Ricci Curvature and Fundamental Groups of Effective Regular Sets

For a Gromov-Hausdorff convergent sequence of closed manifolds $M^n_i\xrightarrow{GH} X$ with ${\rm Ric}\ge -(n-1)$, ${\rm diam}(M_i) ≤ D,$ and ${\rm vol}(M_i) ≥ v > 0,$ we study the relation between $π_1(M_i)$ and $X.$ It was known before that there is a surjective homomorphism $ϕ_i : π_1(M_i) → π_1(X)$ 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, $ψ_i :π_1(\mathcal{R}^◦_{ϵ,δ} )→π_1(M_i).$ These surjective homomorphisms $ϕ_i$ and $ψ_i$ are natural in the sense that their composition $ϕ_i◦ψ_i$ is exactly the homomorphism induced by the inclusion map $R^◦_{ ϵ,δ}\hookrightarrow X.$