Convergence of isometries, with semicontinuity of symmetry of Alexandrov spaces

0.1 MB·English

by  John Harvey

CONVERGENCE OF ISOMETRIES, WITH SEMICONTINUITY
OF SYMMETRY OF ALEXANDROV SPACES
5
1
0 JOHNHARVEY
2

n ABSTRACT. TheequivariantGromov–Hausdorffconvergenceofmetric
a
spaces is studied. Where all isometry groups under consideration are
J
compact Lie, it is shown that an upper bound on the dimension of the
6
groupguaranteesthatthe convergenceis by Lie homomorphisms. Ad-
1
ditionallowerboundsoncurvatureandvolumestrengthenthisresultto

] convergencebymonomorphisms,so thatsymmetriescanonlyincrease
G
onpassingtothelimit.
M
.
h
t
a 1. INTRODUCTION
m
The equivariant Gromov–Hausdorff topology allows one to study the
[

convergenceofmetricspaces whilekeeping track oftheirsymmetries. The
1
v definition of this topology involves functions between the metric spaces
8 and between the groups of isometries. These functions are not required to
4
becontinuous,ortobegrouphomomorphisms.
9
3 The main theorem of this article is that, assuming that all groups under
0
considerationarecompactLiegroups,convergenceisalwaysbyhomomor-
.
1 phismsofLiegroups.
0
5
Main Theorem. Let (X ,p ,G ) be a sequence of pointed group metric
1 i i i
v: spaces, converging to (Xω,pω,Gω) in the equivariant pointed Gromov–
Hausdorfftopology. Assumethat G and G are compact Lie groups,with a
i i
X
uniformupper boundon thedimensionoftheG . Then, for largeenoughi,
i
r
a thefunctionsGi Gω which demonstratetheconvergence maybe chosen
→
tobehomomorphismsofLiegroups.
Note that this result does not require any assumptions about the metric
spacesthemselves(otherthanthatmetricballscenteredatthedistinguished
pointsberelativelycompact,whichisrequiredtodefinethetopology). The
onlyassumptionis onthegroupsthemselves.
ThefunctionsG G arelesstractable. Asimpleexamplesuchas the
ω i
→
convergenceofthecyclicgroupacting onthecircletotheaction ofthefull
Date:January19,2015.
2010MathematicsSubjectClassification. Primary:53C23;Secondary:53C20.
TheauthorissupportedbytheSFB“Groups,GeometryandActions”.
1
2 JOHNHARVEY
circle, (S1,Z ) (S1,S1) as p , shows that homomorphisms need
p
→ → ∞
notexistin thereversedirection.
Where the homomorphisms have a non-trivial kernel, some symmetries
are lost. It is easy to find examples where injectivity fails, either by using
spaces with infinite topology (such as Hawaiian earrings) or by shrinking
theorbitsofthegroupactionsothespacesconvergetotheorbitspacewith
atrivialgroupaction.
If thespaces X are Riemannianmanifoldswith alowersectionalcurva-
i
ture and volume bound, or, more generally, Alexandrov spaces, then it is
shownthatthehomomorphismsarealways injective(Proposition4.1).
It would be of interest to see whether similar results continue to hold
with weaker hypotheses, for example, using non-compact Lie groups, or
onlyconsideringa Ricci curvaturelowerbound.
Finally, in Proposition 4.5, the injectivity result is applied to achieve an
understanding of how much symmetry a closed Riemannian manifold can
retainwhenitsvolumeislargerelativetoitssectionalcurvatureandradius.
2. EQUIVARIANT POINTED GROMOV–HAUSDORFF TOPOLOGY
TheequivariantGromov–Hausdorfftopologyisacoarsemethodofdefin-
ingtheconvergenceofasequenceofmetricspaceswithisometricgroupac-
tions. Ithasbeenusedinthestudyoffundamentalgroupsofcertainclasses
of manifolds [5, 3] and in the study of sequences of Riemannian orbifolds
[4, 12]. A slightly different definition, developed independently by Bestv-
ina [1] and Paulin [13], is used in geometricgroup theory. In the setting of
geometric group theory, the group is generally fixed. However, the defini-
tion used in the current work is designed for situations where the object of
studyis notthegroupitself,butratherthesymmetriesofthespace.
ItisageneralizationofthestandardGromov–Hausdorfftopology(infact,
ametric)onthesetofisometryclassesofcompactmetricspaces[7],which
itself generalizes the Hausdorff metric on the closed subsets of a compact
metricspace.
