Ricci curvature and fundamental groups of effective regular sets

Abstract

For a Gromov-Hausdorff convergent sequence of closed manifolds $M_i^n\stackrel{GH}{⟶}X$ 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 ${\varphi }_i:{\pi }_1(M_i)\to {\pi }_1(X)$ 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 ${\varphi }_i$ and ${\psi }_i$ are natural in the sense that their composition ${\varphi }_i\circ {\psi }_i$ is exactly the homomorphism induced by the inclusion map ${\mathcal{R}}_{\epsilon ,\delta }^{\circ }\hookrightarrow X$.

Submitted to a special issue in honor of Xiaochun Rong on his 70th birthday