Open Sets of Maximal Dimension in Complex Hyperbolic Quasi-Fuchsian Space

Parker, John R.; Platis, Ioannis D.

doi:10.4310/jdg/1146169913

Cited by 19 publications

(32 citation statements)

References 19 publications

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“…This means that z 2 3 = |z 3 | 2 e iθ and so, z This generalises the result of Goldman in [6], used by Parker and Platis in [12], that the pre-image of the origin in H In [6,12] the extra condition y and defined by (z 1 z 2 ) = 0. In other words, the components of the fibre are:…”

Section: The Fibre Over the Originsupporting

confidence: 52%

Action of $\mathbb{R}$-Fuchsian groups on $\mathbb{CP}^2$

Cano¹,

Parker

Seade³

2016

Asian Journal of Mathematics

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. We look at lattices in Iso + (H 2 R ), the group of orientation preserving isometries of the real hyperbolic plane. We study their geometry and dynamics when they act on CP 2 via the natural embedding of SO + (2, 1) → SU(2, 1) ⊂ SL(3, C). We use the Hermitian cross product in C 2,1 introduced by Bill Goldman, to determine the topology of the Kulkarni limit set Λ Kul of these lattices, and show that in all cases its complement Ω Kul has three connected components, each being a disc bundle over H 2 R . We get that Ω Kul coincides with the equicontinuity region for the action on CP 2 . Also, it is the largest set in CP 2 where the action is properly discontinuous and it is a complete Kobayashi hyperbolic space. As a byproduct we get that these lattices provide the first known examples of discrete subgroups of SL(3, C) whose Kulkarni region of discontinuity in CP 2 has exactly three connected components, a fact that does not appear in complex dimension 1 (where it is known that the region of discontinuity of a Kleinian group acting on CP 1 has 0, 1, 2 or infinitely many connected components).

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Section: The Fibre Over the Originsupporting

confidence: 52%

Action of $\mathbb{R}$-Fuchsian groups on $\mathbb{CP}^2$

Cano¹,

Parker

Seade³

2016

Asian Journal of Mathematics

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“…This special case n D 2 was first proved by Parker and Platis [18] although in this case by a somewhat different method.…”

Section: Introductionmentioning

confidence: 99%

Flexing closed hyperbolic manifolds

2007

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We show that for certain closed hyperbolic manifolds, one can nontrivially deform the real hyperbolic structure when it is considered as a real projective structure. It is also shown that in the presence of a mild smoothness hypothesis, the existence of such real projective deformations is equivalent to the question of whether one can nontrivially deform the canonical representation of the real hyperbolic structure when it is considered as a group of complex hyperbolic isometries. The set of closed hyperbolic manifolds for which one can do this seems mysterious. 57M50

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“…Packs are the counterpart of bisectors: in general, a pack is real analytic 3-dimensional submanifold of complex hyperbolic space which is naturally foliated by Lagrangian planes. In what follows we shall review briefly the definition of a pack given in the most general setting in [10]. A weaker definition given by P Will may be found in [17] as well as in [3].…”

Section: Packsmentioning

confidence: 99%

“…We call D Ax.Ã 1 Ã 0 / the spine of P and the Lagrangian planes R x for x 2 R the slices of P . Moreover, by [10,Proposition 3.3], P is homeomorphic to a 3-ball whose boundary lies in @H 2 C and also the complement H 2 C P of P , has two components, each homeomorphic to a 4-ball. Observe that P contains L, the complex line containing , the spine of P .…”

Section: Packsmentioning

confidence: 99%

Quakebend deformations in complex hyperbolic quasi-Fuchsian space

Platis

2008

Geom. Topol.

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We study quakebend deformations in complex hyperbolic quasi-Fuchsian space Q C . †/ of a closed surface † of genus g > 1, that is the space of discrete, faithful, totally loxodromic and geometrically finite representations of the fundamental group of † into the group of isometries of complex hyperbolic space. Emanating from an R-Fuchsian point 2 Q C . †/, we construct curves associated to complex hyperbolic quakebending of and we prove that we may always find an open neighborhood U. / of in Q C . †/ containing pieces of such curves. Moreover, we present generalisations of the well known Wolpert-Kerckhoff formulae for the derivatives of geodesic length function in Teichmüller space. 32G05; 32M05

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Open Sets of Maximal Dimension in Complex Hyperbolic Quasi-Fuchsian Space

Cited by 19 publications

References 19 publications

Action of $\mathbb{R}$-Fuchsian groups on $\mathbb{CP}^2$

Action of $\mathbb{R}$-Fuchsian groups on $\mathbb{CP}^2$

Flexing closed hyperbolic manifolds

Quakebend deformations in complex hyperbolic quasi-Fuchsian space

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