Ricci curvature and isometric actions with scaling nonvanishing property

Abstract

In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the unit ball at the point is not collapsed, that is, $\mathrm{vol}(B_1(p))\ge v>0$. In this paper, we conjecture a new link in terms of isometries: if the maximal displacement of an isometry $f$ on $B_1(p)$ is at least $\delta>0$, then the maximal displacement of $f$ on the rescaled unit ball $r^{-1}B_r(p)$ is at least $\Phi(\delta,n,v)>0$ for all $r\in(0,1)$. We call this scaling $\Phi$-nonvanishing property at $p$. We study the equivariant Gromov-Hausdorff convergence of a sequence of Riemannian universal covers with abelian $\pi_1(M_i,p_i)$-actions $(\widetilde{M}_i,\tilde{p}_i,\pi_1(M_i,p_i))\overset{GH}\longrightarrow(\widetilde{X},\tilde{p},G)$, where $\pi_1(M_i,p_i)$-action is scaling $\Phi$-nonvanishing at $\tilde{p_i}$. We establish a dimension monotonicity on the limit group associated to any rescaling sequence. As one of the applications, we prove that for an open manifold $M$ of non-negative Ricci curvature, if the universal cover $\widetilde{M}$ has Euclidean volume growth and $\pi_1(M,p)$-action on $R^{-1}\widetilde{M}$ is scaling $\Phi$-nonvanishing at $\tilde{p}$ for all $R$ large, then $\pi_1(M)$ is finitely generated.