Zwei Orbittheoreme für Bewegungsgruppen. (Two theorems on orbits of groups of motions) (Q750963)

Let X denote a metric space with metric d, and let Mot(X) be the metrizable topological group of all self isometries of X with the topology of uniform convergence on bounded sets. A subgroup G of Mot(X) is called a motion group on X. The orbit space X/G is endowed with the quotient topology. If orbits are closed then X/G can be given the Hausdorff metric h and we have the following result: (1) \(h(Gx,Gy)=\inf_{S,T\in G}d(Sx,Ty)=d(x,Gy)\). (2) \(Gx=\lim Gx_ n\) iff there exists a sequence \(S_ n\in G\) with \(x=\lim S_ nx_ n\). The action of G on X is discontinuous at x if for a sequence \(T_ n\) of pairwise different elements of G the sequence \(T_ nx\) has no cluster point. It is totally discontinuous at x if there is a neighbourhood U of x such that \(\{\) \(T\in G|\) TU\(\cap U\neq 0\}\) is finite. Result: (3) For a motion group G the action is not totally discontinuous if there is a sequence \(T_ n\in G\) of pairwise different elements with \(x=\lim T_ nx\). Consequence: (4) For motion groups the action is totally discontinuous at x iff it is discontinuous at x. Consequence: (5) If G acts discontinuously at every point, then the orbits are discrete. Consequence: (6) A discontinuously acting motion group is discrete. The converse is false in general. Theorem. Let (X,d) be a metric space in which the compact subsets are exactly the closed and bounded ones. Then the following are equivalent for a motion group on X: (a) G operates discontinuously. (c) G is discrete in Mot(X) in the compact open topology.