On Bieberbach’s analysis of discrete Euclidean groups

On Bieberbach's Analysis of Discrete Euclidean Groups

For a subgroup G of the euclidean group En — 0„ • R" (semidirect product) and a real number r > 0, let G* denote the translation subgroup of G, G, the group generated by all (A, a) in G with || 1 A|| < r (operator norm), and k^r) the maximum number of elements of 0„ with mutual distances > r relative to the metric d{A, B) ||^4 B\\. We give an elementary, largely geometrical proof of the followi...

Fundamental Domains of Discrete Groups Acting on Euclidean Space

Fundamenatal domains of wallpaper groups acting on R2 whose closures are homeomorphic to closed disks can be classified in the same way that Grunbaum classifies isohedral plane tilings in [7]. Almost all of these types of fundamental domains appear as Dirichlet domains, and examples are given.

Euclidean simplices generating discrete reflection groups

Let P be a convex polytope in the spherical space S, in the Euclidean space E, or in the hyperbolic space H. Consider the group GP generated by reflections in the facets of P . We call GP a reflection group generated by P . The problem we consider in this paper is to list polytopes generating discrete reflection groups. The answer is known only for some combinatorial types of polytopes. Already...

Structural Properties of Inner Amenable Discrete Groups

A Theorem on Locally Euclidean Groups

1. Let G be a connected locally compact group and let G' denote the closure of the commutator subgroup of G. G' is called the derived subgroup of G. Consider the derived series of G, that is, the sequence of subgroups Go, Gi, ■ ■ ■ , Gn, ■ ■ ■ where G0 = G and Gn+i = G'nEach Gn is connected and this sequence becomes stationary at some finite stage,2 that is, for some n, Gn = Gn+i. We define Gn ...