On the fundamental group of complete manifolds with almost Euclidean volume growth

TL;DR

This paper proves that complete Riemannian manifolds with nonnegative Ricci curvature and specific volume growth conditions have trivial or finite fundamental groups, advancing understanding of their topological structure.

Contribution

It establishes new results linking volume growth conditions to the finiteness or triviality of the fundamental group in manifolds with nonnegative Ricci curvature.

Findings

Fundamental group is trivial under certain conditions.

Fundamental group is finite with specific volume growth.

Provides new insights into topology of manifolds with nonnegative Ricci curvature.

Abstract

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.

Equations

$${\alpha }_M:=\underset{r\to \infty }{\lim }\frac{V_p(r)}{{\omega }_n{r}^n}.$$

$$0<\frac{V_p(2r)}{V_p(r)}\le 2^n$$

$$\underset{r\to \infty }{\lim }\frac{V_p(2r)}{V_p(r)}=2^n.$$

$$\frac{V_p(2r)}{V_p(r)}>C(n)$$

$$\underset{r\to \infty }{\mathrm{liminf}}\frac{V_p(2r)}{V_p(r)}>C(n),$$

$$\underset{r\to \infty }{\lim }\frac{V_p(r)}{r^{n-ϵ}}>0,$$

$$d_M({\gamma }_k(0),{\gamma }_k(d_k/2))=d_k/2.$$

$$d(x,\gamma (D))\ge (r-1)D+4S_{n}D.$$

$$h:=d_M(\gamma (d/2),\sigma {|}_{[0,d)})\ge S(n)d.$$

$$H=d_M(\gamma (d/2),\sigma (d/2))\ge 2S(n)d,$$

$$h_1\ge \frac{d}{2}-r_0$$

$$h_1\ge H-(\frac{d}{2}-r_0).$$

$$h_1\ge H/2\ge S(n)d.$$

$$h_2\ge H/2\ge S(n)d.$$

$$h=\min (h_1,h_2)\ge S(n)d.$$

$$\frac{V_p(2d)}{V_p(d)}\le (1-(\frac{2S(n)}{3}{)}^n)(2^n-1)+1.$$

$$1\ge \frac{V_y(h)}{V_p(d)}+\frac{V_{{\Sigma }_p(d)}(d)}{V_p(d)}.$$

$$\frac{V_y(h)}{V_p(d)}\ge (\frac{2S(n)}{3}{)}^n.$$

$$V_y(h)\ge V_y(S(n)d)\ge (\frac{2S(n)}{3}{)}^n{V}_y(3d/2).$$

$$\frac{V_{{\Sigma }_p(d)}(d)}{V_p(d)}\ge \frac{1}{2^n-1}(\frac{V_p(2d)}{V_p(d)}-1).$$

$$B_p(2r)\setminus B_p(r)\subset B_{{\Sigma }_p(r)}(2r)\setminus B_{{\Sigma }_p(r)}(r).$$

$$V_p(2r)-V_p(r)$$

$$\frac{V_{{\Sigma }_p(r)}(r)}{V_p(r)}\ge \frac{1}{2^n-1}(\frac{V_p(2r)}{V_p(r)}-1).$$

$$C(n)=(1-(\frac{2S(n)}{3}{)}^n)(2^n-1)+1.$$

$$\frac{V_p(2d_k)}{V_p(d_k)}\le C(n),$$

$$V_p(2r)\ge 2^{n-ϵ}{V}_p(r).$$

$$V_p(r_0)\le (2^{n-ϵ}{)}^{-1}{V}_p(2r_0)\le \cdot \cdot \cdot \le (2^{n-ϵ}{)}^{-k}{V}_p(2^k{r}_0),$$

$$V_p(r)\ge V_p(2^k{r}_0)$$

Full text

1. Introduction

Throughout the paper $M$ denotes a complete noncompact Riemannian $n$-manifold with nonnegative Ricci curvature. Let $V_{p}(r)$ be the volume of the metric ball $B_{p}(r)$ origin at $p$ with radius $r$ in $M$. By Bishop-Gromov volume comparison , $\frac{V_{p}(r)}{\omega_{n}r^{n}}$ is a decreasing function, where $\omega_{n}$ is the volume of unit ball in $\mathbb{R}^{n}$. So the limit is existent as $r$ goes to infinite. Denote the volume growth of $M$ by

[TABLE]

The $\alpha_{M}$ is independent of $p$ and so a global geometric invariant. Moreover, the volume comparison also implies that $0\leq\alpha_{M}\leq 1$ and $\alpha_{M}=1$ if and only if $M$ is isometric to $\mathbb{R}^{n}$. We say that $M$ has Euclidean volume growth (or large volume growth) if $\alpha_{M}>0$.

The volume comparison theorem implies that

[TABLE]

for all $r>0$ and Euclidean volume growth condition implies that

[TABLE]

The main result of this note is

Theorem 1.1**.**

*Given $n$, there is a constant $C(n)<2^{n}$ such that if an open $n$- manifold $M$ satisfies

1)*

[TABLE]

*for some $p\in M$ and all $r>0$, then $M$ is simple connected.