Let be the set of all isometry classes of pointed metric spaces (X,p)
M
(that is, X is a metric space and p is a distinguished point of X) such that
foreach r > 0 theopen ballB(p,r)is relativelycompact.
Now consider the set of triples (X,p,Γ) where (X,p) and Γ is
∈ M
a closed group of isometries acting effectively on X. Such triples will be
referred to as pointed group metric spaces. Say that two pointed group
metric spaces are equivalent if they are equivariantly pointed isometric up
toanautomorphismofthegroup. Let bethesetofequivalenceclasses
eq
M
ofpointedgroup metricspaces.
CONVERGENCEOFISOMETRIES 3
TheequivariantpointedGromov–Hausdorffdistancewasfirstdefinedby
Fukaya [4], and achieved its final form some years later in his work with
Yamaguchi[5].
If(X,p,Γ) , thenlet Γ(r) Γ be γ Γ γp B(p,r) .
eq
∈ M ⊂ { ∈ | ∈ }
Definition 2.1. Let (X,p,Γ),(Y,q,Λ) . An equivariant pointed
eq
∈ M
Gromov–Hausdorffǫ–approximationisatriple(f,φ,ψ)offunctions
f: B(p,1/ǫ) Y
→
φ: Γ(1/ǫ) Λ(1/ǫ)
→
ψ: Λ(1/ǫ) Γ(1/ǫ)
→
suchthat
(1) f(p) = q;
(2) theǫ–neighborhoodofB(p,1/ǫ)containsB(q,1/ǫ);
(3) ifx,y B(p,1/ǫ)then dist(x,y) dist(f(x),f(y)) < ǫ;
∈ | − |
(4) ifγ Γ(1/ǫ), and bothx,γx B(p,1/ǫ),then
∈ ∈
dist(f(γx),φ(γ)f(x)) < ǫ;
(5) ifλ Λ(1/ǫ),and bothx,ψ(λ)x B(p,1/ǫ), then
∈ ∈
dist(f(ψ(λ)x),λf(x)) < ǫ.
Note that these functions need not be morphisms from the relevant cate-
gory. In particular, they need not becontinuous,or respect the group struc-
turein anyway.
TheequivariantpointedGromov–Hausdorffdistanceisdefinedfromthese
approximationsbysettingd ((X,p,Γ),(Y,q,Λ))equaltotheinfinumof
epH
allǫsuchthatequivariantpointedGromov–Hausdorffǫ–approximationsex-
istfrom(X,p,Γ) to(Y,q,Λ)and from(Y,q,Λ)to(X,p,Γ).
By [5, Proposition 3.6], given a sequence in , if the sequence of
eq
M
underlying pointed metric spaces is convergent in the pointed Gromov–
Hausdorfftopology,then there is a subsequencewhich is convergent in the
equivariantpointedGromov–Hausdorfftopology.
By [4, Theorem 2.1], the sequence of orbit spaces corresponding to a
convergent sequence in must itself converge in the usual Gromov–
eq
M
Hausdorfftopology.
3. APPROXIMATING SYMMETRIES WITH GROUP HOMOMORPHISMS
Thissectioncontainstheproofofthefollowingtheorem.
Theorem3.1. Let(X ,p ,G )beasequenceofpointedgroupmetricspaces
i i i
in , converging to (X ,p ,G ) in the equivariant pointed Gromov–
eq ω ω ω
M
Hausdorff topology. Assume that G and G are compact Lie groups, with
i ω
4 JOHNHARVEY
auniformupperboundonthedimensionoftheG . Then,forlargeenoughi,
i
thefunctionsG G which demonstratetheconvergence maybe chosen
i ω
→
tobehomomorphismsofLiegroups.
The proof of this result relies on the center of mass construction from
Grove–Petersen [9], which allows for the construction of continuous maps
fromdiscreteones. Letusbeginthissectionbyreviewingthatconstruction.
The Riemannian manifolds in question will be the compact Lie groups G
i
andG withbi-invariantmetrics.
ω
Let(M,g)beacompleteRiemannianmanifold,withdimM = n,secg
≥
k,vol(M,g) v and diam(M,g) D.
≥ ≤
Aminimalµ–netforM isdefinedtobeasetofpointsinM suchthatthe
µ–ballscoverall ofM butthe µ–ballsaredisjoint.
