Complex hyperbolic quasi-Fuchsian groups and Toledo’s invariant.

*(English)*Zbl 1042.57023The 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 [P.Tukia, Publ. Math., Inst. Hautes Étud. Sci. 61, 171–214 (1985; Zbl 0572.30036) and 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 \(\Sigma_{g,0}\), 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 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 stable due to B. Apanasov [Deformations and stability in complex hyperbolic geometry, Preprint 1997-111, MSRI at Berkeley (1997); and Lect. Notes Ser., Seoul 46, 1–35 (1999; Zbl 0970.53040)]; see also the authors’ paper “Representations of free Fuchsian groups in complex hyperbolic space”, published later in [Topology 39, 33–60 (2000; Zbl 0977.32017)], and E. Falbel and P.-V. Koseleff [Topology 39, 1209–1223 (2000; Zbl 0977.32018)].

The main result of the present paper is the authors’ construction of a continuous family of discrete, faithful, type preserving geometrically finite representations \(\rho_t\) of the fundamental group \(\pi_1(\Sigma_{g,p})\) of a given Riemann surface \(\Sigma_{g,p}\) of genus \(g\) with \(p>0\) punctures and negative Euler characteristic to \(\text{PU}(2,1)\) which interpolates between its \(\mathbb C\)-Fuchsian and \(\mathbb R\)-Fuchsian representations. These representations take every possible (real) value of the Toledo invariant, in contrast with the case of closed surfaces where the Toledo invariant lies in a discrete set and indexes the components of the representation variety.

Due to a well known result [M. H. Millington, J. Lond. Math. Soc., II. Ser. 1, 351–357 (1969; Zbl 0206.36801)], the fundamental group \(\pi_1(\Sigma_{g,p})\) is isomorphic to a finite index subgroup of the modular group PSL\((2,\mathbb Z)\). Because of this fact, the principal ingredient of the proof is a construction of the mentioned continuous family of representations of the modular group which interpolates between its \(\mathbb C\)-Fuchsian and \(\mathbb R\)-Fuchsian representations. The same family of representations (with their fundamental domains in the complex hyperbolic space) was earlier constructed by E. Falbel and P.-V. Koseleff [Math. Res. Lett. 9, 379–391 (2002; Zbl 1008.20038)].

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 stable due to B. Apanasov [Deformations and stability in complex hyperbolic geometry, Preprint 1997-111, MSRI at Berkeley (1997); and Lect. Notes Ser., Seoul 46, 1–35 (1999; Zbl 0970.53040)]; see also the authors’ paper “Representations of free Fuchsian groups in complex hyperbolic space”, published later in [Topology 39, 33–60 (2000; Zbl 0977.32017)], and E. Falbel and P.-V. Koseleff [Topology 39, 1209–1223 (2000; Zbl 0977.32018)].

The main result of the present paper is the authors’ construction of a continuous family of discrete, faithful, type preserving geometrically finite representations \(\rho_t\) of the fundamental group \(\pi_1(\Sigma_{g,p})\) of a given Riemann surface \(\Sigma_{g,p}\) of genus \(g\) with \(p>0\) punctures and negative Euler characteristic to \(\text{PU}(2,1)\) which interpolates between its \(\mathbb C\)-Fuchsian and \(\mathbb R\)-Fuchsian representations. These representations take every possible (real) value of the Toledo invariant, in contrast with the case of closed surfaces where the Toledo invariant lies in a discrete set and indexes the components of the representation variety.

Due to a well known result [M. H. Millington, J. Lond. Math. Soc., II. Ser. 1, 351–357 (1969; Zbl 0206.36801)], the fundamental group \(\pi_1(\Sigma_{g,p})\) is isomorphic to a finite index subgroup of the modular group PSL\((2,\mathbb Z)\). Because of this fact, the principal ingredient of the proof is a construction of the mentioned continuous family of representations of the modular group which interpolates between its \(\mathbb C\)-Fuchsian and \(\mathbb R\)-Fuchsian representations. The same family of representations (with their fundamental domains in the complex hyperbolic space) was earlier constructed by E. Falbel and P.-V. Koseleff [Math. Res. Lett. 9, 379–391 (2002; Zbl 1008.20038)].

Reviewer: Boris N. Apanasov (Norman)

##### MSC:

57S30 | Discontinuous groups of transformations |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

30F30 | Differentials on Riemann surfaces |