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Action of R-Fuchsian groups on CP2.

Cano, Angel and Parker, John R. and Seade, José (2016) 'Action of R-Fuchsian groups on CP2.', Asian journal of mathematics., 20 (3). pp. 449-474.

Abstract

We look at lattices in Iso+(H2R)Iso+(HR2), the group of orientation preserving isometries of the real hyperbolic plane. We study their geometry and dynamics when they act on CP2CP2 via the natural embedding of SO+(2,1)↪SU(2,1)⊂SL(3,C)SO+(2,1)↪SU(2,1)⊂SL(3,C). We use the Hermitian cross product in C2,1C2,1 introduced by Bill Goldman, to determine the topology of the Kulkarni limit set ΛKulΛKul of these lattices, and show that in all cases its complement ΩKulΩKul has three connected components, each being a disc bundle over H2RHR2. We get that ΩKulΩKul coincides with the equicontinuity region for the action on CP2CP2. Also, it is the largest set in CP2CP2 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)SL(3,C) whose Kulkarni region of discontinuity in CP2CP2 has exactly three connected components, a fact that does not appear in complex dimension 11 (where it is known that the region of discontinuity of a Kleinian group acting on CP1CP1 has 00, 11, 22 or infinitely many connected components).

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.4310/ajm.2016.v20.n3.a3
Publisher statement:Copyright © International Press. First published in Asian Journal of Mathematics in 20(3), 2016, published by International Press.
Date accepted:11 March 2015
Date deposited:18 March 2015
Date of first online publication:01 July 2016
Date first made open access:No date available

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