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 \leq \operatorname{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 $\operatorname{dim} (G) \geq \limsup _{i \to \infty} \operatorname{dim} (G_i)$, generalizing a result of Mazur--Rong--Wang to the non-compact setting. The argument also applies in the non-smooth setting of $\operatorname{RCD}$ spaces. As an application, we investigate $\operatorname{RCD}$ spaces with large isometry groups, extending results of Galaz-Garc\'ia--Kell--Mondino--Sosa and Galaz-Garc\'ia--Guijarro.