Equivariant Alexandrov geometry and orbifold finiteness (Q2631018)

The main goal of the paper is a result of the type of Perelman's Stability Theory (see, for instance, [\textit{V. Kapovitch}, in: Metric and comparison geometry. Surveys in differential geometry. Vol. XI. Somerville, MA: International Press. 103--136 (2007; Zbl 1151.53038)]). Let us consider the sequence (\(X_i\)) of Alexandrov spaces of the same dimension with uniform upper diameter and lower curvature bounds and the compact Lie group \(G\) which acts isometrically on each space \(X_i\). If the sequence \((X_i,G)\) converges in the equivariant Gromov-Hausdorff topology to \((X,\Gamma)\) then \(G\) and \(\Gamma\) are isomorphic and moreover, \(X_i\) are equivariantly homeomorphic to \(X\). As a consequence, the author obtains the finiteness of classes of orbifolds (with uniform bound of diameter, curvature, and volume) up to homeomorphisms.