The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature
We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.
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| citation_txt | The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature. Jiayin Pan. SIGMA 16 (2020), 078, 16 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 078, 16 pages
The Fundamental Groups of Open Manifolds
with Nonnegative Ricci Curvature
Jiayin PAN
Department of Mathematics, University of California-Santa Barbara,
Santa Barbara CA 93106, USA
E-mail: j pan@math.ucsb.edu
URL: http://web.math.ucsb.edu/~j_pan/
Received June 02, 2020, in final form August 04, 2020; Published online August 17, 2020
https://doi.org/10.3842/SIGMA.2020.078
Abstract. We survey the results on fundamental groups of open manifolds with nonnegative
Ricci curvature. We also present some open questions on this topic.
Key words: Ricci curvature; fundamental groups
2020 Mathematics Subject Classification: 53C21; 53C23; 57S30
1 Introduction
One fascinating topic in Riemannian geometry is the interplay between curvature and topology.
Our primary interest in this survey is how nonnegative Ricci curvature determines the structure
of fundamental groups of open (complete and non-compact) manifolds.
For comparison, we recall the topological implications of nonnegative sectional curvature for
open manifolds. By Cheeger–Gromoll soul theorem [11], any open manifold M of sec ≥ 0 is
homotopic to a closed totally geodesic submanifold in M . In particular, M has finite topology.
On the fundamental group level, the soul theorem implies that π1(M) is finitely generated and
virtually abelian (contains an abelian subgroup of finite index). In contrast, an open manifold
with Ric ≥ 0 may have infinite second or higher Betti number, even when the manifold has
Euclidean volume growth [3, 26, 36, 38, 39, 49, 50]; the reader may refer to Section 4 of [41] for
a survey on the examples of manifolds with nonnegative Ricci curvature and infinite topology.
Therefore, it is particularly interesting to see whether the structure of fundamental groups,
finite generation and virtual abelianness, still holds for open manifolds with nonnegative Ricci
curvature.
On finite generation, this is the longstanding Milnor conjecture raised in 1968.
Conjecture 1.1 (Milnor [27]). Let M be an open n-manifold of Ric ≥ 0. Then π1(M) is finitely
generated.
The Milnor conjecture remains open.
On virtual abelianness, Wei constructed an open manifold of positive Ricci curvature whose
fundamental group is not virtually abelian [47]. Later, Wilking showed that any finitely gener-
ated virtually nilpotent group can be realized the fundamental group of some open manifold of
positive Ricci curvature [48]. On the other hand, by the work of Gromov and Milnor [20, 27],
nonnegative Ricci curvature implies that any finitely generated subgroup of the fundamental
group is virtually nilpotent. Kapovitch and Wilking proved a stronger result by dropping the
finite generation condition and achieving a uniform bound on the index [23].
This paper is a contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov
on his 75th Birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Gromov.html
mailto:j_pan@math.ucsb.edu
http://web.math.ucsb.edu/~j_pan/
https://doi.org/10.3842/SIGMA.2020.078
https://www.emis.de/journals/SIGMA/Gromov.html
2 J. Pan
We summarize these results in the table below.
let M be an open manifold of sec ≥ 0 Ric ≥ 0
then π1(M) is
finitely generated true unknown
virtually abelian true false
virtually nilpotent true true
Table 1. Nonnegative curvature and fundamental groups.
Given that proving or disproving the Milnor conjecture seems very difficult with the current
understanding of nonnegative Ricci curvature, it is natural to ask the following question:
Question 1.2. For an open manifold of Ric ≥ 0, on what conditions is π1(M) finitely generated?
Though virtual abelianness does not hold for all open manifolds with nonnegative Ricci
curvature, we can ask a similar question:
Question 1.3. For an open manifold of Ric ≥ 0, on what conditions is π1(M) virtually abelian?
In Section 2, we go through some basic tools in Ricci curvature and their applications to fun-
damental groups. This covers Bishop–Gromov relative volume comparison, Cheeger–Gromoll
splitting theorem, and structure results on asymptotic cones. For applications on fundamen-
tal groups, we include the finiteness result by Anderson [2] and Li [24], Milnor’s estimate on
the growth function [27], and fundamental groups of closed manifolds with nonnegative Ricci
curvature [10].
We survey results related the Milnor conjecture in Section 3. We describe Gromov’s short ge-
nerators [19], Kapovitch–Wilking’s bound on the number of short generators [23], and Wilking’s
reduction on the Milnor conjecture [48]. Regarding answers to Question 1.2, we survey the work
by Sormani on manifolds with small linear diameter growth [43], the classification by Liu in
dimension 3 [25], and the recent progress by the author via asymptotic geometry [29, 30, 31].
In Section 4, we first describe the construction by Wei [47] and other related examples by
Nabonnand [28] and Bérard-Bergery [4]. Then we present author’s recent work on Question 1.3
[31, 32, 33].
We also post some further questions and conjectures in Sections 3 and 4.
We mention that there are also many beautiful results on the finite topology of open mani-
folds under additional geometric constraints like sectional curvature, conjugate radius, diameter
growth, etc. We are unable to survey these results here due to the limited space.
2 Some basic tools in Ricci curvature
We review some basic tools in Ricci curvature with a focus on the applications to fundamental
groups of open manifolds with Ric ≥ 0. Though some of the results in this section hold in a more
general setting, we will only consider the case of nonnegative Ricci curvature.
Because the fundamental group acts isometrically on the Riemannian universal cover by path
lifting, these tools are commonly applied to not only the manifold itself but also its universal
cover when we study the fundamental group.
2.1 Volume comparison
One of the basic tools is the Bishop–Gromov relative volume comparison.
