Non-collapsed eGH convergence and dimension

Abstract

Let $(X_i,p_i)$ be a non-collapsing sequence of pointed $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, and $G_i\le \mathrm{Iso}(X_i)$ a sequence of closed subgroups of isometries. We show that if the triples $(X_i,G_i,p_i)$ converge in the equivariant Gromov--Hausdorff sense to a triple $(X,G,p)$, then $\dim (G)\ge {\mathrm{limsup}}_{i\to \infty }\dim (G_i)$, generalizing a result of Mazur--Rong--Wang to the non-compact setting. The argument also applies in the non-smooth setting of $\mathrm{RCD}$ spaces. As an application, we investigate $\mathrm{RCD}$ spaces with large isometry groups, extending results of Galaz-García--Kell--Mondino--Sosa and Galaz-García--Guijarro.

Definition of good approximations was updated to an equivalent one and clarified some references