Balls in complex hyperbolic manifolds

Xie, Bin; Wang, Jieyan; Jiang, Yueping

doi:10.1007/s11425-013-4765-z

“…From this one can compute an explicit lower bound for the volume of a hyperbolic n-manifold. Recently, some analogous results have been obtained in complex and quaternionic settings [5,22] Adeboye obtained an explicit lower bound for the volume of a real hyperbolic orbifold depending on the dimension [1]. The main tool is the spectral radius of the involved matrices.…”

Section: Introductionmentioning

confidence: 77%

On volumes of quaternionic hyperbolic n-orbifolds

Cao

¹

,

Fu

²

2018

Geom Dedicata

1

0

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“…Following the method of [30], Xie, Wang, and Jiang [42] produced a manifold bound in the complex hyperbolic setting. A volume bound for complex hyperbolic n-orbifolds was given in [2].…”

Section: Historymentioning

confidence: 99%

On the volume of orbifold quotients of symmetric spaces

Adeboye¹,

Wang²,

Wei³

2018

Preprint

0

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A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the fundamental domain of any lattice contains a ball whose radius depends only on the group itself. A direct consequence is a positive minimum volume for orbifolds modeled on the corresponding symmetric space. However, sharp bounds are known only for hyperbolic orbifolds of dimensions two and three, and recently for quaternionic hyperbolic orbifolds of all dimensions.As in [1] and [2], this article combines H. C. Wang's radius estimate with an improved upper sectional curvature bound for a canonical left-invariant metric on a real semisimple Lie group and uses Gunther's volume comparison theorem to deduce an explicit uniform lower volume bound for arbitrary orbifold quotients of a given irreducible symmetric spaces of non-compact type. The numerical bound for the octonionic hyperbolic plane is the first such bound to be given. For (real) hyperbolic orbifolds of dimension greater than three, the bounds are an improvement over what was previously known.

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“…We adapt techniques developed by Martin [Mar89a] in the real hyperbolic setting to obtain a bound λ n on the maximal radius of a real hyperbolic n-manifold. These ideas where recently adapted to the complex hyperbolic case by Jiang, Wang and Xie [XWJ14]. The bounds given by the above-mentioned authors, and the one presented in this article, both decrease exponentially with the dimension, though the methods employed do not allow us to discuss their optimality.…”

Section: Introductionmentioning

confidence: 97%

“…We adapt techniques developed by Martin [Mar89a] in the real hyperbolic setting to obtain a bound λ n on the maximal radius of a real hyperbolic n-manifold. These ideas where recently adapted to the complex hyperbolic case by Jiang, Wang and Xie [XWJ14].…”

mentioning

confidence: 99%