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

TL;DR

This paper investigates the topological and geometric limits of simply connected manifolds with Ricci curvature bounds and discrete isometric group actions, showing that the limit space's fundamental group is generated by loops in maximal torus orbits.

Contribution

It proves that the limit of such manifolds under equivariant Gromov-Hausdorff convergence has a simply connected quotient and a fundamental group generated by loops in maximal torus orbits.

Findings

The identity component of the limit group is a nilpotent Lie group.

The quotient space by a maximal torus is simply connected.

Fundamental group is generated by loops in torus orbits.

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 $y$.