# A Restriction for Singularities on Collapsing Orbifolds

Authors: Yu Ding
Journal: ISRN Geometry
 (2011)

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## Abstract

Every pointp in an orbifold X has a neighborhood that is homeomorphic to Gp∖Br(0), where Gp is a finite group acting on Br(0)⊂ℝn, so that Gp(0)=0. Assume X is a Riemannian orbifold with isolated singularities that is collapsing, that is, X admits a sequence of metrics gi with uniformly bounded curvature, so that, for any x∈X, the volume of B1(x), with respect to the metric gi, goes to 0 as i→∞. For such X, we prove that |Gp|≤(2π/0.47)n(n−1) for all singularities p∈X.


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## Body

## 1. Introduction

Ann-dimensional Riemannianorbifold, X, is a metric space so that the following is true: for any x∈X, there exists r=r(x)>0 and a Riemannian metric g̃x on B2r(0)⊂ℝn, a finite group Gx (the isotropy group) acting on (Br(0),g̃x) by isometries, so that Gx(0)=0, and there is an isometry ιx:Br(x)→Gx∖Br(0) with ιx(x)=0 (see [1]). We call x∈X aregular point if |Gx|=1; otherwise, x is a singular point. We say the curvature of X satisfies(1.1)|KX|≤κ2,
if the sectional curvature K of every (Br(0),g̃x) above satisfies |K|≤κ2. We say X is collapsing, if X admits a sequence of metrics, gi, with uniformly bounded curvature, so that, for any x∈X,(1.2)lim⁡i→∞Volgi(B1(x))=0.As an example, consider the standardℤm=ℤ/mℤ action on the sphere S2.The quotient orbifoldXm=ℤm∖S2 will be arbitrarily collapsed when m→∞ (see Figure 1). However, for any fixedm, Xm can be collapsed only to a certain degree; it does not support a sequence of collapsing metrics. In fact, for each one of the two singularities on Xm, there is a neighborhood that is isometric to ℤm∖ℝ2, where ℝ2 is equipped with some ℤm invariant metric. Therefore if gi is a collapsing sequence of metrics on Xm, we get a corresponding sequence g̃i of pullback metrics on S2; every g̃i is smooth. Observe Vol(S2,g̃i)=mVol(Xm,gi), where m is fixed and limi→∞Vol(Xm,gi)=0, thus limi→∞Vol(S2,g̃i)=0. If the diameter of (Xm,gi) stays bounded, we immediately get a contradiction to the Gauss-Bonnet theorem; in general, we can use the result in [2] to conclude that S2 admits an F-structure, in particular the Euler characteristic χ(S2) vanishes—this is a contradiction since clearly χ(S2)=2.Figure 1On the other hand, consider the double of a 2-dimensional rectangle. Clearly it admits a flat metric, thus we obtain a sequence of collapsing metrics by rescale. Notice, in this example, for each of the four singularities, the isotropy groupGx has order 2, a quite small number.Intuitively, these examples suggest that when an orbifold is collapsing, a conelike singularity cannot be too “sharp,” that is, there should be some bound in|Gx|. The main result of this paper is as follows.Theorem 1.1.
AssumeX is a compact, collapsing orbifold, p∈X is an isolated singularity. Then |Gp|≤(2π/0.47)n(n-1).IfX has an isolated singularity p, then the dimension of X must be even, and Gp⊂SO(n). Theorem 1.1 fails if we drop the requirement that x is anisolatedsingularity; for example, we can take any orbifold X′ and let X=X×S1; by shrinking the S1 factor, we see X is collapsing while there is no restriction on singularities of X′. The bound |Gp|≤(2π/0.47)n(n-1) has its root in the Bieberbach theorem of crystallographic groups and Gromov's almost flat manifold theorem.Clearly, Theorem1.1 is a corollary of the following.Theorem 1.2.
For anyL>0, there is ϵ=ϵ(n,L) so that if X is an orbifold with all singularities q∈X satisfying |Gq|<L, Vol(B1(q))<ϵ, then |Gp|≤(2π/0.47)n(n-1) for any isolated singularity p∈X.Remark 1.3.
