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ISRN Geometry
VolumeΒ 2011Β (2011), Article IDΒ 502814, 7 pages
http://dx.doi.org/10.5402/2011/502814
Research Article

A Restriction for Singularities on Collapsing Orbifolds

Department of Mathematics and Statistics, California State University, Long Beach, CA 90840, USA

Received 8 August 2011; Accepted 5 September 2011

Academic Editors: S.Β Kar, U.Β LindstrΓΆm, and E. H.Β Saidi

Copyright Β© 2011 Yu Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Every point 𝑝 in an orbifold 𝑋 has a neighborhood that is homeomorphic to πΊπ‘β§΅π΅π‘Ÿ(0), where 𝐺𝑝 is a finite group acting on π΅π‘Ÿ(0)βŠ‚β„π‘›, 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 π‘₯βˆˆπ‘‹, the volume of 𝐡1(π‘₯), with respect to the metric 𝑔𝑖, goes to 0 as π‘–β†’βˆž. For such 𝑋, we prove that |𝐺𝑝|≀(2πœ‹/0.47)𝑛(π‘›βˆ’1) for all singularities π‘βˆˆπ‘‹.

1. Introduction

An 𝑛-dimensional Riemannian orbifold, 𝑋, is a metric space so that the following is true: for any π‘₯βˆˆπ‘‹, there exists π‘Ÿ=π‘Ÿ(π‘₯)>0 and a Riemannian metric ̃𝑔π‘₯ on 𝐡2π‘Ÿ(0)βŠ‚β„π‘›, a finite group 𝐺π‘₯ (the isotropy group) acting on (π΅π‘Ÿ(0),̃𝑔π‘₯) by isometries, so that 𝐺π‘₯(0)=0, and there is an isometry πœ„π‘₯βˆΆπ΅π‘Ÿ(π‘₯)→𝐺π‘₯β§΅π΅π‘Ÿ(0) with πœ„π‘₯(π‘₯)=0 (see [1]). We call π‘₯βˆˆπ‘‹ a regular point if |𝐺π‘₯|=1; otherwise, π‘₯ is a singular point. We say the curvature of 𝑋 satisfies||𝐾𝑋||β‰€πœ…2,(1.1) if the sectional curvature 𝐾 of every (π΅π‘Ÿ(0),̃𝑔π‘₯) above satisfies |𝐾|β‰€πœ…2. We say 𝑋 is collapsing, if 𝑋 admits a sequence of metrics, 𝑔𝑖, with uniformly bounded curvature, so that, for any π‘₯βˆˆπ‘‹,limπ‘–β†’βˆžVol𝑔𝑖𝐡1ξ€Έ(π‘₯)=0.(1.2)

As an example, consider the standard β„€π‘š=β„€/π‘šβ„€ action on the sphere 𝑆2.

The quotient orbifold π‘‹π‘š=β„€π‘šβ§΅π‘†2 will be arbitrarily collapsed when π‘šβ†’βˆž (see Figure 1). However, for any fixed π‘š, π‘‹π‘š 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 π‘‹π‘š, there is a neighborhood that is isometric to β„€π‘šβ§΅β„2, where ℝ2 is equipped with some β„€π‘š invariant metric. Therefore if 𝑔𝑖 is a collapsing sequence of metrics on π‘‹π‘š, we get a corresponding sequence ̃𝑔𝑖 of pullback metrics on 𝑆2; every ̃𝑔𝑖 is smooth. Observe Vol(𝑆2,̃𝑔𝑖)=π‘šVol(π‘‹π‘š,𝑔𝑖), where π‘š is fixed and limπ‘–β†’βˆžVol(π‘‹π‘š,𝑔𝑖)=0, thus limπ‘–β†’βˆžVol(𝑆2,̃𝑔𝑖)=0. If the diameter of (π‘‹π‘š,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 πœ’(𝑆2) vanishesβ€”this is a contradiction since clearly πœ’(𝑆2)=2.

502814.fig.001
Figure 1

On 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 group 𝐺π‘₯ has order 2, a quite small number.

