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

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