Convergence of symmetries: nilpotency, dimension, and perfectness

Let $(X_i,p_i)$ be a 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 groups 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 $\mathfrak{g}$, the Lie algebra of $G$, admits an ideal $\mathfrak{h}⊴\mathfrak{g}$ with $\dim (\mathfrak{h})\le {\mathrm{lim\hspace{0.17em}sup}}_i\dim (G_i)$ and $\mathfrak{g}\mathrm{/}\mathfrak{h}$ nilpotent. Moreover, if the sequence $(X_i,p_i)$ is non-collapsing, we show that $\mathfrak{h}$ can be taken of dimension ${\mathrm{lim\hspace{0.17em}sup}}_i\dim (G_i)$.