Abstract
We consider discrete, faithful, type-preserving representations of the fundamental group of a punctured Riemann surface into PU(21), the holomorphic isometry group of complex hyperbolic space. Our main result is that there is a continuous family of such representations which interpolates between ℂ-Fuchsian representations and ℝ-Fuchsian representations. Moreover, these representations take every possible (real) value of the Toledo invariant. This contrasts with the case of closed surfaces where ℂ-Fuchsian and ℝ-Fuchsian representations lie in different components of the representation variety. In that case the Toledo invariant lies in a discrete set and indexes the components of the representation variety.
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Gusevskii, N., Parker, J.R. Complex Hyperbolic Quasi-Fuchsian Groups and Toledo's Invariant. Geometriae Dedicata 97, 151–185 (2003). https://doi.org/10.1023/A:1023616618854
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DOI: https://doi.org/10.1023/A:1023616618854