The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature 3
Theorem 2.1 (Bishop–Gromov [5, 21]). Let M be an open n-manifold of Ric ≥ 0 and let
x ∈M . Then
vol(Br(x)) ≤ vol(Bn
r (0)),
vol(BR(x))
vol(Br(x))
≤
vol(Bn
R(0))
vol(Bn
r (0))
=
Rn
rn
for all R > r > 0, where Br(x) is the metric r-ball centered at x and Bn
r (0) is the r-ball in the
standard Euclidean space Rn.
Theorem 2.1 implies that on an open manifold M of Ric ≥ 0, the limit
lim
r→∞
vol(Br(x))
vol(Bn
r (0))
= L
always exists with L ∈ [0, 1], where x ∈M . Moreover, this limit does not depend on the choice
of x. When L > 0, we say that M has Euclidean volume growth of constant L.
If L equals the maximum value 1, then M is isometric to the standard Euclidean space Rn.
When M has almost max volume growth, that is, L is sufficiently close to 1, there is a corre-
sponding almost rigidity result: Perelman proved that such a manifold is contractible [35]; later
Cheeger and Colding proved that it is diffeomorphic to Rn [6].
For L > 0, Li and Anderson independently proved that π1(M) is finite.
Theorem 2.2 (Li [24], Anderson [2]). Let M be an open n-manifold of Ric ≥ 0. If M has
Euclidean volume growth of constant L, then π1(M) is finite with order at most 1/L.
In particular, if L > 1/2, then M is simply connected. Li’s proof uses heat kernel on the uni-
versal cover. We briefly go through Anderson’s proof, which utilizes the volume comparison and
a method to calculate the volume via Dirichlet domain. As π1(M,x) acts freely, discretely, and
isometrically on the Riemannian universal cover M̃ , we can construct the Dirichilet domain D
centered at x̃, where x̃ ∈ M̃ is a lift of x. D is a fundamental domain; in particular, under
π1(M,x)-action, γ · (Br(x̃)∩D) and γ′ · (Br(x̃)∩D) are disjoint whenever γ and γ′ are different
elements in π1(M,x). Also, D satisfies vol(Br(x)) = vol(Br(x̃) ∩ D) for all r > 0. To prove
Theorem 2.2, we take a finite subset S ⊆ π1(M,x) and choose d > 0 such that d(γx̃, x̃) ≤ d for
all γ ∈ S. Let
S(k) = {γ ∈ π1(M,x) | γ can be expressed as a word in S with word length ≤ k},
where k ∈ N. Note that d(γx̃, x̃) ≤ kd for all γ ∈ S(k). For any r > 0, we estimate the volume
of S(k) · (Br(x̃) ∩D):
#S(k) · vol(Br(x)) = #S(k) · vol(Br(x̃) ∩D) = vol(S(k) · (Br(x̃) ∩D)) ≤ vol(Br+kd(x̃)).
It follows from volume comparison that
#S(k) ≤ vol(Br+kd(x̃))
vol(Br(x))
≤
vol(Bn
r+kd(0))
vol(Br(x))
.
Let r →∞, we see that
#S(k) ≤ lim
r→∞
vol(Bn
r+kd(0))
vol(Bn
r (0))
· vol(Bn
r (0))
vol(Br(x))
=
1
L
.
Since this estimate holds for any k ∈ N and any finite subset S ⊆ π1(M,x), Theorem 2.2 follows.
This proof also indicates that if M and M̃ have the same volume growth rate, then π1(M) is
finite as well.
Later, this finiteness result was extended to any manifold whose volume growth has order
n− ε for some sufficiently small ε > 0.
4 J. Pan
Theorem 2.3 (Wu [51]). Given n, there exists a constant ε(n) > 0 such that the following
holds.
Let M be an open n-manifold of Ric ≥ 0. If there is x ∈M such that
lim
r→∞
vol(Br(x))
rn−ε(n)
∈ (0,∞),
then π1(M) is finite.
Also see Wan’s work on finiteness under a relative volume growth condition [46], which
implies Theorem 2.3. It is unknown whether the volume growth condition in Theorem 2.3 can
be weakened to lim inf or any ε ∈ (0, 1).
When π1(M,x) is infinite, Milnor proved that the volume comparison provides an estimate on
the growth function of any finitely generated subgroup, which in turn implies virtual nilpotency
by Gromov’s work.
Theorem 2.4. Let M be a complete (open or closed) n-manifold of Ric ≥ 0. Then any finitely
generated subgroup of π1(M)
(1) has polynomial growth with degree at most n (Milnor [27]),
(2) is virtually nilpotent (Gromov [20]).
We can derive Theorem 2.4(1) from Anderson’s proof. Let S be a generating set of the finitely
generated subgroup. By taking r = 1, then we can estimate #S(k) by
#S(k) ≤
vol(Bn
1+kd(0))
vol(B1(x))
,
where the right hand side is a polynomial of k with degree n.
Theorem 2.4(2) is useful to prove that certain fundamental group is virtually abelian under
additional conditions: without lose of generality, we can start with a nilpotent one. It also
follows from Theorem 2.4(2) that every finitely generated subgroup is finitely presented [52].
We end this section with Kapovitch–Wilking’s result, which significantly improves Theo-
rem 2.4(2).
Theorem 2.5 (Kapovitch–Wilking [23]). Given n, there exists a constant C(n) such that the
following holds.
Let M be an open n-manifold of Ric ≥ 0. Then π1(M) contains a nilpotent subgroup N of
index at most C(n). Moreover, any finitely generated subgroup of N has nilpotency length at
most n.
2.2 Splitting theorem
A line is an isometric embedding γ : R→M ; in other words,
d(γ(s), γ(t)) = |s− t|
for all s, t ∈ R. Cheeger–Gromoll splitting theorem is a fundamental tool in nonnegative Ricci
curvature.
