Ricci Curvature and Fundamental Groups of Effective Regular Sets

Abstract

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 $R^◦_{ ϵ,δ}\hookrightarrow X$.

Keywords

Ricci curvature / fundamental groups / Gromov-Hausdorff convergence

Acknowledgment

The author is partially supported by National Science Foundation (Grant No. DMS-2304698) and Simons Foundation Travel Support for Mathematicians. The author would like to thank Xiaochun Rong and Guofang Wei for many fruitful discussions through the years. The author would like to thank Dimitri Navarro for suggestions on an early draft of this paper.

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