  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.
