Transformation Groups

, Volume 1, Issue 1, pp 71–82

Free quotients and the first betti number of some hyperbolic manifolds

  • Alexander Lubotzky

DOI: 10.1007/BF02587736

Cite this article as:
Lubotzky, A. Transformation Groups (1996) 1: 71. doi:10.1007/BF02587736


In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number. The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the non-arithmetic lattices inSO(n,1) constructed by Gromov and Piatetski-Shapiro [GPS] and to groups generated by reflections. In all these cases we actually show that Γ=π1(M) has a finite index subgroup which is mapped onto a nonabelian free group.

Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

  • Alexander Lubotzky
    • 1
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael

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