2)*

[TABLE]

for some $p\in M$, then the fundamental group $\pi_{1}(M)$ is finite.

We should note that even though $M$ has Euclidean volume growth, one can not deduce that $M$ is simple connected. So formula (1.1) holding for all $r>0$ is important.

Set $\epsilon(n)=n-\log_{2}\frac{C(n)+2^{n}}{2}$. We see immediately that 2) of Theorem 1.1 implies the following

Corollary 1.2**.**

Given $n$, there is a constant $\epsilon(n)$ such that if an open $n$- manifold $M$ satisfies

[TABLE]

for some $p\in M$, $0\leq\epsilon<\epsilon(n)$, then $\pi_{1}(M)$ is finite.

This shows a gap phenomenon for a well-known result of Peter Li [3] and Anderson [1] states that $\pi_{1}(M)$ is finite provided $M$ has Euclidean volume growth.

On the other hand, Anderson has proved that (see Theorem 1.1 in [1]) under condition (1.3), every finitely generated subgroup of $\pi_{1}(M)$ is actually of polynomial growth of order $\leq\epsilon<1$. In [6] Bingye Wu proved that under condition (1.3) $\pi_{1}(M)$ is finitely generated. But every infinite group of finitely generated has polynomial growth of order at least 1 (I thank the referee for pointing out this fact. See Section 3). So Corollary 1.2 is also a consequence of Anderson and Wu’s results.

Acknowledgment: I would like to thank the referee for his (her) invaluable suggestions. The referee’s explanation clarify some understanding of mine on Anderson’s paper [1].

2. A related volume ratio

In this section we prove an estimate on the volume ratio $V_{p}(2r)/V_{p}(r)$ related to certain generated elements of $\pi_{1}(M)$ (Lemma 2.4 below). The main ingredients are Sormani’s uniform cut lemma [5] and some ideas due to Shen [4].

2.1. A uniform estimate

Let $g\in\pi_{1}(M,p)$ and $\pi:\widetilde{M}\rightarrow M$ be the universal cover. Following [5], we say that $\gamma$ is a minimal representative geodesic loop (based at p) of $g$ if $\gamma=\pi\circ\tilde{\gamma}$, where $\tilde{\gamma}$ is a minimal geodesic connecting $\tilde{p}$ and $g\tilde{p}$. So $L(\gamma)=d_{\widetilde{M}}(\tilde{p},g\tilde{p})$.

Given a group $G$, we say that $\{g_{1},g_{2},g_{3},\cdot\cdot\cdot\}$ is an ordered set of independent generators of $G$ if every $g_{i}$ can not be expressed as the previous generators and their inverses, $g_{1},g_{1}^{-1},\cdot\cdot\cdot,g_{i-1},g_{i-1}^{-1}$.

In [5] Sormani proved the following two lemmas.

Lemma 2.1**.**

(halfway lemma) There exists an ordered set of independent generators $\{g_{1},g_{2},\cdot\cdot\cdot\}$ of $\pi_{1}(M,p)$ with minimal representative geodesic loops $\gamma_{k}$ of length $d_{k}$ such that

[TABLE]

In particular, we have a sequence of such generators if $\pi_{1}(M,p)$ is infinitely generated.

Lemma 2.2**.**

(uniform cut lemma) Let $M^{n}$ ($n\geq 3$) be a complete manifold with $Ric\geq 0$. Let $\gamma$ be a noncontractible geodesic loop based at $p_{0}$ of length $L(\gamma)=2D$ such that

1. (1)

If $\sigma$ based at $p_{0}$ is a loop homotopic to $\gamma$, then $L(\sigma)\geq 2D$; 2. (2)

The $\gamma$ is minimal on $[0,D]$ and $[D,2D]$.

Then there is a constant $S_{n}$ depending on $n$ such that if $x\in\partial B_{p_{0}}(rD)$ where $r\geq 1+2S_{n}$, then

[TABLE]

The main idea of proof of uniform cut lemma is to lift geodesic loop to the universal covering space and research carefully the related excess function. It contains a nice application of Abresch-Gromoll’s estimate on excess function [2]. The above two lemmas allow her to show that Milnor conjecture holds for the manifold with so called small linear diameter growth.

Let $\gamma$ be a minimal representative geodesic loops based at $p$ of $L(\gamma)=d$ satisfying Lemma 2.1. The below estimate is important for our purpose.

Lemma 2.3**.**

Let $\sigma$ be a geodesic issuing from $p$ such that $\sigma(t)$ is minimal on $[0,d)$. Then there is a constant $S(n)$ such that

[TABLE]

Proof.

