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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)].

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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