Discrete Subgroups of the Euclidean Group

Abstract

Since the Euclidean group $\mathbb{E}$ has continuously many elements:

$$ \left| \mathbb{E} \right| = \left| \mathbb{R} \right| = c, $$

it has 2^csubsets, and each of these generates a subgroup. So, to avoid bewilderment, it is necessary to place some restriction on the class of subgroups to be investigated. One such restriction, and a very natural one, was foreshadowed at the end of Chapter 1.A subgroup G of $\mathbb{E}$ is said to be discrete if, for any point O in ℝ², every circle centre O contains only finitely many points of the orbit $OG = \{ Og|g \in G\}$. This means that OG has no accumulation points, and it follows that around every point O ∈ ℝ² there is a circle (of positive radius) containing no point of OG other than O itself. A practical consequence is that, for any point O ∈ ℝ² not fixed by G, there is an element in OG of minimal positive distance from O.