On volumes of quaternionic hyperbolic n-orbifolds

Cao, Wensheng; Fu, Jianli

doi:10.48550/arxiv.1704.03052

Cited by 1 publication

(2 citation statements)

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“…Compared to the curvature estimates in [1] and [2], the above estimate is more general and much improved. It also improves the estimate in [10].…”

Section: The Sectional Curvatures Of Semi-simple Lie Groupsmentioning

confidence: 72%

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On the volume of orbifold quotients of symmetric spaces

Adeboye¹,

Wang²,

Wei³

2018

Preprint

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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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“…Compared to the curvature estimates in [1] and [2], the above estimate is more general and much improved. It also improves the estimate in [10].…”

Section: The Sectional Curvatures Of Semi-simple Lie Groupsmentioning

confidence: 72%

“…A volume bound for complex hyperbolic n-orbifolds was given in [2]. Recent results for quaternionic hyperbolic space are a manifold bound due to Cao [9] and an orbifold bound due to Cao and Fu [10].…”

Section: Historymentioning

confidence: 99%