Theorem 2.6 (Cheeger–Gromoll [10]). Let M be an open n-manifold of Ric ≥ 0. If M contains
a line, then M splits isometrically as a metric product R×N .
The splitting theorem implies the virtual abelianness of π1(M) when M is closed and has
nonnegative Ricci curvature.
The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature 5
Theorem 2.7 (Cheeger–Gromoll [10]). Let M be a closed n-manifold of Ric ≥ 0. Then π1(M)
is virtually abelian.
In fact, as a consequence of Theorem 2.6, the universal cover of M must split isometrically
as Rk ×N , where N is a compact manifold. Consequently, its isometry group splits as well
Isom
(
M̃
)
= Isom
(
Rk
)
× Isom(N),
where Isom(N) is a compact Lie group. Also, without lose of generality, we can assume
that π1(M) is nilpotent by Theorem 2.4. Since we can view π1(M) as a subgroup of Isom
(
M̃
)
,
Theorem 2.7 follows from the algebraic fact that any nilpotent subgroup of Isom
(
Rk
)
× K is
virtually abelian, where K is a compact Lie group.
We mention that it was conjectured by Fukaya and Yamaguchi that the index of the abelian
subgroup in Theorem 2.7 can be uniformly bounded by some constant C(n) [18]. This conjecture
is open even for nonnegative sectional curvature.
Theorem 2.7 has an analog for open manifolds. For each element γ ∈ π1(M,x), among all
the loops based at x representing γ, we can choose one of the minimal length. This loop is
a geodesic loop and we call it a representing geodesic loop of γ.
Theorem 2.8. Let M be an open n-manifold of Ric ≥ 0. If for some x ∈ M , all representing
geodesic loops of elements in π1(M,x) are contained in a bounded set, then π1(M) is virtually
abelian.
Though in this case, the universal cover may not be the metric product of Rk and a compact
space, it can be shown that π1(M,x) is contained in Isom
(
Rk
)
× K for some compact Lie
group K, which is sufficient for virtual abelianness (see Appendix A of [32] for a proof).
The assumption in Theorem 2.8 always holds when sec ≥ 0 by setting x in a soul of M . This
follows from Cheeger–Gromoll soul theorem and Sharafutdinov retraction [11, 40]. On the other
hand, the assumption in Theorem 2.8 is very restricted for manifolds with nonnegative Ricci
curvature. In fact, if M has positive Ricci curvature and an infinite fundamental group, then
it follows from the Cheeger–Gromoll splitting theorem that the representing geodesic loops cγ
will always escape from any bounded sets as γ exhausts π1(M,x) [45]. In Section 4.2, we will
see the author’s work [32, 33] generalizing Theorem 2.8.
2.3 Asymptotic geometry
The Gromov–Hausdorff distance between two metric spaces measures how close they look alike.
Using relative volume comparison Theorem 2.1, Gromov proved a precompactness result. This
concept of convergence revolutionizes the Riemannian geometry.
Theorem 2.9 (Gromov [21]). Let (Mi, xi) be a sequence of complete n-manifolds of Ric ≥
−(n − 1), then after passing to a subsequence, it converges in the (pointed) Gromov–Hausdorff
topology to a length metric space (X,x).
The limit space X is called a Ricci limit space. Cheeger, Colding, and Naber developed the
fundamental theory on the structure of these Ricci limit spaces [6, 7, 8, 9, 12, 13, 14, 15].
In the context of an open manifold M with Ric ≥ 0, we can use Gromov–Hausdorff conver-
gence to study its asymptotic geometry. For any sequence ri →∞, passing to a subsequence if
necessary, we obtain a convergent sequence(
r−1i M,x
) GH−→ (Y, y).
The limit (Y, y) is called an asymptotic cone of M , or a tangent cone of M at infinity. It does
not depend on the base point x, but it may depend on the scaling sequence ri.
6 J. Pan
Cheeger and Colding substantially generalized the splitting theorem to Ricci limit spaces [6].
For convenience, we denote M(n, 0) the set of all Ricci limit spaces coming from sequences of
complete n-manifolds of Ric ≥ 0.
Theorem 2.10 (Cheeger–Colding [6]). Let X ∈ M(n, 0). If X contains a line, then X splits
isometrically as a metric product R×N .
One important class of asymptotic cones is metric cones.
Theorem 2.11 (Cheeger–Colding [6]). Let M be an open n-manifold of Ric ≥ 0. Suppose
that M has Euclidean volume growth. Then any asymptotic cone (Y, y) of M is a metric cone
with vertex y.
For a metric cone C(Z) ∈ M(n, 0), diam(Z) ≥ π is equivalent to C(Z) containing a line.
Then by Theorem 2.10, C(Z) splits isometrically as Rk × C(Z ′), where diam(Z ′) < π.
Since we want to study π1(M,x), which acts on the universal cover M̃ as isometries, it is
natural to apply the asymptotic method to
(
M̃, π1(M,x)
)
as well. This involves the equivariant
Gromov–Hausdorff convergence, introduced by Fukaya and Yamaguchi [17, 18]. Let ri → ∞,
after passing to a subsequence, we can obtain the diagram below:(
r−1i M̃, x̃, π1(M,x)
) GH−−−−→
(‹Y , ỹ, G)yπ yπ(
r−1i M,x
) GH−−−−→
(
Y = ‹Y /G, y). (2.1)
The π1(M,x)-actions on r−1i M̃ passes a limit G-action on ‹Y , where G is some closed subgroup
of Isom
(‹Y ). It follows from Theorem 2.12 below that G is a Lie group.
Theorem 2.12 (Colding–Naber [15]). Let X ∈M(n, 0). Then Isom(X) is a Lie group.
