Proof of the rigidity theorem

  • Werner Ballmann
  • Mikhael Gromov
  • Viktor Schroeder
Part of the Progress in Mathematics book series (PM, volume 61)


Let V* = X*/Γ* and V = X/Γ be as in Theorem 14.3. By our assumption there exists an isomorphism θ. Γ* → Γ. Therefore V* and V are homotopy equivalent. Let f̄: V* → V and ḡ: V → V* be maps such that ḡ∘f̄ and ḡ∘f̄ are homotopic to the identities on V* and V. Let f: X* → X and g: X → X* be the lifts to the covering spaces. Then there are constants l,b > 0 such that f and g are (l,b)-pseudoisometries. Clearly f(γ*x*) = θ(γ*)f(x*) for x* ∈ X* and γ* ∈ Γ*, and g(γx) = θ -1(γ)g(x) for x € X and γ ∈ Γ. Furthermore there is a constant A > 0 such that d(x*,gfx*) ⩽ A and d(x,fgx) ⩽ A for x* ∈ X* and x ∈ X.


Convergent Subsequence Unique Fixed Point Geodesic Segment Weyl Chamber Unit Speed 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Werner Ballmann
    • 1
    • 2
  • Mikhael Gromov
    • 3
  • Viktor Schroeder
    • 4
    • 5
  1. 1.Dept. of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Math. Institut der UniversitätBonnWest Germany
  3. 3.Inst. des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  4. 4.Math. Institut der UniversitätMünsterGermany
  5. 5.Math. Institut der UniversitätBaselSwitzerland

Personalised recommendations