Ricci curvature and fundamental groups of effective regular sets

Abstract

For a Gromov-Hausdorff convergent sequence of closed manifolds $M_i^n\overset{GH}\longrightarrow X$ with $\mathrm{Ric}\ge-(n-1)$, $\mathrm{diam}(M_i)\le D$, and $\mathrm{vol}(M_i)\ge v>0$, we study the relation between $\pi_1(M_i)$ and $X$. It was known before that there is a surjective homomorphism $\phi_i:\pi_1(M_i)\to \pi_1(X)$ by the work of Pan-Wei. In this paper, we construct a surjective homomorphism from the interior of the effective regular set in $X$ back to $M_i$, that is, $\psi_i:\pi_1(\mathcal{R}_{\epsilon,\delta}^\circ)\to \pi_1(M_i)$. These surjective homomorphisms $\phi_i$ and $\psi_i$ are natural in the sense that their composition $\phi_i \circ \psi_i$ is exactly the homomorphism induced by the inclusion map $\mathcal{R}_{\epsilon,\delta}^\circ \hookrightarrow X$.