Complex hyperbolic quasi-Fuchsian groups and Toledo's invariant
type: Item
instance of: scholarly article
title: Complex hyperbolic quasi-Fuchsian groups and Toledo's invariant
zbMATH Open document ID: 1042.57023
DOI: 10.1023/A:1023616618854
author: Nikolay Gusevskii
author: John R. Parker
published in: Geometriae Dedicata
publication date: 2003-08-06T00:00:00Z
review text: The paper studies discrete, faithful, type preserving geometrically finite representations \(\rho(\pi_1)\) of the fundamental group \(\pi_1\) of a Riemann surface \(\Sigma_{g,p}\) of genus \(g\) with \(p\) punctures (and negative Euler characteristic) into \(\text{PU}(2,1)\), the holomorphic isometry group of complex hyperbolic space \(H^2_{\mathbb C}\). Due to the Tukia-Apanasov isomorphism theorem [\textit{P.Tukia}, Publ. Math., Inst. Hautes Étud. Sci. 61, 171--214 (1985; Zbl 0572.30036) and \textit{B. Apanasov}, Russ. Acad. Sci. Dokl. Math. 56, 757--760 (1997; Zbl 0976.53053)], the limit set of any such (quasi-Fuchsian) group \(\rho(\pi_1)\) is a Jordan curve at infinity of \(H^2_{\mathbb C}\) (one point compactification of the Heisenberg group). If \(\rho(\pi_1)\) preserves a totally geodesic subspace of \(H^2_{\mathbb C}\) (either a totally real Lagrangian subspace \(H^2_{\mathbb R}\) or a complex hyperbolic 1-space \(H^1_{\mathbb C}\)) it is called correspondingly \(\mathbb R\)-Fuchsian or \(\mathbb C\)-Fuchsian. These two representations are very different in the sense of their deformations. Namely, for compact surfaces \(\Sigma_{g,0}\), \textit{D. Toledo} proved by introducing his ``Toledo invariant'' \(\tau(\rho)\) [J. Differ. Geom. 29, 125--133 (1989; Zbl 0676.57012)] that any deformation of a \(\mathbb C\)-Fuchsian group cannot produce non-trivial quasi-Fuchsian ones, i.e. it gives only \(\mathbb C\)-Fuchsian groups. On the other hand, due to \textit{B. Apanasov} [J. Reine Angew. Math. 492, 75--91 (1997; Zbl 0891.53055)] there are plenty of non-trivial quasi-Fuchsian deformations of any \(\mathbb R\)-Fuchsian group. It is well known that the compactness condition for \(\Sigma(g,p)\) is important for the mentioned ``local rigidity'' of \(\mathbb C\)-Fuchsian representations: for punctured surfaces (\(p>0\)) there are non-trivial quasi-Fuchsian deformations of corresponding \(\mathbb C\)-Fuchsian groups \(\rho(\pi_1)\) and they are not quasiconformally stab…
reviewed by: Boris N. Apanasov
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 30F30
zbMATH DE Number: 1960049
zbMATH Keywords: complex hyperbolic space
zbMATH Keywords: quasi-Fuchsian group
zbMATH Keywords: Toledo invariant
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1960049