Convergence of group actions in metric measure geometry (Q6652120)

In this paper the authors study the convergence of metric measure spaces equipped with measure-preserving isometric group actions. In so doing, the authors redefine the notion of box and observable distance due to \textit{M. Gromov} [Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. 3rd printing. Basel: Birkhäuser (2007; Zbl 1113.53001)] to their setting.\N\NThe authors first establish that their definitions of box and observable distances on the space of equivariant isomorphism classes of a metric measure space with isometric group action yield well-defined metrics. They go on to show that if a sequence of metric measure spaces with isometric actions \((X_n,G_n)\) converges to a metric measure space with isometric action \((Y,H)\) in the box topology that \(X_n/G_n\) converges to \(Y/H\) in the box topology. They establish an analogous result for the observable topology by bounding the \(\kappa\)-observable distance between quotient in terms of the observable diameter of the spaces and the observable distance between them. They then show that the projection obtained by forgetting the group action is a proper map with respect to the box metric. They culminate their paper by furnishing it with an example for which a sequence of lens spaces converges to a cone of the infinite-dimensional complex projective space \(\mathbb{C}P^\infty\).\N\NThe novelty of the authors results stems from the use of optimal transport in the definitions of box and observable distance in the equivariant case. The analysis for the equivariant case proves far more delicate than it does for standard metric measure spaces.