A criterion of proper action on some compact extensions of ℝn and applications

Abstract

Let K be a compact subgroup of [Formula: see text] and [Formula: see text] the semi-direct product group. Let H be a closed subgroup of G and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. We establish a geometrical criterion of the proper action of [Formula: see text] on [Formula: see text], which requires an accurate description of the structure of closed connected subgroups of Euclidean motion groups. As a consequence, we establish a criterion for the existence of a compact Clifford–Klein form [Formula: see text] and study the Calabi–Markus phenomenon.