Bieberbach groups and a proof of the Margulis Lemma

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


We will describe groups operating on the Euclidean space ℝn. The isometry group of (ℝn is the semidirect product Iso(ℝn) = 0(n) ⋉ ℝn, where 0(n) is the orthogonal group. An element (A,a) ∈ Iso(ℝn) acts by (A,a)x = Ax + a, thus (A,a)(B,b) = (AB,Ab + a) and ρ: Iso(ℝn) → 0(n), ρ(A,a) = A is a homomorphism. The orthogonal map A is called the rotational part, a the translational part of (A,a). An isometry is called a translation, if it has the form (E,a) where E is the identity. We identify the translational subgroup with ℝn. By 6.7, every γ ∈ Iso(ℝn) is semisimple and MIN(γ) is an affine subspace of ℝn. An isometry γ is a translation if and only if MIN(γ) = ℝn. A discrete group Γ of Iso(ℝn) is called crystallographic, if ℝn/Γ is compact. The main result about crystallographic groups is the following well known theorem.


Discrete Group Abelian Subgroup Isometry Group Finite Index Parallel Transport 
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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

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