2
It is shown in [9] that certain constants r,R > 0 and N N exist which
∈
depend only on n,k,v and D, but not on the manifold M itself, so that the
followinghold:
(1) For any minimal µ–net, a ball of radius µ will have non-empty in-
tersection with at most N of the µ–balls centered on the members
oftheµ–net. N dependsonlyon n,k andD.
(2) Let p ,...p M, and let λ ,...,λ > 0 be weights, so that
1 m 1 m
∈
Σλ = 1. Letη < r(1+R+ +Rm 1) 1. Ifdist(p ,p ) < η,i,j =
i − − i j
···
1,...,m, then a center of mass (p ,...p ,λ ,...,λ ) is defined
1 m 1 m
C
which depends continuously on the p and the λ , is unchanged on
i i
droppingany pointwith weight 0, and satisfies dist( ,p ) < η(1+
i
C
R+ +Rm) foreach i.
···
Thebi-invariantmetriconaLiegroup,however,hasnoplaceinthedefi-
nition of equivariant Gromov–Hausdorffconvergence. The natural geome-
tryon thegroup ofisometriesderivesfromhowit actson themetricspace.
Definition 3.2. Let a compact Lie group G act on a pointed metric space
(X,p) . Then for each R > 0 the action pseudoseminormon G is the
∈ M
continuousmap : G(R) [0, )givenby
k·kR → ∞
g = sup dist(x,gx) x,gx B(p,R) .
k kR { | ∈ }
This is well defined, since for elements of the balls centered at p are
M
relatively compact. g = 0 for any isometry g which fixes the ball of
k kR
radius R. However, the compactness of G implies that, once R is large
enough, the pseudoseminorm vanishes only at the identity. It is also clear
that g = g 1
k kR k − kR
The inequality gh g h need not be satisfied, since this
k kR ≤ k kRk kR
supremummightbeachievedforsomex B(p,R)suchthathx / B(p,R).
∈ ∈
However, if X is compact, then once X B(p,R) this inequality is satis-
⊂
fied.
CONVERGENCEOFISOMETRIES 5
Theactionpseudoseminormcanbeusedtodefinearight-invariantaction
pseudosemimetricon G by d (g,h) = gh 1 . (Of course, the construc-
R k − kR
tioncould alsobecarried outleft-invariantly.) Theprecedingcommentson
the pseudoseminorm easily imply that d is non-negative and symmetric.
R
Once R is sufficiently large, d (g,h) = 0 = g = h and d becomes a
R R
⇒
semimetric. IfX is compact, d will also satisfythe triangleinequalityfor
R
largeR, and becomeatruemetric, asin theauthor’searlier work[11].
ProofofTheorem 3.1. Assume, by passing to a subsequence if necessary,
that d ((X ,p ,G ),(X ,p ,G )) < 1/i. Note that for large enough i,
epH i i i ω ω ω
G (i) = G .
i i
Choosebi-invariantRiemannianmetricsσ oneachG ,fori = 1,2,...,ω
i i
so that there is a uniform upper bound on the diameter of the groups. Let
d betheinduceddistancefunctions.
σi
Let d be the action pseudosemimetric on G with respect to B(p ,i),
i i i
and let dω be the action pseudosemimetric on G with respect to B(p ,i).
i ω ω
Assumethat i is so large that dω is a semimetric, but note that there can be
i
nosimilarguaranteeford .
i
Consider a sequence of triples f : B(p ,i) B(p ,i), φ : G G ,
i ω i i ω i
→ →
ψ : G G ,equivariantpointedGromov–Hausdorff1/i–approximations
i i ω
→
demonstratingtheconvergence.
Lemma 3.3. Thefunctionsψ maybechosentobecontinuous.
i
Proof. For each i N, let ν > 0 be such that d (g,h) < 2ν =
∈ i σi i ⇒
d (g,h) < 1/i. Let A beaminimalν –netin (G ,σ ).
i i i i i
Let η > 0 converge to 0, but let each η be large enough that dω(g,h) <
i i i
4/i = d (g,h) < η . This is possible because dω is increasing with
⇒ σ i i
respect toi. Chooseasequenceofminimalη –nets B (G ,σ ).
i i ω ω
⊂
BytheupperboundonthedimensionofG ,andsincek = 0isauniform
i
lowerboundonthecurvature,andD = 1isanupperboundonthediameter,
there is some N such that for large enough i, a ball of radius ν in G non-
i i
triviallyintersectsonlyN oftheν –ballscentered at elementsofA .
i i
Let r,R be the constants from the center of mass construction [9] which
are appropriate for (G ,σ ). Write K = 1+R+ +RN. Note that for
ω ω
···
largeenoughi, 3η < r/K.
