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Representations of free Fuchsian groups in complex hyperbolic space. (English) Zbl 0977.32017
Let $$S$$ be a Riemann surface of finite type with genus $$g$$ and $$n>0$$ punctured, where $$2g-2+n>0$$. We denote by $$H^m_\mathbb{C}$$ the complex hyperbolic $$m$$-space. A representation of $$\pi_1(S)$$ into the isometry group of the hyperbolic 2-space is called Fuchsian if it is discrete and faithful. The intersection of a complex line with complex hyperbolic 2-space is called a complex geodesic and which carries the structure of the Poincaré model of the complex hyperbolic 1-space. Then we have a natural inclusion of their isometry groups $$\text{Isom} (H^1_\mathbb{C}) \to\text{Isom}(H^2_\mathbb{C})$$. A representation of $$\pi_1(S)\to\text{Isom}(H^2_\mathbb{C})$$ obtained by restricting this inclusion to a Fuchsian representation $$\pi_1(S)$$ is called $$\mathbb{C}$$-Fuchsian.
The authors study the deformation space of a $$\mathbb{C}$$-Fuchsian representation $$\rho_0: \pi_1 (S)\to \text{Isom} (H^2_\mathbb{C})$$ and the following problem is dealt with in this paper: Given a $$\mathbb{C}$$-Fuchsian representation $$\rho_0: \pi_1(S)\to \text{Isom} (H^2_\mathbb{C})$$ with the property that the quotient of the invariant complex geodesic by the group has finite area. Then, are there nearby representations $$\rho$$ that do not preserve a complex geodesic? In other words, let $$\Gamma_0$$ be a finitely generated Fuchsian group of the first kind leaving the complex geodesic $$H^1_\mathbb{C}\subset H^2_\mathbb{C}$$ invariant, that is, $$\Gamma$$ acts on $$H^1_\mathbb{C}$$ with finite area quotient. Assume that the quotient space $$H^1_\mathbb{C}/ \Gamma$$ is not compact. Then does $$\Gamma_0$$ admit nearby quasi-Fuchsian deformation? In the present paper the authors show that such deformations exist. They construct an example of such a $$\Gamma_0$$ in the case where $$g=0$$ and $$n=3$$. In fact, they construct a discrete $$\mathbb{C}$$-Fuchsian ideal triangle group $$\Gamma_0$$ with the above property.
This problem is deeply concerned with the properties of Toledo’s invariant [D. Toledo, Math. Scand. 45, 13-26 (1979; Zbl 0435.58008) and J. Differ. Geom. 29, No. 1, 125-133 (1989; Zbl 0676.57012)].

##### MSC:
 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 57M05 Fundamental group, presentations, free differential calculus
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