Non-collapsed eGH convergence and dimension

Let (Xᵢ,pᵢ) be a non-collapsing sequence of pointed n-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, and Gᵢ ≤ Iso (Xᵢ) a sequence of closed subgroups of isometries. We show that if the triples (Xᵢ, Gᵢ, pᵢ) converge in the equivariant Gromov--Hausdorff sense to a triple $(X,G,p)$, then dim (G) ≥ i → ∞ dim (Gᵢ), 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.