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Complex hyperbolic quasi-Fuchsian groups and Toledo’s invariant. (English) Zbl 1042.57023
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 [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)].

##### 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
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