Geometriae Dedicata

, Volume 97, Issue 1, pp 151–185

Complex Hyperbolic Quasi-Fuchsian Groups and Toledo's Invariant

  • Nikolay Gusevskii
  • John R. Parker

DOI: 10.1023/A:1023616618854

Cite this article as:
Gusevskii, N. & Parker, J.R. Geometriae Dedicata (2003) 97: 151. doi:10.1023/A:1023616618854


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.

complex hyperbolic spacequasi-Fuchsian groupToledo invariant

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Nikolay Gusevskii
    • 1
  • John R. Parker
    • 2
  1. 1.Departamento de MatematicaUniversidade Federal de Minas GeraisBelo Horizonte–MGBrasil
  2. 2.Department of Mathematical SciencesUniversity of DurhamDurhamEngland