Theorem 2.12 was first proved for the non-collapsing case by Cheeger–Colding [8], then the
general case by Colding–Naber [15].
If X ∈ M(n, 0) is a metric cone, then due to the splitting X = Rk × C(Z ′), where Z ′ has
diameter less than π, the isometry group of X splits as well
Isom(X) = Isom
(
Rk
)
× Isom(Z ′).
3 Finite generation
3.1 Gromov’s short generators and their properties
Gromov introduced a geometric method to choose generators of π1(M,x) successively [19]. We
first choose γ1 ∈ π1(M,x)− {e} such that
d(γ1x̃, x̃) = min
γ∈π1(M,x)−{e}
d(γx̃, x̃).
Supposing that γk is chosen for some k ∈ N, we pick γk+1 ∈ π1(M,x)−Hk such that
d(γk+1x̃, x̃) = min
γ∈π1(M,x)−Hk
d(γx̃, x̃),
where Hk = 〈γ1, . . . , γk〉 is the subgroup generated by first k generators. It follows from the
choice of these generators that
d(γix̃, γj x̃) ≥ max{d(γix̃, x̃), d(γj x̃, x̃)}
The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature 7
for all i, j. When M has sec ≥ 0, applying Toponogov comparison theorem to the geodesic
triangle with vertices x̃, γix̃, and γj x̃, one can see that θi,j , the angle of this geodesic triangle
at x̃, is at least π/3. A standard packing argument then leads to Gromov’s bound on the number
of short generators.
Theorem 3.1 (Gromov [19]). Given n, there exists a constant C(n) such that the following
holds.
Let M be an open n-manifold of sec ≥ 0 and let x ∈M . Then the number of short generators
of π1(M,x) is bounded by C(n).
For Ricci curvature, Kapovitch and Wilking proved the following:
Theorem 3.2 (Kapovitch–Wilking [23]). Given n, there exists a constant C(n) such that the
following holds.
Let M be an open n-manifold of Ric ≥ 0 and let x ∈M . Let Γ be a finitely generated subgroup
of π1(M,x). Then there is a point q ∈ B1(x) such that the number of short generators of Γ at q
is bounded by C(n).
The statement proved by Kapovitch and Wilking is indeed local. Note that unlike Theo-
rem 3.1, Theorem 3.2 gives a bound on the number of short generators at a point near x, not
exactly at x. The proof of Theorem 3.2 is much more complicated than the sectional curvature
case. It starts with a contradicting sequence
(Mi, xi)
GH−→ (X,x)
so that the number of short generators at all points in B1(xi) is larger than i. The proof involves
an induction argument on the dimension of X in the Colding–Naber sense and a successive blow-
up argument to increase the dimension of the limit space. In this process, one has to shift base
points multiple times. Thus this method cannot bound the number of short generators at
a priorly fixed reference point.
Note that if a bound C(n) can be obtained at the base point x, then finite generation would
follow directly by taking the first C(n) + 1 many short generators to generate a subgroup. Also
see the work by Rong and the author where a bound at the base point is achieved under certain
conditions on π1(M,x)-action [34]. On the other hand, Theorem 3.2 does immediately imply
that any finitely generated subgroup of π1(M) can be generated by at most C(n) many elements.
However, this is far away from the Milnor conjecture. A typical example with this algebraic
property is the set of rational numbers Q as an additive group: Q is not finitely generated, but
every finitely generated subgroup is cyclic. An even simpler example is the cyclic dyadic rationals
Γ =
{ p
2i
∣∣ p ∈ Z, i ∈ N
}
.
A set of short generators of Γ could potentially be γ1 = 1/2, . . . , γi = 1/
(
2i
)
, etc. Ruling
out these abelian groups as fundamental groups is crucial. In fact, with the help of Wilking’s
reduction, it is sufficient to consider abelian ones.
Theorem 3.3 (Wilking [48]). Let M be an open n-manifold of Ric ≥ 0. If π1(M) is not finitely
generated, then π1(M) has an abelian subgroup that is not finitely generated.
Another geometric property of short generators is about its representing geodesic loops. Let c
be a unit speed representing geodesic loop of a short generator γ and let r be the length of c, then
by the choice of short generators, c must be minimal up to its halfway, that is, d(x,m) = r/2,
where m = c(r/2) is the midpoint of c. This implies that m is a cut point of x; in particular,
for y ∈M with d(x, y) > r/2, we have
d(x,m) + d(y,m)− d(x, y) > 0.
8 J. Pan
Applying the Abresch–Gromoll inequality [1] to the universal cover, Sormani established a quan-
titative version of the above estimate [43]: for all y ∈M with d(x, y) ≥ r/2 + S(n)r, we have
d(x,m) + d(y,m)− d(x, y) ≥ 2S(n)r,
where S(n) > 0 is a small constant depending only on n. If π1(M,x) is not finitely generated,
then M has the above uniform cut estimate at a sequence of midpoints mi with d(x,mi)→∞.
With this, Sormani confirmed the Milnor conjecture when M has small linear diameter growth.
Theorem 3.4 (Sormani [43]). Given n, there is a constant S(n) > 0 such that the following
holds.
Let M be an open n-manifold of Ric ≥ 0. If M has small linear diameter growth
lim sup
r→∞
diam(∂Br(x))
r
≤ S(n),
then π1(M) is finitely generated.
The diameter here is extrinsic so diam(∂Br(x)) ≤ 2r always holds. Theorem 3.4 also covers
manifolds with linear volume growth [42], that is,
lim sup
r→∞
vol(Br(x))
r
<∞.
If π1(M,x) is not finitely generated, then we have a sequence of scales ri = d(γix̃, x̃) → ∞,
where {γi}i is a set of short generators. Using this sequence ri, we can see the consequences of
infinite generation through asymptotic geometry.
