Related papers: On fundamental groups of RCD spaces

In this paper, we study some structure properties on the (revised) fundamental group of RCD(0,N) spaces. Our main result generalizes earlier work of Sormani on Riemannian manifolds with nonnegative Ricci curvature and small linear diameter…

The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a…

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at…

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and…

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of…

We say that a metric space $X$ is $(\epsilon,G)$-homogeneous if $G<Iso(X)$ is a discrete group of isometries with $diam(X/G)<\epsilon$.\ A sequence of $(\epsilon_i,G_i)$-homogeneous spaces $X_i$ with $\epsilon_i\to0$ is called a sequence of…

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an…

Let $M$ be an open (complete and non-compact) manifold with $\mathrm{Ric}\ge 0$ and escape rate not $1/2$. It is known that under these conditions, the fundamental group $\pi_1(M)$ has a finitely generated torsion-free nilpotent subgroup…

For any complete $n$-dim Riemannian manifold $M^n$ with nonnegative Ricci curvature, Kapovitch and Wilking proved that any finitely generated subgroup of the fundamental group $\pi_1(M^n)$ can be generated by $C(n)$ generators. Inspired by…

For a polycyclic group $\Lambda$, $\text{rank} (\Lambda )$ is defined as the number of $\mathbb{Z}$ factors in a polycyclic decomposition of $\Lambda$. For a finitely generated group $G$, $\text{rank} (G)$ is defined as the infimum of $…

Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of…

We consider a RCD$(-(N-1),N)$ space $(X,d,\mathcal{H}^N)$ with local bounded covering geometry. The first result is related to Gromov's almost flat manifold theorem. Specifically, if for every point $\tilde{p}$ in the universal cover…

In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called $RCD^*(K,N)$ spaces) with \emph{non-empty} one dimensional regular sets. In particular, we prove…

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose…

We extend the celebrated rigidity of the sharp first spectral gap under $Ric\ge0$ to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to…

Let $M$ be a 4-dimensional open manifold with nonnegative Ricci curvature. In this paper, we prove that if the universal cover of $M$ has Euclidean volume growth, then the fundamental group $\pi_1(M)$ is finitely generated. This result…

A space is defined to be "$n$-spheroidal" if it has the homotopy type of an $n$-dimensional CW-complex $X$ with $H_{n}(X, \mathbb{Z})$ not zero and finitely generated. A group $G$ is called "$n$-spheroidal" if its classifying space $K(G,1)$…

We show that if a sequence $M_n$ of closed aspherical $d$-dimensional Riemannian manifolds with Ricci curvature uniformly bounded below and diameter uniformly bounded above collapses, then for all large enough $n$, the fundamental groups…

In this article, we prove that the fundamental group $\pi_1(M)$ of a complete open manifold $M$ with nonnegative Ricci curvature is finitely generated, under the condition that the Riemannian universal cover $\tilde M$ satisfies an "almost…

We prove that the group of isometries of a metric measure space that satisfies the Riemannian curvature condition, $RCD^*(K,N),$ is in fact a Lie group. We obtain an optimal upper bound on its dimension and classify the spaces where this…