The bound|Gp|≤(2π/0.47)n(n-1) in Theorem 1.1 is not sharp. When n=2, it is not hard to see that either |Gp|=2 or X is a flat orbifold. Therefore by Polya and Niggli's classification of crystallographic groups on ℝ2 [3, page 105] or [4, page 228], we actually have |Gp|≤6 for collapsing 2 orbifolds.Anilmanifold, Γ∖N, is the quotient of the (left) action of a discrete, uniform subgroup Γ⊂N, on a simply connected nilpotent Lie group N. Left invariant vector fields (LIVFs) can be defined on Γ∖N. An affine diffeomorphism of Γ∖N is a diffeomorphism that maps any local LIVF to some local LIVF. In general, a right invariant vector field (RIVF) cannot be defined globally in Γ∖N, unless this vector field is in the center of the Lie algebra of N. However, the right invariant vector fields, not the left invariant ones, are Killing fields of left invariant metrics on Γ∖N. Aninfranil orbifold is the quotient of a nilmanifold by the action of a finite group H of affine diffeomorphisms. If the action H is free, we get aninfranil manifold.In our previous work, [5], we generalized the Cheeger-Fukaya-Gromov nilpotent Killing structure [6] and the Cheeger-Gromov F-structure, [2, 7], to collapsing orbifolds. In particular, sufficiently collapsed X can be decomposed into a union oforbits. Each orbit 𝒪p is the orbit of the action of a sheaf 𝔫 of nilpotent Lie algebras, which comes from local RIVFs on a nilmanifold fibration in the frame bundle FX. Therefore every 𝒪p is an infranil orbifold. The proof of Theorem 1.2 is based on the relation between singularities on X and singularities within an orbit 𝒪p in X, as well as the nilmanifold fibration on FX.X is calledalmost flat, if(1.3)sup⁡|KX|1/2⋅DiamX≤δn,
where DiamX is the diameter of X, δn is a small constant that depends only on n. In [8], Gromov proved that an almost flatmanifoldM has a finite, normal covering space M̃=Γ∖N that is a nilmanifold. Subsequently, Ruh [9] proved that M is diffeomorphic to Λ∖N, where Λ⊃Γ is a discrete subgroup in the affine transformation group of N. In [10], Ghanaat generalized this to an almost flat orbifold X, under the assumption that X isgood in the sense of Thurston [1], that is, X is theglobal quotient of a simply connected manifold M. There are examples of orbifolds that are not good, see [1]. In fact, without much effort, one can remove the assumption that X is good.Proposition 1.4.
IfX is an almost flat orbifold, then X is an infranil orbifold.
Precisely, there is a nilmanifoldX̃=Γ∖N, a finite group H acting on X̃ by affine diffeomorphism, so that X is diffeomorphic to H∖X̃. The order of H is bounded by cn≤(2π/0.47)n(n-1)/2. Moreover, there is a sequence of metrics gj so that Diam(X,gj)→0.The proof is almost the same as [11, 12]; the only difference is one must replace the exponential map by the develop map (see [5, 13]) and modify the definition of Gromov product in [11] accordingly.The proof of Theorem1.2 does not depend on Proposition 1.4. On the other hand, Proposition 1.4 implies Theorem 1.2 for almost flat orbifolds immediately, even without the assumption that the singularities are isolated.Remark 1.5.
Ifp∈X is an isolated singularity, then, near p, X is homeomorphic to (and in the metric sense, close to) a metric cone over a space form of dimension n-1. When n=4, the 4-1=3-dimensional space forms were first classified by Threlfall and Seifert, they used the fact that SO(4) is locally isomorphic to SO(3)×SO(3); [3, chapter 7] or [4] for details.Remark 1.6.
By the work of Anderson, Gao, Nakajima, Tian, Yang, and others, orbifolds with discrete singularities appear naturally as Gromov-Hausdorff limits of noncollapsing Einstein metrics with a uniformLn/2 curvature bound; see [14] for a recent survey. In particular, for Kähler-Einstein metrics, there is a complex structure on the limit X.


