Ricci Limit Spaces Are Semi-locally Simply Connected

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.