Proposition 3.5. Let M be an open n-manifold of Ric ≥ 0. Suppose that π1(M) is not finitely
generated with a set of short generators {γ1, . . . , γi, . . . }. Then in the diagram (2.1) with ri =
d(γix̃, x̃)→∞,
(1) Y is not polar at y [43];
(2) the orbit G · ỹ is not connected [30].
Y being polar at y means that for any z ∈ Y −{y}, there is a ray starting at y going through z.
Recall that short generators {γ1, . . . , γi, . . . } provides a sequence of representing geodesic loops
based at x, on which the uniform cut estimate holds at the midpoint mi. Passing this estimate
to the asymptotic cone Y , we get a limit cut point in Y . This leads to (1) above. For (2), let Hi
be the subgroup of π1(M,x) that is generated by the first i many short generators. Then we
consider the convergence(
r−1i M̃, x̃,Hi, π1(M,x)
) GH−→
(‹Y , ỹ,H,G).
The orbit H · ỹ contains the connected component of G · ỹ that includes ỹ. γi+1 converges to an
element γ∞ ∈ G with d(H · ỹ, γ∞ỹ) = 1; in particular, G · ỹ is not connected.
We mention that Sormani observed another way to see the consequence of infinite generation
from afar by using the loop to infinity property [44]. An open manifold satisfies the loop to
infinity property, if for any noncontractible loop c and any compact subset K, there exists
a loop in M −K that is freely homotopic to c. This property gives a surjective inclusion map
i? : π1(M−K, y)→ π1(M,y), where K is compact and y ∈M−K. Sormani proved that for any
open manifold M of Ric ≥ 0, either M has the loop to infinity property or a double cover of M
splits off a line isometrically; in particular, manifolds of positive Ricci curvature always have this
property. As a result, if there is an example M with Ric > 0 and an infinitely generated π1(M),
then we can slide all these infinitely many generators to the end of M .
The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature 9
3.2 3-manifolds
The Milnor conjecture is true in dimension 3. Schoen and Yau proved that any open 3-manifold of
positive Ricci curvature is diffeomorphic to R3 [37]. Using Schoen–Yau’s minimal surface theory
and Perelman’s solution to Poincaré conjecture, Liu classified open 3-manifold of nonnegative
Ricci curvature [25]: either it is diffeomorphic to R3 or its universal cover splits off a line
isometrically; in particular, this confirms the Milnor conjecture in dimension 3.
Theorem 3.6 (Liu [25]). The Milnor conjecture holds in dimension 3.
Later, the author gave an alternative proof of the finite generation of π1(M) by using asymp-
totic geometry and short generators [30]. Here we give an outline of the proof by the author.
With Wilking’s reduction and a topological result in dimension 3 by Evans and Moser [16],
if π1(M) is not finitely generated, then without lose generality we can assume that it is a sub-
group of Q; in particular, π1(M) is torsion-free. This implies a geometric feature on the asymp-
totic cones: for any equivariant asymptotic cone
(‹Y , ỹ, G) of
(
M̃,Γ
)
, the limit orbit G ·y cannot
be discrete.
Suppose that π1(M,x) is not finitely generated. Let {γ1, . . . , γi, . . . } be a set of short gen-
erators and let ri = d(γix̃, x̃) → ∞. We consider the corresponding asymptotic cones as in the
diagram (2.1). To derive a contradiction, the proof eliminates all the possibilities of asymp-
totic cones ‹Y and Y regarding their dimensions in the Colding–Naber sense [15], which are
integers 1, 2 or 3.
Case 1. dim
(‹Y ) = 3. This case occurs if and only if M̃ has Euclidean volume growth.
Hence ‹Y is a metric cone. If ‹Y splits off an R2 or R3 factor, then Cheeger–Colding theory
implies that M̃ is isometric to R3 [7]. If ‹Y splits off exactly one R factor, then the orbit G · ỹ
is contained in a line. It follows from Theorem 3.5(2) that the orbit must be discrete, which
contradicts the non-discreteness that we derived from the reduction. If ‹Y does not split off any
line, then the orbit G · ỹ must be the cone tip ỹ; a contradiction to both non-discreteness from
the reduction and the non-connectedness from Theorem 3.5(2).
Case 2. dim
(‹Y ) = dim(Y ) = 2. Since Y = ‹Y /G, it is reasonable to believe that G is a
discrete group, thus the orbit G · y must be discrete as well. This again contradicts the non-
discreteness. Note that ruling out this case does not require that Y and ‹Y come from the scaling
sequence ri, the length of short generators.
Case 3. dim(Y ) = 1. A Ricci limit space of dimension 1 is either a ray or a line [22].
Combined with Proposition 3.5(1), the (Y, y) must be a ray with y not being the starting point.
By choosing a suitable rescaling si → ∞, we can obtain a different asymptotic cone Y ′ with
dim(Y ′) ≥ 2. Together with the results from cases 1 and 2, one can derive a contradiction as
well.
3.3 Equivariant stability at infinity
With the idea of investigating equivariant asymptotic cones, the author proved new partial
results on the Milnor conjecture [29, 31]. These results are related to certain geometric stability
condition on the universal cover at infinity and the question below:
Question 3.7. Does geometric stability at infinity implies equivariant stability at infinity?
To understand how Question 3.7 is related to the Milnor conjecture, we think of an example:
the universal cover has a unique asymptotic cone as a standard Euclidean space Rk for some k.
