相关论文: Local estimate of fundamental groups

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…

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 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…

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…

It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. The main result of this paper is a counterexample, which provides an example $M^7$ with ${\rm…

In this note we discuss the fundamental groups and diameters of positively Ricci curved $n$-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of…

It follows from the work of Kapovitch and Wilking that a closed manifold with nonnegative Ricci curvature has an almost nilpotent fundamental group. Leftover questions and conjectures have asked if in this context the fundamental group is…

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…

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci…

In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The…

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…

Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In…

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.

Let $M$ be a complete Riemannian metric of sectional curvature within $[-a^2,-1]$ whose fundamental group contains a $k$-step nilpotent subgroup of finite index. We prove that $a\ge k$ answering a question of M. Gromov. Furthermore, we show…

In this note, we will give an positive answer to Pan-Rong's conjecture that for an open manifold with nonnegative Ricci curvature, if its universal cover has Euclidean volume growth, then its fundamental group is finitely generated.…

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G…

A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes…

We study a notion of curvature for finitely generated groups which serves as a role of Ricci curvature for Riemannian manifolds. We prove an analog of Cheeger-Gromoll splitting theorem. As a consequence, we give a geometric characterization…

We obtain results about fundamental groups of $RCD^{\ast}(K,N)$ spaces previously known under additional conditions such as smoothness or lower sectional curvature bounds. For fixed $K \in \mathbb{R}$, $N \in [1,\infty )$, $D > 0 $, we show…

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…