On the rigidity of discrete isometry groups of negatively curved spaces

Sa'ar Hersonsky, Frédéric Paulin

Research output: Contribution to journalArticlepeer-review

51 Scopus citations

Abstract

We prove an ergodic rigidity theorem for discrete isometry groups of CAT(-1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2-polyhedra, hyperbolic Bruhat-Tits buildings and rank one symmetric spaces. We prove that two negatively curved Riemannian metrics, with conical singularities of angles at least 2π, on a closed surface, with boundary map absolutely continuous with respect to the Patterson-Sullivan measures, are isometric. For that, we generalize J.-P. Otal's result to prove that a negatively curved Riemannian metric, with conical singularities of angles at least 2π, on a closed surface, is determined, up to isometry, by its marked length spectrum.

Original languageEnglish
Pages (from-to)349-388
Number of pages40
JournalCommentarii Mathematici Helvetici
Volume72
Issue number3
DOIs
StatePublished - 1 Jan 1997
Externally publishedYes

Keywords

  • Discrete group
  • Divergence group
  • Hyperbolic building
  • Marked length spectrum
  • Negative curvature
  • Patterson-sullivan measure
  • Rigidity
  • Surface with conical singularities

ASJC Scopus subject areas

  • Mathematics (all)

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