## 2. Proof of Theorem1.2

IfX is an infranil orbifold, then it is easy to obtain the bound in Theorem 1.1. Since the proof contains some ideas for the general case, we give full details.Lemma 2.1.
AssumeX is an infranil orbifold. Then |Gx|≤(2π/0.47)n(n-1)/2.Proof.
AssumeX=Λ∖N, where N is a simply connected nilpotent Lie group, Λ is a discrete group of affine diffeomorphisms on N so that X=Λ∖N is compact. If N is abelian, then X is a flat orbifold, Λ is a discrete group of isometries on N=ℝn that acts properly discontinuously. So the conclusion follows from (the proof of) Bieberbach's theorem on crystallographic groups. In fact, it is well known that the maximal rotational angle of any λ∈Λ is either 0 or at least 1/2. Thus the bound comes from a standard packing argument; notice n(n-1)/2=dimSO(n) and the bi-invariant metric on SO(n) has positive curvature.
We prove the general case by induction on dimension ofX. Remember that Λ contains a normal subgroup Γ of finite index, so that Γ is a uniform, discrete subgroup of N and X is the quotient of the Λ/Γ action on the nilmanifold X̃=Γ∖N. Clearly Gx embeds in Λ/Γ, that is, Gx={λ¯∈Λ/Γ|λ¯x̃=x̃}; here we choose a point x̃ in X̃=Γ∖N that projects to x∈Λ∖N.
LetC be the center of N, then C is connected, of positive dimension. Since any λ∈Λ is affine diffeomorphism, λ moves a C-coset in N to a C-coset. Therefore Λ/Γ acts on the nilmanifold X̃*=(Γ/(Γ∩C))∖(N/C), the quotient X* is an infranil orbifold of lower dimension. Let π:X̃→X̃* be the projection, and assume π(x̃)=x̃*. Thus we have a homomorphism
(2.1)h:Gx⟶Gx*.X̃ is a torus bundle over X̃*, the fiber is T=(Γ∩C)∖C. Assume λ¯∈Λ/Γ is in Kerh, the kernel of h, then λ¯ fixes every T fiber in X̃. If, in addition, λ¯ fixesevery point in the T fiber passing through x̃, we claim λ¯ must be identity. In fact, on N we have λ(z)=a·A(z), where a∈N and A is a Lie group automorphism of N; if λ¯ fixes every point inoneT fiber, then A is identity on the center C⊂N. This implies that λ¯ is atranslation on every T fiber. Since λ¯ is of finite order and fixes every point inoneT-fiber, λ¯ must be identity. Therefore any element λ¯∈Kerh is decided by its restriction on the T fiber passing through x; so Kerh is isomorphic to a finite group of affine diffeomorphisms on T that fixes x̃∈T, thus |Kerh| can be bounded by Bieberbach's theorem. Since
(2.2)|Gx|≤|Gx*|⋅|Ker⁡h|,
the conclusion follows by induction.In [5], the existence of nilpotent Killing structure of Cheeger-Fukaya-Gromov [6] is generalized to sufficiently collapsed orbifolds. We briefly review this construction.As in the manifold case, one can define theframe bundleFX of an orbifold X. If Br(x)⊂X is isometric to Gx∖Br(0), where Gx is a finite group acting on Br(0)⊂ℝn, then locally FX is Gx∖FB(0,r), where FB(0,r) is the orthonormal frame bundle over Br(0), and Gx acts on FB(0,r) by differential, that is, τ∈Gx moves a frame u to τ*u. Therefore FX is a manifold; strictly speaking, FX is not a fiber bundle. Let π:FX→X be the projection.Moreover, there is a naturalSO(n) action on FX; on the frames over regular points, this SO(n) action is the same one as in the manifold case; however, at (the frames over) singular points, this action isnot free. As in the work of Fukaya [15], see also [5], any Gromov-Hausdorff limit Y of a collapsing sequence FXi is a manifold. Following [6], for sufficiently collapsing orbifolds, locally we have an SO(n)-equivariant fibration(2.3)Z⟶FX⟶fY,
