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.