i
Define a map α: A B by mapping p A to an element of B
i i i i
→ ∈
nearest (in the σ metric) to ψ (p). If, for some p,q A , d (p,q) <
ω i ∈ i σi
2ν , then d (α(p),α(q)) < 3η . There is an induced map between the
i σω i
Euclidean spaces RAi RBi, where the coordinate associated to any q
→ ∈
B isobtainedby summingtheco-ordinates foreach elementofα 1(q).
i −
Then a continuous map ψ˜ : G G may be defined by composing
i i ω
mapsG RAi RBi G . →
i ω
→ → →
6 JOHNHARVEY
Let A = (cid:8)p1,...,pℓ(cid:9) and choose smooth functions fj: G [0, ),
i i i i i → ∞
each having theirsupportin theball ofradius ν around pj whichsum to 1.
i i
The map (G,ρ ) RAi is given by g (cid:0)fj(g)(cid:1)l . Note that points in
i → 7→ i j=1
the image of this map have at most N non-zero coordinates. It is possible
to assume that fj(pk) = 0 whenever j = k. Let us make this assumption,
i i 6
so that elements of A are mapped to points with only one non-zero co-
i
ordinate.
The map from RAi RBi is that induced by α, and the map from
RBi (G ,ρ ) is giv→en by the center of mass construction. Note that
ω ω
→
in the domain points have at most N non-zero coordinates, and the corre-
spondingelementsofB areat pairwisedistanceat most3η ,sothismapis
i i
defined once i is so large that 3η < r/K. The set A is mapped to B by
i i i
ψ˜.
i
To complete the proof, it is necessary to verify that ψ˜ will serve as part
i
oftheequivariantpointedGromov–Hausdorffapproximation.
Let p1,...,pm be those elements of A within ν of g in the σ metric.
i i i i i
Their images ψ˜(p1),...,ψ˜(pm) B are then at most 4η from ψ (g) in
i i i i ∈ i i i
the σ metric. The point ψ˜(g) is obtained from the elements of B via
ω i i
the center of mass construction, and so is at most 3η K from those points
i
withnon-zerocoordinates. Thisgivesaglobalboundofη (3K +4)forthe
i
differencebetween ψ and ψ˜ in theσ metric.
i i ω
Now for each j N, consider dω(ψ (g),ψ˜(g)). By continuity of the
∈ j i i
semimetricdω, forlargeenough i,
j
d (ψ (g),ψ˜(g)) < η (3K +4) = dω(ψ (g),ψ˜(g)) < 1/j.
σω i i i ⇒ j i i
Then the triplef : B(p ,j/2) B(p ,j/2), φ : G G , ψ˜ : G
i ω i i ω i i i
→ → →
G isanequivariantpointedGromov–Hausdorff(2/j)–approximation. (cid:3)
ω
Returning to the proof of the theorem, by the monotonicity of the ac-
tionsemimetricsonG ,andtheircontinuitywithrespecttothebi-invariant
ω
metric, it is clear that for large i the (now assumed to be continuous) map
ψ will be an almost homomorphism in the sense of Grove–Karcher–Ruh
i
[8]. That is to say, for each g,h G , d (ψ (gh)ψ (h) 1,ψ (g)) q for
∈ i σω i i − i ≤
a fixed small q. By [8, Theorem 4.3], there is then a continuous group ho-
momorphism within 1.36q of ψ , and again by continuity of dω, for large
i j
enoughithishomomorphismmaybeused inplaceofψ . (cid:3)
i
4. IMMERSED SUBGROUPS
In this section, we address the question of when the homomorphism ψ
i
can be chosen to be a monomorphism, so that G can be identified with a
i
CONVERGENCEOFISOMETRIES 7
subgroup of G . In other words, we wish to understand for which conver-
ω
gentsequencesin symmetriesare preserved,andforwhichsequences
eq
M
symmetriesarelost.
As an application, Proposition 4.5 bounds the symmetries of manifolds
ofalmostmaximalvolumefor theircurvatureandradius.
Let H be the kernel of ψ . Then (X ,H ) (X ,1) in . It follows
i i i i ω eq
→ M
that X /H X in the Gromov–Hausdorff topology. Combining these
i i ω
→
two convergences, it is clear that the projection map X X /H is a
i i i
→
Gromov–Hausdorffǫ –approximationforsomeǫ 0,sothatthediameter
i i
→
oftheorbitsofthekernelconverges to0.