We have seen in Proposition 3.5(2) that if π1(M,x) is not finitely generated, then(
r−1i M̃, x̃, π1(M,x)
) GH−→
(
Rk, 0, G
)
10 J. Pan
has a limit orbit G · 0 that is not connected. If we change the scale ri to sri, where s > 0, then
the corresponding limit
(
s−1Rk, 0, G
)
is not equivariantly isometric to
(
Rk, 0, G
)
, because the
distance between two components of G · 0 is changed by a scale of s−1. On the other hand,
because we assumed that the asymptotic cone of M̃ is independent of the scaling sequence, one
may wonder whether the equivariant limit is independent of the scales as well. If this turns out
to be true, then any equivariant asymptotic cone at infinity ought to have a connected orbit
at the base point and finite generation would follow. Moreover, it can be shown that if G is
nilpotent and (sRk, 0, G) are all equivariantly isometric to each other for all scales s > 0, then
the G = Rl ×K, where K-action fixes 0 and the subgroup Rl × {e} acts as translations. Hence
one may expect that if M̃ has a unique asymptotic cone as Rk, then any equivariant asymptotic
cone
(
Rk, 0, G
)
of
(
M̃, π1(M)
)
should have a splitting structure G = Rl×K as described above,
where π1(M) is nilpotent.
The author studied Question 3.7 when the Riemannian universal cover satisfies one of the
conditions below.
� M̃ is k-Euclidean at infinity, where k is an integer. This means that any asymptotic cone
of M̃ is a metric cone that splits off exactly Rk.
� M̃ is (C(X), εX)-stable at infinity, where C(X) ∈ M(n, 0) is a metric cone and εX is
a constant only depending on the cross-section X. This means that any asymptotic cone
of M̃ is a metric cone, whose cross-section is εX -close to X with respect to Gromov–
Hausdorff distance.
Theorem 3.8 ([29, 31]). Let M be an open n-manifold of Ric ≥ 0. Suppose that one of the
following conditions holds on the Riemannian universal cover M̃ :
(1) M̃ is k-Euclidean at infinity for some 0 ≤ k ≤ n; or
(2) M̃ is (C(X), εX)-stable at infinity for some metric cone C(X) ∈M(n, 0) and a sufficiently
small εX > 0.
Then π1(M) is finitely generated.
Both cases include the case that M has nonnegative sectional curvature, with which M̃ has
a unique asymptotic cone as a metric cone. They also cover the case that M̃ has Euclidean
volume growth and a unique asymptotic cone. Another special case for (2) is when M̃ has
Euclidean volume growth of constant at least 1 − ε(n) for some small ε(n) > 0; in this case,
due to Cheeger–Colding almost maximal volume rigidity [6], any asymptotic cone of M̃ has
a cross-section being Gromov–Hausdorff close to the unit (n− 1)-dimensional sphere.
We view the condition in Theorem 3.8 as a geometric stability condition of M̃ at infinity. As
indicated, the key to proving finite generation is establishing corresponding equivariant stability
at infinity. For simplicity, we only state the result of equivariant stability when M̃ has a unique
cone as a metric cone C(Z) here.
Theorem 3.9. Let M be an open n-manifold with Ric ≥ 0 and a nilpotent fundamental group.
Suppose that the universal cover M̃ has a unique asymptotic cone as a metric cone C(Z). Then
there exist a closed nilpotent subgroup K of Isom(Z) and an integer l ∈ [0, n] such that any
equivariant asymptotic cone
(‹Y , ỹ, G) of
(
M̃, π1(M,x)
)
is
(
C(Z), z,Rl × K
)
, where K-action
fixes z and the subgroup Rl × {e} acts as translations in the Euclidean factor of C(Z).
To prove Theorem 3.9, it is essential to treat the set of all equivariant asymptotic cones of(
M̃, π1(M,x)
)
, denoted as Ω
(
M̃, π1(M,x)
)
, as a whole, not as individual spaces. It is known
that Ω
(
M̃, π1(M,x)
)
is compact and connected in the pointed equivariant Gromov–Hausdorff
topology. This fact serves as an intuition behind the proof.
The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature 11
The proof of Theorem 3.9 is a contradicting argument. Suppose that there are two dif-
ferent equivariant asymptotic cones (C(Z), z,G1) and (C(Z), z,G2). Using the connectedness
of Ω
(
M̃, π1(M,x)
)
, we can find a chain of spaces in Ω
(
M̃, π1(M,x)
)
connecting (C(Z), z,G1)
and (C(Z), z,G2), such that the adjacent spaces in the chain are very close in the equivariant
Gromov–Hausdorff topology. Roughly speaking, we find a contradiction somewhere in the chain.
The author developed two technical tools to achieve this. One is a critical rescaling argument.
We view (C(Z), z,G1) and (C(Z), z,G2) as the limits coming from two different scales(
r−1i M̃, x̃, π1(M,x)
) GH−→ (C(Z), z,G1),
(
s−1i M̃, x̃, π1(M,x)
) GH−→ (C(Z), z,G2).
Adjusting the scales, we can assume that one is the rescaling of the other by some ti →∞. The
critical rescaling argument chooses a suitable intermediate scaling sequence li →∞ with li ≤ ti
and looks for a contradiction in the corresponding limit coming from li. This argument helps
reduce the problem to a compact one regarding isometric actions on the cross-section Z. The
other tool is an equivariant Gromov–Hausdorff distance gap between isometric actions on Z. To
describe this, we think of a chain of spaces {(Z,Hj)}mj=1, where each Hj is a closed subgroup
of Isom(Z). The equivariant Gromov–Hausdorff distance gap implies that if the distance between
adjacent spaces is sufficiently small along the chain, then all (Z,Hj) have to be the same space.
Theorem 2.12 is crucial in establishing such a distance gap.
Based on [29, 31], we ask the following question on open manifolds whose Riemannian uni-
versal covers have Euclidean volume growth.
Question 3.10. When the Riemannian universal cover M̃ has Euclidean volume growth, is
there any equivariant stability among all equivariant asymptotic cones of
(
M̃, π1(M,x)
)
?
