Subgroups of 𝑃𝑆𝐿(3,ℂ) with four lines in general position in its limit set

Barrera, Waldemar; Cano, Angel; Navarrete, Juan Pablo

doi:10.1090/s1088-4173-2011-00231-1

Cited by 8 publications

(3 citation statements)

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“…Perhaps the most surprising feature of this result is part (2), namely that that L x ∪ C x , which is a copy of RP 2 , tends to L ξ , which is a copy of CP 1 . The way this happens is the following.…”

Section: Introductionmentioning

confidence: 93%

“…• One: For instance a lattice in PU (2,1) where Ω Kul is exactly a copy of H • Two: For instance all the C-Fuchsian lattices described in Section 1.3. More generally, all groups constructed by suspending a Fuchsian subgroup of PSL(2, C) of the first kind.…”

Section: An Immediate Consequence Of Theorem 3 Ismentioning

confidence: 99%

“…• Four: For instance the examples in [2] of complex Kleinian groups with exactly four lines in general position in the Kulkarni limit set.…”

Section: An Immediate Consequence Of Theorem 3 Ismentioning

confidence: 99%

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

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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: Introductionmentioning

confidence: 93%

Section: An Immediate Consequence Of Theorem 3 Ismentioning

confidence: 99%