where the fiber Z is a nilmanifold, Y is a smooth manifold with controlled geometry.As in [6], we can put acanonical affine structure on the Z fibers, that is, a canonical way to construct a diffeomorphism from a fiber Z to the nilmanifold Γ∖N. In particular, there is a sheaf 𝔫, of a nilpotent Lie algebra of vector fields on FX. Sections of 𝔫 are local right invariant vector fields on the nilmanifold fibers Z. By integrating 𝔫, we get alocal action of a simply connected nilpotent Lie group, 𝔑, on FX. Therefore we also call a Z fiber anorbit,  and we can write Z=𝒪̃.The fibrationf:FX→Y is SO(n)-equivariant, so any Q∈SO(n) moves a Z fiber to a (perhaps another) Z fiber by affine diffeomorphism. Moreover, the SO(n) action on 𝔫 is locally trivial, that is, if A∈𝔰𝔬(n) is sufficiently small, then eA∈SO(n) moves a section, 𝔫(U), of 𝔫 on any open set U⊂FX, to itself (over U∩UeA) ( [6, Proposition 4.3]). In particular, the sheaf 𝔫 induces a sheaf, which we also denote by 𝔫, on the orbifold X away from the singular points. An orbit 𝒪̃q̃=Z on X projects down to anorbit𝒪q on X.Assumeq̃∈FX is any frame over q∈X. Let(2.4)I(q)={Q∈SO(n)∣Zq̃Q=Zq̃}
be the the isotropy group of an orbit Zq̃=𝒪q̃⊂FX. We will simply write I(q) by I. Let I0 be the identity component of I. It can be shown that, restricted on Z=𝒪̃q̃, the action of I0 is identical to the action of a torus, and the Lie algebra of this torus, I0, is in the center of 𝔫 (see [5, 6] for more details). Consider the nilmanifold(2.5)Ǒq̃=Õq̃I0.
Therefore Zq̃=𝒪̃q̃ is a torus bundle over 𝒪̌q̃. Notice, on Z=𝒪̃q̃, I moves I0 fibers to I0 fibers, thus the orbit 𝒪q is the quotient of 𝒪̌q̃ by the action of the finite group I/I0. Therefore 𝒪q is an infranil orbifold. In particular, the singularities within 𝒪q satisfy the bound in Lemma 2.1.It is important to remark that the above structure is not trivial.Lemma 2.2.
LetL be any integer. Then there is ϵ=ϵ(n,L), so that if X is an orbifold with |Gx|≤L, Vol(B1(x))≤ϵ for all x∈X, then every 𝔫-orbit 𝒪 on X is of positive dimension.Proof (sketch).
For any unit vectorA∈𝔰𝔬(n), the bound in |Gx| implies that etA does not have fixed point in 𝒪̃q̃ unless t=0 or |t|>cL-1. However, for sufficiently collapsed orbifolds, there is a vector B in the center of 𝔫 so that B generates a closed loop in 𝒪̃q̃ that is shorter than cL-1, therefore B cannot be in the Lie algebra of I0, which is in both 𝔰𝔬(n) and the center of 𝔫. Thus the orbit 𝒪̃q̃ is not contained in a single SO(n) orbit in FX, so 𝒪q is of positive dimension in X=FX/SO(n) (see [5] for more details).Proof of Theorem1.2.
Assumep∈X is an isolated singular point, p̃∈π-1(p) is in FX. Zp̃=𝒪̃p̃ is the fiber that projects to 𝒪p. Let I(p), I0, 𝒪̌q̃ be as above. Let
(2.6)Kp={Q∈SO(n)∣p̃Q=p̃}.
Thus Kp is a subgroup of I, and |Kp|=|Gp|. Let
(2.7)Kp-={Q∈Kp∣(p̃′I0)Q=(p̃′I0)∀p̃′∈Zp̃}.
Thus Kp- is a normal subgroup of Kp.
Lemma 2.3.
IfQ∈Kp fixes every point in Zp̃, then Q is the identity in SO(n).
Proof.
PotentiallyQ may fix every point in Zp̃ while moving some points of FX that are outside Zp̃. We will rule out this possibility.
By assumption,p is an isolated singularity. For any Q that is not identity, the connected component of the fixed point set of Q that passes through p̃ must project to p under π:FX→X, because away from π-1(p) the SO(n) action is free. Therefore π(Zp̃)=p is a single point in X, and this contradicts the fact that the 𝔫-orbits on X are of positive dimension; see Lemma 2.2.
In particular, we have afaithful representation of Kp in the affine group of Zp̃, that is, we can identify Kp with therestricted action of the group KponZp̃.Take any Q∈Kp- that is not identity in SO(n). By definition Q fixes p̃. If Q fixesevery point in p̃I0, as in Lemma 2.1, Q is a translation oneveryI0 fiber; because Q is of finite order and Q moves every I0 fiber to itself, Q necessarily fixes every point in Z; thus Q is identity. So Q rotates the tangent plane of p̃I0 at p̃. Therefore Kp- is isomorphic to a finite group of affine diffeomorphisms on the torus p̃I0. By the Bieberbach theorem,
(2.8)|Kp-|≤(2π0.47)k(k-1)/2,k=dim⁡I0.
Recall that Bieberbach's theorem implies that all finite subgroups of SL(n,ℤ) have a uniform upper bound in order.We have
(2.9)Op=H(Zp̃/I0),
where H=I/I0 is a finite group. Let Hp be the subgroup of H that fixes p. Now we get an embedding(2.10)KpKp-⊂Hp.
By Lemma 2.1,
(2.11)|Hp|≤(2π0.47)i(i-1)/2,i=dim⁡Op.
Thus
(2.12)|Gp|=|Kp|=|KpKp-|⋅|Kp-|≤(2π0.47)n(n-1).



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