Discontinuous motion groups of different degrees in metric spaces (Q1347008)

This is an exhaustive survey-style analysis of the equivalencies, dependencies and independencies between 16 conditions for discontinuous groups of isometries of two- and three-dimensional Euclidean space. The conditions are variations on the local finiteness condition for orbits of a group \(\mathbf B\) of isometries of a metric space \(({\mathcal X}, d)\) (the set \(S(X,P,r) := \{\tau(P) \mid d(\tau(P), X) < r\), \(\tau \in {\mathbf B}\}\) is finite, for all \(X\), \(P\) in \(\mathcal X\) and \(r > 0\)) and the condition of discreteness of points in their orbit (for all \(P \in {\mathcal X}\), \(S(P, P, r)\) is a singleton for sufficiently small \(r > 0\)).