We set $h_{1}\coloneqq d_{M}(\gamma(d/2),\sigma|_{[0,d/2)})$ and $h_{2}\coloneqq d_{M}(\gamma(d/2,\sigma|_{[d/2,d)}).$ By Lemma 2.2, we have

[TABLE]

where $S(n)$ is a universal constant, $S(n)=\frac{n}{n-1}\frac{1}{4}\frac{1}{3^{n}}(\frac{n-2}{n-1})^{n-1}.$

Suppose that $h_{1}=d_{M}(\gamma(d/2),\sigma(r_{0}))$. By the triangle inequality one has

[TABLE]

and

[TABLE]

Then

[TABLE]

We also note that $d_{M}(\gamma(d/2),\sigma(d))\geq d/2.$ So similarly one has

[TABLE]

It follows that

[TABLE]

∎

2.2. A volume’s ratio

Continuing with notations $p,d$ in Lemma 2.3, we shall prove

Lemma 2.4**.**

We have the following ratio of volume

[TABLE]

Before giving the proof of Lemma 2.4, (following [4]) we introduce some necessary notations. Let $\Sigma_{p}$ be a close subset of unit tangent sphere $S_{p}M\subset T_{p}M$. Let $B_{\Sigma_{p}}(r)$ be the set of points $x\in B_{p}(r)$ such that there exists a minimal geodesic $\gamma$ from $p$ to $x$ with $\gamma^{{}^{\prime}}(0)\in\Sigma_{p}$. We write $V_{\Sigma_{p}}(r)$ for the volume of $B_{\Sigma_{p}}(r)$.

We denote by $\Sigma_{p}(r)$ the set of unit vectors $v\in S_{p}M$ such that $\gamma(t)=\exp_{p}(tv)$ is minimal on $[0,r)$.

Proof.

of Lemma 2.4. We write $y=\gamma(d/2)$. Since $h\leq d/2$, we have $B_{p}(d)\supset B_{y}(h)\cup B_{\Sigma_{p}(d)}(d).$ By the definition of $h$, this gives $V_{p}(d)\geq V_{y}(h)+V_{\Sigma_{p}(d)}(d),$ i.e.

[TABLE]

Claim 1:

[TABLE]

By the Bishop-Gromov comparison theorem, $\mu^{n}V_{p}(r)\geq V_{p}(\mu r)$ for $\mu\geq 1$. By Lemma 2.3 we have $h\geq S(n)d$. So

[TABLE]

Since $B_{y}(3d/2)\supset B_{p}(d)$, we have $V_{y}(3d/2)\geq V_{p}(d)$. Thus we obtain (2.2).

Claim 2:

[TABLE]

Following the observation of Shen (c.f. [4] Lemma 2.4), we see that

[TABLE]

Then we have

[TABLE]

The second in equality follows from the generalized volume comparison (Lemma 2.2 of [4]). Thus

[TABLE]

Jointing formulas (2.1), (2.2) and (2.3), we establish the lemma. ∎

3. A proof of theorem 1.1

We set

[TABLE]

If $\frac{V_{p}(2r)}{V_{p}(r)}>C(n)$ for all $r>0$, then there is no nontrivial generator satisfying Lemma 2.4. So $M$ is simple connected. Thus the first part of Theorem 1.1 is proved.

The proof of second part of Theorem 1.1 is divided into two steps.

Firstly, $\pi_{1}(M,p)$ is finitely generated. We argue by contradiction. Assume $\pi_{1}(M,p)$ is infinitely generated, then by Lemma 2.2, there is a sequence $\{d_{k}\}$, $d_{k}\rightarrow\infty$ as $k\rightarrow\infty$ satisfying Lemma 2.4, i.e.

[TABLE]

for all $k\geq 1$. This contradicts to condition (1.2).

Secondly, condition (1.2) implies that $V_{p}(r)\geq C\cdot r^{n-\epsilon}$ for some $\epsilon<1$ and sufficiently large $r$. So by Anderson’s result [1], $\pi_{1}(M)$ has polynomial growth of order $\leq\epsilon<1$.

The form of $C(n)$ allows us to write $C(n)=2^{n-\epsilon},\epsilon<1$. By condition (1.2), there exists $r_{0}>0$, for all $r\geq r_{0}$, one has

[TABLE]

So

[TABLE]

$k\in N$. For any $r\geq r_{0}$, we can assume $r\in[2^{k}r_{0},2^{k+1}r_{0}]$ for some $k\in N$. Then

[TABLE]

It follows that $V_{p}(r)\geq C\cdot r^{n-\epsilon}$ for all $r>r_{0}$.

An algebraic fact: If $\Gamma$ is an infinite group with generators $S=\{g_{1},\cdot\cdot\cdot,g_{k}\}$, then $\sharp U(r)\geq r$ for all $r\in N$, where $U(r)$ is the set of elements with word length $\leq r$ with respect to $S$. In particular, $\Gamma$ has polynomial growth of order at least 1. (This proof is provided by referee) To see this we argue by contradiction. Let $r$ be the smallest integer so that $\sharp U(r)<r$, then $r-1\leq\sharp U(r-1)\leq\sharp U(r)<r$. This shows $U(r)=U(r-1)$. In other words, any word of length $r$ can be expressed as a word of length $\leq r-1$. It follows that $\Gamma=U(r-1)$, which is finite, a contradiction.

The second part of Theorem 1.1 follows from above immediately.

Remark 3.1*.*

Our proof of finite generation of $\pi_{1}(M)$ is much different to Wu’s arguments under condition (1.3). Wu’s proof was based on the estimate of ordered set of independent generators with minimal representative geodesic loops.