One may think of the Euclidean volume growth condition as certain geometric stability
condition at infinity as well, but a very weak one: we only know that all asymptotic cones are
metric cones whose cross sections share the same (n − 1)-dimensional Hausdorff volume. An
affirmative answer to Question 3.10 would imply the finite generation for this class of manifolds.
4 Virtual abelianness
4.1 Wei’s construction and related examples
Wei first constructed an open manifold of positive Ricci curvature whose fundamental group
is not virtually abelian, as a torsion-free nilpotent group [47]. Based on this construction,
later Wilking showed that any finitely generated virtually nilpotent group can be realized as
the fundamental group of some open manifold with positive Ricci curvature [48]. We briefly
describe Wei’s construction and other related examples.
Let
(
Sp−1, ds2
)
be the unit (p− 1)-dimensional sphere and let (N, g0) be a closed manifold.
A doubly warped product [0,∞)×f Sp−1 ×h N has the metric
g = dr2 + f(r)2ds2 + h(r)2g0,
It is diffeomorphic to Rp ×N if the warping functions f and h are smooth and satisfy
f(0) = 0, f ′(0) = 1, f ′′(0) = 0, h(0) > 0, h′(0) = 0
and f(r) > 0, h(r) > 0 for all r > 0.
Nabonnand constructed a doubly warped product [0,∞)×f S2×h S1 of positive Ricci curva-
ture [28]. Later, Bergery generalized this method to a doubly warped product M = [0,∞) ×f
Sp−1 ×h N of positive Ricci curvature, where N is any closed manifold with nonnegative Ricci
12 J. Pan
curvature [4]. Note that π1(M) = π1(N) is always virtually abelian. In this construction,
warping function h(r) can be adjusted so that h(r)→ 0 or h(r)→ c > 0 as r →∞.
We illustrate Wei’s example. Let ‹N be a simply connected nilpotent Lie group and let Γ
be a lattice in ‹N . When ‹N has nilpotency length ≥ 2, Γ is not virtually abelian and thus the
compact quotient manifold N = ‹N/Γ does not admit a metric of nonnegative Ricci curvature.
However, N admits a family of metrics gr with almost nonnegative sectional curvature. Wei
constructed a warped product M = [0,∞) ×f Sp−1 × Nr of positive Ricci curvature with the
metric
g = dr2 + f(r)2ds2 + gr.
This metric is not a doubly warped product in the usual sense since the metric gr decays at
different rates for different steps of N . Note that π1(M) = π1(N) is not virtually abelian. In
Wei’s construction, diam(N, gr) has polynomial decay as r →∞.
We mention that one can construct a doubly warped product [0,∞)×fSp−1×gS1 with positive
Ricci curvature and a logarithm decaying h(r) (see Appendix B of [32]). However, unlike Wei’s
construction, it is impossible to use logarithm decaying functions to warp a nilpotent manifold
M = [0,∞)×f Sp−1 ×Nr while keeping positive Ricci curvature. We will see the reason in the
next section.
4.2 Escape rate
In the light of Theorem 2.8, we see that the virtual abelianness is related to where the repre-
senting geodesic loops of π1(M,x) are positioned. We also mentioned in Section 2.2 that the
escape phenomenon is prevalent for nonnegative Ricci curvature. Motivated by these, the author
introduced the escape rate to quantify this phenomenon by comparing the size of representing
geodesic loops to their length [32].
Definition 4.1. Let (M,x) be an open manifold with an infinite fundamental group. We define
the escape rate of (M,x), a scaling invariant, as
E(M,x) = lim sup
|γ|→∞
dH(x, cγ)
|γ|
,
where γ ∈ π1(M,x), |γ| = length(cγ), and dH is the Hausdorff distance.
As a loop based at x, its size dH(x, cγ) is at most half of its length. Hence E(M,x) always
ranges from 0 to 1/2. When all representing geodesic loops stay inside a bounded set or escape
at a sublinear rate compared to their length, we have E(M,x) = 0.
Regarding examples as warped products [0,∞)×fSp−1×hN , their escape rates are determined
by the warping function h(r). As diam(Nr) decreases to 0, the representing geodesic loop will
take advantage of the thin end to shorten its length, which will in turn enlarge its size. In other
words, one should expect that the faster diam(Nr) decays, the larger the escape rate is. In
fact, if h(r) has polynomial decay, as in Wei’s construction, then E(M,x) is positive; if h(r) has
logarithm decay or converges to a positive constant, then E(M,x) = 0.
It turns out that this simple geometric quantity can measure the structure of fundamental
group. First recall that if π1(M,x) is not finitely generated, then we have infinitely many
short generators whose representing geodesic loops are minimal up to halfway. This shows that
E(M,x) = 1/2 if π1(M) is not finitely generated. The author proved that zero escape rate
implies virtual abelianness, which substantially generalizes Theorem 2.8.
Theorem 4.2 ([32]). Let (M,x) be an open n-manifold of Ric ≥ 0. If E(M,x) = 0, then
π1(M,x) is virtually abelian.
The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature 13
We outline the proof of Theorem 4.2, which goes through the asymptotic geometry. A crucial
step is showing that the following statements are equivalent:
(1) E(M,x) = 0;
(2) in any equivariant asymptotic cone
(‹Y , ỹ, G) of
(
M̃, π1(M,x)
)
, the orbit G · ỹ is geodesic
in ‹Y , that is, its intrinsic and extrinsic metric agree;
(3) in any equivariant asymptotic cone
(‹Y , ỹ, G) of
(
M̃, π1(M,x)
)
, the orbit G · ỹ is geodesic
in ‹Y and is isometric to a standard Euclidean space.