Intuitively, these examples suggest that when an orbifold is collapsing, a conelike singularity cannot be too β€œsharp,” that is, there should be some bound in |𝐺π‘₯|. The main result of this paper is as follows.

Theorem 1.1. Assume 𝑋 is a compact, collapsing orbifold, π‘βˆˆπ‘‹ is an isolated singularity. Then |𝐺𝑝|≀(2πœ‹/0.47)𝑛(π‘›βˆ’1).

If 𝑋 has an isolated singularity 𝑝, then the dimension of 𝑋 must be even, and πΊπ‘βŠ‚π‘†π‘‚(𝑛). Theorem 1.1 fails if we drop the requirement that π‘₯ is an isolated singularity; for example, we can take any orbifold π‘‹ξ…ž and let 𝑋=𝑋×𝑆1; by shrinking the 𝑆1 factor, we see 𝑋 is collapsing while there is no restriction on singularities of π‘‹ξ…ž. The bound |𝐺𝑝|≀(2πœ‹/0.47)𝑛(π‘›βˆ’1) has its root in the Bieberbach theorem of crystallographic groups and Gromov's almost flat manifold theorem.

Clearly, Theorem 1.1 is a corollary of the following.

Theorem 1.2. For any 𝐿>0, there is πœ–=πœ–(𝑛,𝐿) so that if 𝑋 is an orbifold with all singularities π‘žβˆˆπ‘‹ satisfying |πΊπ‘ž|<𝐿, Vol(𝐡1(π‘ž))<πœ–, then |G𝑝|≀(2πœ‹/0.47)𝑛(π‘›βˆ’1) for any isolated singularity π‘βˆˆπ‘‹.

Remark 1.3. The bound |𝐺𝑝|≀(2πœ‹/0.47)𝑛(π‘›βˆ’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.

A nilmanifold, Γ⧡𝑁, is the quotient of the (left) action of a discrete, uniform subgroup Ξ“βŠ‚π‘, 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. An infranil 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 an infranil 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 of orbits. Each orbit π’ͺ𝑝 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 π’ͺ𝑝 is an infranil orbifold. The proof of Theorem 1.2 is based on the relation between singularities on 𝑋 and singularities within an orbit π’ͺ𝑝 in 𝑋, as well as the nilmanifold fibration on 𝐹𝑋.

𝑋 is called almost flat, if||𝐾sup𝑋||1/2β‹…Diam𝑋≀𝛿𝑛,(1.3) where Diam𝑋 is the diameter of 𝑋, 𝛿𝑛 is a small constant that depends only on 𝑛. In [8], Gromov proved that an almost flat manifold 𝑀 has a finite, normal covering space 𝑀=Γ⧡𝑁 that is a nilmanifold. Subsequently, Ruh [9] proved that 𝑀 is diffeomorphic to Λ⧡𝑁, where Ξ›βŠƒΞ“ 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 is good in the sense of Thurston [1], that is, 𝑋 is the global 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. If 𝑋 is an almost flat orbifold, then 𝑋 is an infranil orbifold.
Precisely, there is a nilmanifold 𝑋=Γ⧡𝑁, a finite group 𝐻 acting on 𝑋 by affine diffeomorphism, so that 𝑋 is diffeomorphic to 𝑋𝐻⧡. The order of 𝐻 is bounded by 𝑐𝑛≀(2πœ‹/0.47)𝑛(π‘›βˆ’1)/2. Moreover, there is a sequence of metrics 𝑔𝑗 so that Diam(𝑋,𝑔𝑗)β†’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 Theorem 1.2 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. If π‘βˆˆπ‘‹ 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 π‘›βˆ’1. When 𝑛=4, the 4βˆ’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 uniform 𝐿𝑛/2 curvature bound; see [14] for a recent survey. In particular, for KΓ€hler-Einstein metrics, there is a complex structure on the limit 𝑋.

2. Proof of Theorem 1.2

If 𝑋 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. Assume 𝑋 is an infranil orbifold. Then |𝐺π‘₯|≀(2πœ‹/0.47)𝑛(π‘›βˆ’1)/2.

