Kaleidoscopical configurations in G-spaces

Abstract

Let G be a group and X be a G-space with the action G× X → X, (g, x) → gx. A subset F of X is called a kaleidoscopical configuration if there exists a coloring X: X → C such that the restriction of X on each subset gF, g ∈ G, is a bijection. We present a construction (called the splitting construction) of kaleidoscopical configurations in an arbitrary G-space, reduce the problem of characterization of kaleidoscopical configurations in a finite Abelian group G to a factorization of G into two subsets, and describe all kaleidoscopical configurations in isometrically homogeneous ultrametric spaces with finite distance scale. Also we construct 2c (unsplittable) kaleidoscopical configurations of cardinality c in the Euclidean space Rn.