Ricci Limit Spaces Are Semi-locally Simply Connected

TL;DR

This paper proves that Ricci limit spaces are semi-locally simply connected and establishes a generalized Margulis lemma for these spaces, advancing understanding of their local topology and geometric structure.

Contribution

It demonstrates semi-local simple connectivity of Ricci limit spaces and proves a generalized Margulis lemma for these spaces, extending geometric analysis tools.

Findings

Ricci limit spaces are semi-locally simply connected

Existence of small loops contractible within larger neighborhoods

Generalized Margulis lemma holds for Ricci limit spaces

Abstract

Let $(X,p)$ be a Ricci limit space. We show that for any $\epsilon > 0$ and $x \in X$, there exists $r< \epsilon$, depending on $\epsilon$ and $x$, so that any loop in $B_{r}(x)$ is contractible in $B_{\epsilon}(x)$. In particular, $X$ is semi-locally simply connected. Then we show that the generalized Margulis lemma holds for Ricci limit spaces of $n$-manifolds.