orbifolds, Pac. J. Math., Volume 237 (2008) no. 1, pp. 1-19 | Article | MR 2415204

[2] Ilesanmi Adeboye; Guofang Wei On volumes of hyperbolic orbifolds, Algebr. Geom. Topol., Volume 12 (2012) no. 1, pp. 215-233 | Article | MR 2916274

[3] Ilesanmi Adeboye; Guofang Wei On volumes of complex hyperbolic orbifolds, Mich. Math. J., Volume 63 (2014) no. 2, pp. 355-369 | Article | MR 3215554

[4] Mikhail Belolipetsky On volumes of arithmetic quotients of SO (1,n), Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 3 (2004) no. 4, pp. 749-770

[5] Mikhail Belolipetsky Hyperbolic orbifolds of small volume (2014) (https://arxiv.org/abs/1402.5394)

[6] Mikhail Belolipetsky; Vincent Emery On volumes of arithmetic quotients of PO (n,1) , n odd, Proc. Lond. Math. Soc., Volume 105 (2012) no. 3, pp. 541-570

[7] Su-Shing Chen; L. Greenberg Hyperbolic spaces, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, 1974, pp. 49-87 | Zbl 0295.53023

[8] Kevin Corlette Archimedean superrigidity and hyperbolic geometry, Ann. Math., Volume 135 (1992) no. 1, pp. 165-182

[9] Douglas R. Farenick; Barbara A. F. Pidkowich The spectral theorem in quaternions, Linear Algebra Appl., Volume 371 (2003), pp. 75-102 | Zbl 1030.15015

[10] Shmuel Friedland; Sa’ar Hersonsky Jorgensen’s inequality for discrete groups in normed algebras, Duke Math. J., Volume 69 (1993) no. 3, pp. 593-614

[11] Matthieu Gendulphe Systole et rayon interne des variétés hyperboliques non compactes, Geom. Topol., Volume 19 (2015) no. 4, pp. 2039-2080

[12] Alfred Gray Tubes, Progress in Mathematics, Volume 221, Birkhäuser, 2004, xiii+280 pages | Zbl 1048.53040

[13] Mikhail Gromov Hyperbolic groups, Essays in group theory (Mathematical Sciences Research Institute Publications) Volume 8, Springer, 1987, pp. 75-263

[14] Mikhail Gromov; Richard Schoen Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. Math., Inst. Hautes Étud. Sci. (1992) no. 76, pp. 165-246

[15] Marc Hindry Arithmétique, Tableau Noir, Volume 102, Calvage et Mounet, 2008, xvi+328 pages | Zbl 1135.11001

[16] Michael Kapovich Representations of polygons of finite groups, Geom. Topol., Volume 9 (2005), pp. 1915-1951 | Zbl 1163.20029

[17] David A. Každan; Grigoriĭ A. Margulis A proof of Selberg’s conjecture, Math. USSR, Sb., Volume 4 (1968) no. 1, pp. 147-152 | Zbl 0241.22024

[18] Inkang Kim; John R. Parker Geometry of quaternionic hyperbolic manifolds, Math. Proc. Camb. Philos. Soc., Volume 135 (2003) no. 2, pp. 291-320

[19] Gaven J. Martin Balls in hyperbolic manifolds, J. Lond. Math. Soc., Volume 40 (1989) no. 2, pp. 257-264

[20] Gaven J. Martin On discrete Möbius groups in all dimensions: a generalization of Jørgensen’s inequality, Acta Math., Volume 163 (1989) no. 3-4, pp. 253-289

[21] John R. Parker On the volumes of cusped, complex hyperbolic manifolds and orbifolds, Duke Math. J., Volume 94 (1998) no. 3, pp. 433-464 | Article | MR 1639519

[22] William P. Thurston The geometry and topology of 3-manifolds, 1980 (electronic edition of the 1980 lecture notes, distributed by Princeton University. Available at http://library.msri.org/books/gt3m/)

[23] Hsien-chung Wang Discrete nilpotent subgroups of Lie groups, J. Differ. Geom., Volume 3 (1969), pp. 481-492 | MR 0260930

[24] BaoHua Xie; JieYan Wang; YuePing Jiang Balls in complex hyperbolic manifolds, Sci. China, Math., Volume 57 (2014) no. 4, pp. 767-774

[25] Fuzhen Zhang Quaternions and matrices of quaternions, Linear Algebra Appl., Volume 251 (1997), pp. 21-57