Ricci limit spaces are semi-locally simply connected

Abstract

Let $(X, p)$ be a Ricci limit space. We show that for any $\epsilon \gt 0$ and $x \in X$, there exists $r \lt \epsilon$, 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.