Stability and equivariant Gromov-Hausdorff convergence (Q6593634)

In this well-written paper the author proves an equivariant Gromov-Hausdorff stability theorem for Alexandrov spaces of curvature bounded below, generalizing work of \textit{J. Harvey} [J. Geom. Anal. 26, No. 3, 1925--1945 (2016; Zbl 1350.53046)]. Precisely, the following theorem is proved, where \(\mathcal{A}(n,k,D)\) stands for the class of compact \(n\)-dimensional Alexandrov spaces of \(\mathrm{curv}\geq k\) and \(\mathrm{diam}\leq D\).\N\NTheorem A. Let \(\{X_k\}_{k=1}^{\infty}\) and \(X\) be in \( \mathcal{A}(n,k,D)\). Fix \(m\in \mathbb{N}\) and assume that \(G_k\) is an \(m\)-dimensional closed subgroup of \(\mathrm{Isom}(X_k)\). Further assume that the sequence \(\{(X_k,G_k)\}_{k=1}^{\infty}\) converges in the equivariant Gromov-Hausdorff sense to \((X,G)\), where \(G\) is an \(m\)-dimensional closed subgroup of \(\mathrm{Isom}(X)\). Then for all \(k\) large enough, there exists a Lie group isomorphism \(\theta_k: G_k \to G\) and a homeomorphism \(\xi_k: X_k\to X\) such that \(\chi_k\circ g_k = \theta_k(g_k)\circ \xi_k\) for all \(g_k\in G_k\).\N\NIn Harvey's work, \(G_k=G\) for all \(k\), whereas in the article under review, the groups acting are allowed to vary.\N\NAs a corollary (Corollary B), the author obtains the stability of the associated Borel constructions in this setting, that is, under the same hypothesis as the previous result, for \(k\) large enough, \(EG\times_{G_k} X_k\) is homeomorphic to \(EG\times_{G}X\), where \(EG\) is the universal space of \(G\).\N\NFinally, the author gives sufficient conditions in Theorem C for continuous equivariant Gromov-Hausdorff approximations to exist, in the spirit of \textit{S. V. Ivanov} [St. Petersbg. Math. J. 9, No. 5, 945--959 (1997; Zbl 0896.53034); translation from Algebra Anal. 9, No. 5, 65--83 (1997)].\N\NIt is worth pointing out that the author defines the useful notion of an ``almost commutative diagram'' which simplifies the proofs and is interesting on its own.