The rigidity of some finite group actions on CAT($\kappa$) spaces

Abstract

In this paper, we first prove the optimal lower bound for Alexandrov angle rigidity of torsion elliptic isometries on any complete CAT($\kappa$) space, which, when attained, leads to an embedded 2-flat in the tangent cone invariant under the induced action of the isometry. Next, we will prove similar result for action of symmetry groups of either a regular orthoplex, a regular hypercube, a regular dodecahedron or a regular icosahedron on a set of points in any complete CAT($\kappa$) space in a way corresponds to the set of vertices of the polytope, the angle made at the circumcenter by any pair of points corresponding to an edge is bounded below by that of the edge in the polytope. As a result, we give a condition for the convex hull of the set of points to be isometric to a corresponding regular polytope in a model space of constant curvature $\kappa$.