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The Space of Hyperbolic Manifolds and the Volume Function

  • Riccardo Benedetti
  • Carlo Petronio
Part of the Universitext book series (UTX)

Abstract

In the whole of this chapter we shall always suppose manifolds are connected and oriented. It follows from the Gauss-Bonnet formula B.3.3 (for n = 2) and from the Gromov-Thurston theorem C.4.2 (for n ≥ 3) that the volume of a hyperbolic manifold is a topological invariant. Moreover B.3.3 implies that such an invariant is (topologically) complete for n = 2 in the compact case, and it may be proved that in the finite-volume case it becomes complete together with the number of cusp ends (“punctures”). Hence the problem of studying the volume function arises quite naturally: this is the aim of the present chapter.

Keywords

Volume Function Hyperbolic Manifold Solid Torus Hyperbolic Structure Geometric Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Riccardo Benedetti
    • 1
  • Carlo Petronio
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di PisaPisaItaly

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