Proof. Assume 𝑋=Λ⧡𝑁, where 𝑁 is a simply connected nilpotent Lie group, Ξ› 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, Ξ› 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 πœ†βˆˆΞ› is either 0 or at least 1/2. Thus the bound comes from a standard packing argument; notice 𝑛(π‘›βˆ’1)/2=dim𝑆𝑂(𝑛) and the bi-invariant metric on 𝑆𝑂(𝑛) has positive curvature.
We prove the general case by induction on dimension of 𝑋. Remember that Ξ› contains a normal subgroup Ξ“ of finite index, so that Ξ“ is a uniform, discrete subgroup of 𝑁 and 𝑋 is the quotient of the Ξ›/Ξ“ action on the nilmanifold 𝑋=Γ⧡𝑁. Clearly 𝐺π‘₯ embeds in Ξ›/Ξ“, that is, 𝐺π‘₯={πœ†βˆˆΞ›/Ξ“|πœ†Μƒπ‘₯=Μƒπ‘₯}; here we choose a point Μƒπ‘₯ in 𝑋=Γ⧡𝑁 that projects to π‘₯βˆˆΞ›β§΅π‘.
Let 𝐢 be the center of 𝑁, then 𝐢 is connected, of positive dimension. Since any πœ†βˆˆΞ› is affine diffeomorphism, πœ† moves a 𝐢-coset in 𝑁 to a 𝐢-coset. Therefore Ξ›/Ξ“ acts on the nilmanifold ξ‚π‘‹βˆ—=(Ξ“/(Ξ“βˆ©πΆ))⧡(𝑁/𝐢), the quotient π‘‹βˆ— is an infranil orbifold of lower dimension. Let ξ‚ξ‚π‘‹πœ‹βˆΆπ‘‹β†’βˆ— be the projection, and assume πœ‹(Μƒπ‘₯)=Μƒπ‘₯βˆ—. Thus we have a homomorphism β„ŽβˆΆπΊπ‘₯⟢𝐺π‘₯βˆ—.(2.1)𝑋 is a torus bundle over ξ‚π‘‹βˆ—, the fiber is 𝑇=(Ξ“βˆ©πΆ)⧡𝐢. Assume πœ†βˆˆΞ›/Ξ“ is in Kerβ„Ž, the kernel of β„Ž, then πœ† fixes every 𝑇 fiber in 𝑋. If, in addition, πœ† fixes every point in the 𝑇 fiber passing through Μƒπ‘₯, we claim πœ† must be identity. In fact, on 𝑁 we have πœ†(𝑧)=π‘Žβ‹…π΄(𝑧), where π‘Žβˆˆπ‘ and 𝐴 is a Lie group automorphism of 𝑁; if πœ† fixes every point in one 𝑇 fiber, then 𝐴 is identity on the center πΆβŠ‚π‘. This implies that πœ† is a translation on every 𝑇 fiber. Since πœ† is of finite order and fixes every point in one 𝑇-fiber, πœ† must be identity. Therefore any element πœ†βˆˆKerβ„Ž is decided by its restriction on the 𝑇 fiber passing through π‘₯; so Kerβ„Ž is isomorphic to a finite group of affine diffeomorphisms on 𝑇 that fixes Μƒπ‘₯βˆˆπ‘‡, thus |Kerβ„Ž| can be bounded by Bieberbach's theorem. Since ||𝐺π‘₯||≀||𝐺π‘₯βˆ—||β‹…||||Kerβ„Ž,(2.2) 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 the frame bundle 𝐹𝑋 of an orbifold 𝑋. If π΅π‘Ÿ(π‘₯)βŠ‚π‘‹ is isometric to 𝐺π‘₯β§΅π΅π‘Ÿ(0), where 𝐺π‘₯ is a finite group acting on π΅π‘Ÿ(0)βŠ‚β„π‘›, then locally 𝐹𝑋 is 𝐺π‘₯⧡𝐹𝐡(0,π‘Ÿ), where 𝐹𝐡(0,π‘Ÿ) is the orthonormal frame bundle over π΅π‘Ÿ(0), and 𝐺π‘₯ acts on 𝐹𝐡(0,π‘Ÿ) by differential, that is, 𝜏∈𝐺π‘₯ moves a frame 𝑒 to πœβˆ—π‘’. Therefore 𝐹𝑋 is a manifold; strictly speaking, 𝐹𝑋 is not a fiber bundle. Let πœ‹βˆΆπΉπ‘‹β†’π‘‹ be the projection.

