On the limit of simply connected manifolds with discrete isometric cocompact group actions

Abstract

We study complete, connected and simply connected $n$-dim Riemannian manifold $M$ satisfying Ricci curvature lower bound. Further more, suppose that $M$ admits discrete isometric group actions $G$ so that the diameter of the quotient space $\mathrm{diam}(M/G)$ is bounded. In particular, for any $n$-manifold $N$ satisfying $\mathrm{diam}(N) \le D$ and $\mathrm{Ric} \ge -(n-1)$, the universal cover and fundamental group $(\widetilde{N},G)$ satisfies the above condition. Let $\{(M_i,p_i)\}_{i \in \mathbb{N}}$ be a sequence of complete, connected and simply connected $n$-dim Riemmannian manifolds satisfying $\mathrm{Ric} \ge -(n-1)$. Let $G_i$ be a discrete subgroup of $\mathrm{Iso}(M_i)$ such that $\mathrm{diam}(M_i/G_i) \le D$ where $D>0$ is fixed. Passing to a subsequence, $(M_i, p_i,G_i)$ equivariantly pointed-Gromov-Hausdorff converges to $(X,p,G)$. Then $G$ is a Lie group by Cheeger-Colding and Colding-Naber. We shall show that the identity component $G_0$ is a nilpotent Lie group. Therefore there is a maximal torus $T^k$ in $G$. Our main result is that $X/T^k$ is simply connected. Moreover, $\pi_1(X,p)$ is generated by loops contained in the $T^k$-orbits up to conjugation; each of these loops can be represented by $\alpha^{-1} \cdot \beta \cdot \alpha$ where $\alpha$ is a curve from $y$ to $p$ for some $y \in X$, and $\beta$ is a loop at $y$ contained in the $T^k$-orbit of $y$.