On the Zariski Closure of the Linear Part of a Properly Discontinuous Group of Affine Transformations

2002/02/01 by Herbert Abels, G. A. Margulis, G. A. Soifer · 51 citations
Mathematics · #Finite Group Theory Research #Geometric and Algebraic Topology #Advanced Algebra and Geometry #Mathematics #Affine transformation #Closure (psychology) #Group (periodic table) #Affine group #Combinatorics #Space (punctuation) #Pure mathematics #Physics #Quantum mechanics

paper · doi:10.4310/jdg/1090351104

published in Journal of Differential Geometry 60(2) (Lehigh University)

openalex publication_date 2002/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

Let Γ be a subgroup of the group of affine transformations of the affine space ℝ²ⁿ⁺¹. Suppose Γ acts properly discontinuously on ℝ²ⁿ⁺¹. The paper deals with the question which subgroups of \rmGL(2n +1,ℝ) occur as Zariski closure ℓ (Γ) of the linear part of such a group Γ. The two main results of the paper say that \rmSO(n + 1, n) does occur as ℓ (Γ) of such a groupΓ if n is odd, but does not if n is even.

Citations

• Properly Discontinuous Groups of Affine Transformations: A Survey
• The fundamental group of a compact flat Lorentz space form is virtually polycyclic
• The structure of complete locally affine manifolds
• Fundamental polyhedra for Margulis space-times
• �ber die Bewegungsgruppen der Euklidischen R�ume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich
• Semigroups containing proximal linear maps
• Free subgroups in linear groups
• Three-dimensional affine crystallographic groups
• Über die Bewegungsgruppen der Euklidischen Räume

Cited by

• Affine manifold [wikipedia]