Moreover, there is a natural 𝑆𝑂(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 is not free. As in the work of Fukaya [15], see also [5], any Gromov-Hausdorff limit π‘Œ of a collapsing sequence 𝐹𝑋𝑖 is a manifold. Following [6], for sufficiently collapsing orbifolds, locally we have an 𝑆𝑂(𝑛)-equivariant fibrationπ‘βŸΆπΉπ‘‹π‘“βŸΆπ‘Œ,(2.3) where the fiber 𝑍 is a nilmanifold, π‘Œ is a smooth manifold with controlled geometry.

As in [6], we can put a canonical 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 a local action of a simply connected nilpotent Lie group, 𝔑, on 𝐹𝑋. Therefore we also call a 𝑍 fiber an orbit, and we can write π’ͺ𝑍=.

The fibration π‘“βˆΆπΉπ‘‹β†’π‘Œ is 𝑆𝑂(𝑛)-equivariant, so any π‘„βˆˆπ‘†π‘‚(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 π΄βˆˆπ”°π”¬(𝑛) is sufficiently small, then π‘’π΄βˆˆπ‘†π‘‚(𝑛) moves a section, 𝔫(π‘ˆ), of 𝔫 on any open set π‘ˆβŠ‚πΉπ‘‹, to itself (over π‘ˆβˆ©π‘ˆπ‘’π΄) ( [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 π’ͺΜƒπ‘ž=𝑍 on 𝑋 projects down to an orbit π’ͺπ‘ž on 𝑋.

Assume Μƒπ‘žβˆˆπΉπ‘‹ is any frame over π‘žβˆˆπ‘‹. Let𝐼(π‘ž)=π‘„βˆˆπ‘†π‘‚(𝑛)βˆ£π‘Μƒπ‘žπ‘„=π‘Μƒπ‘žξ€Ύ(2.4) be the the isotropy group of an orbit π‘Μƒπ‘ž=π’ͺΜƒπ‘žβŠ‚πΉπ‘‹. We will simply write 𝐼(π‘ž) by 𝐼. Let 𝐼0 be the identity component of 𝐼. It can be shown that, restricted on π’ͺ𝑍=Μƒπ‘ž, 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π’ͺΜƒπ‘ž=π’ͺΜƒπ‘žπΌ0.(2.5) Therefore π‘Μƒπ‘ž=π’ͺΜƒπ‘ž is a torus bundle over π’ͺΜƒπ‘ž. Notice, on π’ͺ𝑍=Μƒπ‘ž, 𝐼 moves 𝐼0 fibers to 𝐼0 fibers, thus the orbit π’ͺπ‘ž is the quotient of π’ͺΜƒπ‘ž by the action of the finite group 𝐼/𝐼0. Therefore π’ͺπ‘ž is an infranil orbifold. In particular, the singularities within π’ͺπ‘ž satisfy the bound in Lemma 2.1.

It is important to remark that the above structure is not trivial.

Lemma 2.2. Let 𝐿 be any integer. Then there is πœ–=πœ–(𝑛,𝐿), so that if 𝑋 is an orbifold with |𝐺π‘₯|≀𝐿, Vol(𝐡1(π‘₯))β‰€πœ– for all π‘₯βˆˆπ‘‹, then every 𝔫-orbit π’ͺ on 𝑋 is of positive dimension.

Proof (sketch). For any unit vector π΄βˆˆπ”°π”¬(𝑛), the bound in |𝐺π‘₯| implies that 𝑒𝑑𝐴 does not have fixed point in π’ͺΜƒπ‘ž unless 𝑑=0 or |𝑑|>π‘πΏβˆ’1. However, for sufficiently collapsed orbifolds, there is a vector 𝐡 in the center of 𝔫 so that 𝐡 generates a closed loop in π’ͺΜƒπ‘ž that is shorter than π‘πΏβˆ’1, therefore 𝐡 cannot be in the Lie algebra of 𝐼0, which is in both 𝔰𝔬(𝑛) and the center of 𝔫. Thus the orbit π’ͺΜƒπ‘ž is not contained in a single 𝑆𝑂(𝑛) orbit in 𝐹𝑋, so π’ͺπ‘ž is of positive dimension in 𝑋=𝐹𝑋/𝑆𝑂(𝑛) (see [5] for more details).