The proof of (2)⇒ (3) relies on the Cheeger–Colding splitting theorem and a critical rescaling
argument, which is the type of argument we mentioned before in Section 3.3 when explaining
Theorem 3.9. The details are distinct from the proof of Theorem 3.9 due to different contexts.
To derive virtual abelianness from (3), we take a nilpotent subgroup N in π1(M,x) of finite
index. For any ri →∞, we consider(
r−1i M̃, x̃, N, π1(M,x)
) GH−→
(‹Y , ỹ,H,G).
It can be shown that H acts on G·ỹ = H ·ỹ = Rk by translations. This limiting behavior restricts
N -action at large scale on M̃ : they acts as almost translations in the sense that d
(
γ2x̃, x̃
)
is
close to twice of d(γx̃, x̃) when γ has large displacement at x̃. It was observed in [31] that this
feature implies virtual abelianness.
We explain why the Heisenberg 3-group H3 cannot act this way isometrically. For conve-
nience, we write |γ| = d(γx̃, x̃). The commutator calculus in H3 has
[
gk, h
]
= [g, h]k =
[
g, hk
]
for all g, h ∈ H3 and k ∈ N. Now let g, h ∈ H3 with [g, h] 6= e. We choose a large integer k such
that R ≤
∣∣[g, h](k
2)
∣∣ =
∣∣[gk, hk]∣∣. We set α = gk and β = hk, then continue to raise the power.
For any integer p and l = 2p, we have∣∣[α, β](l
2)
∣∣ =
∣∣[αl, βl]∣∣ ≤ 2 · 2p(|α|+ |β|).
On the other hand, the almost translation at large scale condition implies that∣∣[α, β](l
2)
∣∣ ≥ (1.9)2p|[α, β]|.
This leads to a contradiction when p is sufficiently large.
For manifolds in Theorem 3.8, as a consequence of their equivariant stability at infinity, they
have zero escape rate [32]. We can further bound the index of the abelian subgroup if in addition
the universal cover has Euclidean volume growth.
Theorem 4.3 ([31]). Let M be an open n-manifold of Ric ≥ 0. Suppose that one of the following
conditions holds on the Riemannian universal cover M̃ :
(1) M̃ is k-Euclidean at infinity for some 0 ≤ k ≤ n; or
(2) M̃ is (C(X), εX)-stable at infinity for some metric cone C(X) ∈M(n, 0) and a sufficiently
small εX > 0.
Then π1(M) is virtually abelian. If in addition M̃ has Euclidean volume growth of constant at
least L > 0, then the index can be bounded by some constant C(n,L).
For manifolds whose universal covers have Euclidean volume growth, if one can answer Ques-
tion 3.10 affirmatively, then that would confirm E(M,x) = 0 and the conjecture below.
14 J. Pan
Conjecture 4.4. Given n and L ∈ (0, 1], there exists a constant C(n,L) such that the following
holds.
Let M be an open n-manifold of Ric ≥ 0. If the Riemannian universal cover of M has
Euclidean volume growth of constant at least L, then π1(M) is finitely generated and contains
an abelian subgroup of index at most C(n,L).
As mentioned in Section 3.3, the case L being close to 1 is indeed included in Theorems 3.8(2)
and 4.3(2).
Theorem 4.2 is further generalized to an escape rate gap by the author.
Theorem 4.5 ([33]). Given n, there exists a constant ε(n) > 0 such that the following holds.
Let (M,x) be an open n-manifold of Ric ≥ 0. If E(M,x) ≤ ε(n), then π1(M,x) is virtually
abelian.
To end this section, we pose the question whether escape rate can detect nilpotent groups of
different steps.
Question 4.6. For each n, are there positive constants ε(n, k) strictly increasing in k such that
the following holds?
For any open n-manifold (M,x) of Ric ≥ 0, if E(M,x) ≤ ε(n, k), then π1(M) contains a free
nilpotent subgroup of nilpotency length at most k with finite index.
Acknowledgements
The author is partially supported by AMS Simons travel grant. The author would like to thank
Guofang Wei and Christina Sormani for their encouragement and suggestions when preparing
this survey.
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https://arxiv.org/abs/math.DG/0510139
1 Introduction
2 Some basic tools in Ricci curvature
2.1 Volume comparison
2.2 Splitting theorem
2.3 Asymptotic geometry
3 Finite generation
3.1 Gromov's short generators and their properties
3.2 3-manifolds
3.3 Equivariant stability at infinity
4 Virtual abelianness
4.1 Wei's construction and related examples
4.2 Escape rate
References
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| id | nasplib_isofts_kiev_ua-123456789-210770 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T11:48:02Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Pan, Jiayin 2025-12-17T14:32:01Z 2020 The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature. Jiayin Pan. SIGMA 16 (2020), 078, 16 pages 1815-0659 2020 Mathematics Subject Classification: 53C21; 53C23; 57S30 arXiv:2006.00745 https://nasplib.isofts.kiev.ua/handle/123456789/210770 https://doi.org/10.3842/SIGMA.2020.078 We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic. The author is partially supported by the AMS Simons Travel Grant. The author would like to thank Guofang Wei and Christina Sormani for their encouragement and suggestions when preparing this survey. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature Article published earlier |
| spellingShingle | The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature Pan, Jiayin |
| title | The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature |
| title_full | The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature |
| title_fullStr | The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature |
| title_full_unstemmed | The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature |
| title_short | The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature |
| title_sort | fundamental groups of open manifolds with nonnegative ricci curvature |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210770 |
| work_keys_str_mv | AT panjiayin thefundamentalgroupsofopenmanifoldswithnonnegativericcicurvature AT panjiayin fundamentalgroupsofopenmanifoldswithnonnegativericcicurvature |