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

TL;DR

This paper establishes optimal angle rigidity bounds for certain isometries and symmetry group actions on CAT(κ) spaces, linking geometric invariants to polytope structures in curved spaces.

Contribution

It introduces new bounds for Alexandrov angle rigidity and characterizes when convex hulls of symmetric point sets are isometric to regular polytopes in curved spaces.

Findings

Optimal lower bounds for Alexandrov angle rigidity are proven.

Conditions are identified for convex hulls to be isometric to regular polytopes.

Results connect symmetry group actions with geometric and polyhedral structures in CAT(κ) 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 $\kappa$.