Proof of Theorem 1.2. Assume π‘βˆˆπ‘‹ is an isolated singular point, Μƒπ‘βˆˆπœ‹βˆ’1(𝑝) is in 𝐹𝑋. 𝑍̃𝑝=π’ͺ̃𝑝 is the fiber that projects to π’ͺ𝑝. Let 𝐼(𝑝), 𝐼0, π’ͺΜƒπ‘ž be as above. Let 𝐾𝑝={π‘„βˆˆπ‘†π‘‚(𝑛)βˆ£Μƒπ‘π‘„=̃𝑝}.(2.6) Thus 𝐾𝑝 is a subgroup of 𝐼, and |𝐾𝑝|=|𝐺𝑝|. Let πΎβˆ’π‘=ξ€½π‘„βˆˆπΎπ‘βˆ£ξ€·Μƒπ‘ξ…žπΌ0𝑄=Μƒπ‘ξ…žπΌ0ξ€Έβˆ€Μƒπ‘ξ…žβˆˆπ‘Μƒπ‘ξ€Ύ.(2.7) Thus πΎβˆ’π‘ is a normal subgroup of 𝐾𝑝.
Lemma 2.3. If π‘„βˆˆπΎπ‘ fixes every point in 𝑍̃𝑝, then 𝑄 is the identity in 𝑆𝑂(𝑛).
Proof. Potentially 𝑄 may fix every point in 𝑍̃𝑝 while moving some points of 𝐹𝑋 that are outside 𝑍̃𝑝. 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 ̃𝑝 must project to 𝑝 under πœ‹βˆΆπΉπ‘‹β†’π‘‹, because away from πœ‹βˆ’1(𝑝) the 𝑆𝑂(𝑛) action is free. Therefore πœ‹(𝑍̃𝑝)=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 a faithful representation of 𝐾𝑝 in the affine group of 𝑍̃𝑝, that is, we can identify 𝐾𝑝 with the restricted action of the group 𝐾𝑝 on 𝑍̃𝑝.Take any π‘„βˆˆπΎβˆ’π‘ that is not identity in 𝑆𝑂(𝑛). By definition 𝑄 fixes ̃𝑝. If 𝑄 fixes every point in ̃𝑝𝐼0, as in Lemma 2.1, 𝑄 is a translation on every 𝐼0 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 ̃𝑝𝐼0 at ̃𝑝. Therefore πΎβˆ’π‘ is isomorphic to a finite group of affine diffeomorphisms on the torus ̃𝑝𝐼0. By the Bieberbach theorem, ||πΎβˆ’π‘||≀2πœ‹ξ‚0.47π‘˜(π‘˜βˆ’1)/2,π‘˜=dim𝐼0.(2.8) Recall that Bieberbach's theorem implies that all finite subgroups of 𝑆𝐿(𝑛,β„€) have a uniform upper bound in order.We have π’ͺ𝑝=𝐻𝑍̃𝑝/𝐼0ξ€Έ,(2.9) where 𝐻=𝐼/𝐼0 is a finite group. Let 𝐻𝑝 be the subgroup of 𝐻 that fixes 𝑝. Now we get an embeddingπΎπ‘πΎβˆ’π‘βŠ‚π»π‘.(2.10) By Lemma 2.1, ||𝐻𝑝||≀2πœ‹ξ‚0.47𝑖(π‘–βˆ’1)/2,𝑖=dimπ’ͺ𝑝.(2.11) Thus ||𝐺𝑝||=||𝐾𝑝||=||||πΎπ‘πΎβˆ’π‘||||β‹…||πΎβˆ’π‘||≀2πœ‹ξ‚0.47𝑛(π‘›βˆ’1).(2.12)

Acknowledgment

The author is grateful to Professor Tian for very helpful suggestions.

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