Non-collapsed eGH convergence and dimension

Let $(X_i,p_i)$ be a non-collapsing sequence of pointed $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, and $G_i\le \text{Iso}(X_i)$ 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 $\text{dim}(G)\ge {\mathrm{lim\hspace{0.17em}sup}}_{i\to \mathrm{\infty }}\text{dim}(G_i)$, generalizing a result of Harvey to the non-compact setting. The argument also applies in the non-smooth setting of RCD spaces. As an application, we investigate RCD spaces with large isometry groups, extending results of Galaz-García–Kell–Mondino–Sosa and Galaz-García–Guijarro.