Asian Journal of Mathematics

Volume 20 (2016)

Number 3

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

Pages: 449 – 474

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n3.a3

Authors

Angel Cano (Instituto de Mathemáticas, UNAM, Cuernavaca, Morelos, México)

John R. Parker (Department of Mathematical Sciences, Durham University, Durham, England)

José Seade (Instituto de Mathemáticas, UNAM, Cuernavaca, Morelos, México)

Abstract

We look at lattices in $\mathrm{Iso}_{+} (\mathbf{H}^2_{\mathbb{R}})$, the group of orientation preserving isometries of the real hyperbolic plane. We study their geometry and dynamics when they act on $\mathbb{CP}^2$ via the natural embedding of $\mathrm{SO}_{+} (2, 1) \hookrightarrow \mathrm{SU}(2, 1) \subset \mathrm{SL}(3, \mathbb{C})$. We use the Hermitian cross product in $\mathbb{C}^{2,1}$ introduced by Bill Goldman, to determine the topology of the Kulkarni limit set $\Lambda_{\mathrm{Kul}}$ of these lattices, and show that in all cases its complement $\Omega_{\mathrm{Kul}}$ has three connected components, each being a disc bundle over $H^2_{\mathbb{R}}$. We get that $\Omega_{\mathrm{Kul}}$ coincides with the equicontinuity region for the action on $\mathbb{CP}^2$. Also, it is the largest set in $\mathbb{CP}^2$ 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 $\mathrm{SL}(3, \mathbb{C})$ whose Kulkarni region of discontinuity in $\mathbb{CP}^2$ 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 $\mathbb{CP}^1$ has $0$, $1$, $2$ or infinitely many connected components).

Keywords

Fuchsian group, limit set, complex projective space

2010 Mathematics Subject Classification

22E40, 37B05

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Published 12 July 2016