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.

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DOI : https://doi.org/10.5802/afst.1586
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     title = {Maximal radius of quaternionic hyperbolic manifolds},
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Philippe, Zoé. 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. http://archive.numdam.org/articles/10.5802/afst.1586/

[1] Adeboye, Ilesanmi Lower bounds for the volume of hyperbolic n-orbifolds, Pac. J. Math., Volume 237 (2008) no. 1, pp. 1-19 | Article | MR 2415204 | Zbl 1149.57026

[2] Adeboye, Ilesanmi; Wei, Guofang On volumes of hyperbolic orbifolds, Algebr. Geom. Topol., Volume 12 (2012) no. 1, pp. 215-233 | Article | MR 2916274 | Zbl 1255.57015

[3] Adeboye, Ilesanmi; Wei, Guofang On volumes of complex hyperbolic orbifolds, Mich. Math. J., Volume 63 (2014) no. 2, pp. 355-369 | Article | MR 3215554 | Zbl 1368.57011

[4] Belolipetsky, Mikhail On volumes of arithmetic quotients of SO (1,n), Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 3 (2004) no. 4, pp. 749-770 | Numdam | MR 2124587 | Zbl 1170.11307

[5] Belolipetsky, Mikhail Hyperbolic orbifolds of small volume (2014) (https://arxiv.org/abs/1402.5394) | MR 3728640 | Zbl 1373.22017

[6] Belolipetsky, Mikhail; Emery, Vincent On volumes of arithmetic quotients of PO (n,1) , n odd, Proc. Lond. Math. Soc., Volume 105 (2012) no. 3, pp. 541-570 | Article | MR 2974199 | Zbl 1327.22013

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

[8] Corlette, Kevin Archimedean superrigidity and hyperbolic geometry, Ann. Math., Volume 135 (1992) no. 1, pp. 165-182 | Article | MR 1147961 | Zbl 0768.53025

[9] Farenick, Douglas R.; Pidkowich, Barbara A. F. The spectral theorem in quaternions, Linear Algebra Appl., Volume 371 (2003), pp. 75-102 | Article | MR 1997364 | Zbl 1030.15015

[10] Friedland, Shmuel; Hersonsky, Sa’ar Jorgensen’s inequality for discrete groups in normed algebras, Duke Math. J., Volume 69 (1993) no. 3, pp. 593-614 | Article | MR 1208812 | Zbl 0799.30033

[11] Gendulphe, Matthieu Systole et rayon interne des variétés hyperboliques non compactes, Geom. Topol., Volume 19 (2015) no. 4, pp. 2039-2080 | Article | MR 3375522 | Zbl 1326.57037

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

[13] Gromov, Mikhail Hyperbolic groups, Essays in group theory (Mathematical Sciences Research Institute Publications), Volume 8, Springer, 1987, pp. 75-263 | Article | MR 919829

[14] Gromov, Mikhail; Schoen, Richard 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 | Article | MR 1215595 | Zbl 0896.58024

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

[16] Kapovich, Michael Representations of polygons of finite groups, Geom. Topol., Volume 9 (2005), pp. 1915-1951 | Article | MR 2175160 | Zbl 1163.20029

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

[18] Kim, Inkang; Parker, John R. Geometry of quaternionic hyperbolic manifolds, Math. Proc. Camb. Philos. Soc., Volume 135 (2003) no. 2, pp. 291-320 | MR 2006066 | Zbl 1048.32017

[19] Martin, Gaven J. Balls in hyperbolic manifolds, J. Lond. Math. Soc., Volume 40 (1989) no. 2, pp. 257-264 | Article | MR 1044273 | Zbl 0709.30040

[20] Martin, Gaven J. On discrete Möbius groups in all dimensions: a generalization of Jørgensen’s inequality, Acta Math., Volume 163 (1989) no. 3-4, pp. 253-289 | Article | Zbl 0698.20037

[21] Parker, John R. On the volumes of cusped, complex hyperbolic manifolds and orbifolds, Duke Math. J., Volume 94 (1998) no. 3, pp. 433-464 | Article | MR 1639519 | Zbl 0951.32019

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

[24] Xie, BaoHua; Wang, JieYan; Jiang, YuePing Balls in complex hyperbolic manifolds, Sci. China, Math., Volume 57 (2014) no. 4, pp. 767-774 | MR 3178024 | Zbl 1306.32014

[25] Zhang, Fuzhen Quaternions and matrices of quaternions, Linear Algebra Appl., Volume 251 (1997), pp. 21-57 | Article | MR 1421264 | Zbl 0873.15008

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