Bieberbach groups and a proof of the Margulis Lemma
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.
KeywordsDiscrete Group Abelian Subgroup Isometry Group Finite Index Parallel Transport
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