If H is non-trivial, then some symmetries are lost, and there are two
i
very natural ways for such actions to arise. The first is where the sequence
ofspaces(X ,p ,H )isobtainedbyshrinkingthegrouporbits,suchasina
i i i
Cheegerdeformation.
However, this can also occur where the space has an unusual topology.
Consider, for example, the infinite wedge of 2-spheres with a Hawaiian
earringtopology. LetX bethisspaceendowedwithametricwhichhasno
ω
non-trivialisometries. Let X be isometricto X , except that the ith sphere
i ω
hasaroundmetric. ThentheisometrygroupofX isO(2),andthelimitof
i
(X ,O(2))is(X ,1).
i ω
Thefollowingpropositionshowsthatgivenalowercurvaturebound,un-
less the sequence collapses symmetries are always preserved. A suitable
sense of lower curvature bound is that from Alexandrov geometry. An
Alexandrov space of curvature k is a generalization of a Riemannian
≥
manifold with sectional curvature k. Very roughly, it is a metric space
≥
in which triangles are “fatter” than triangles with the same side-lengths in
constant curvature k. The reader is referred to [2] for the definition and
basicideas.
Thesubspaceof underconsiderationis then
eq
M
Ωn = (X,p,G) X has curvature k,dimX = n,Gis compact .
k { ∈ Meq | ≥ }
Proposition4.1. Let(X ,p ,G )beasequenceconverginginΩn. Thenthe
i i i k
limitgroupofisometriescontainsan isomorphicimageof G , forlargei.
i
Proof. LetH E G bethekernelofthehomomorphismgivenbyTheorem
i i
3.1, so that (X ,p ,H ) (X,p,1). Then X /H X. However, X /H
i i i i i i i
→ →
is also a sequence of Alexandrov spaces with curvature bounded below by
k,andsobythecontinuityofHausdorffmeasureonΩn [2],H istrivialfor
k i
largei. (cid:3)
LetusrestrictourattentionfurthertothoseAlexandrovspaceswithcom-
pactisometrygroups: thespace Alexc(k,n) givenby
⊂ M
(X,p) X hascurvature k,dimX = n,IsomX iscompact .
{ ∈ M | ≥ }
8 JOHNHARVEY
Recallingthatthesymmetrydegreeofametricspaceisthedimensionofits
fullisometrygroup, Proposition4.1 hasthefollowingcorollary:
Corollary4.2(Semicontinuityofsymmetries). Thesymmetrydegreeisup-
persemicontinuouson Alexc(k,n).
Thefollowingquestionis natural:
Question4.3. TowhatextentcanthelowercurvatureboundsinProposition
4.1and itsCorollary 4.2berelaxed?
For example, if the X are manifolds with only a lower Ricci curva-
i
turebound,that propertyis notnecessarilyinherited bythequotientspaces
X /H ,and so theproofofthePropositionwouldfail.
i i
4.1. Voluminous manifolds. According to Grove and Petersen [10], the
volume of a manifold can be controlled by a lower bound on the sectional
curvatureand by theradius. TheradiusofametricspaceX istheinvariant
radX = min max dist(p,q).
p q
Theorem 4.4 (Grove–Petersen [10]). Fix a real number k, a positive r (
≤
π/√k ifk > 0),andanintegern 2. Thenthereisanw = w(k,r,n) > 0
≥
suchthatifM isaRiemanniann-manifoldwithsecM kandradM r,
≥ ≤
then volM w(k,r,n). Furthermore, if volM is sufficiently close to
w(k,r,n)the≤nM istopologicallyeitherSn orRPn.
Itisclearthatanupperboundexists. Asimplevolumecomparisongives
that volM vn(r), the volume of the ball of radius r in the simply con-
≤ k
nected n–dimensionalspaceformofconstantcurvaturek.
In the case of k > 0 and π < radM π , the bound is not sharp.
2√k ≤ √k
Here some volume must always be lost, and the manifolds with volume
closetotheboundw arealwayshomeomorphictospheres. However,when
k π , bothtopologiesadmitmetricswithvolumecloseto w.
≤ 2√k
Riemannian manifolds can only achieve this volume bound in the case
of round spheres and projective spaces. However, the key observation in
provingthetheoremisthatthereisalwaysanAlexandrovspaceofcurvature
k, radius r and dimension n which achieves the volume w(k,r,n).
≥ ≤
ThenanysequenceofRiemannian manifoldswithvolumeconvergingtow
musthaveoneofthesespacesasitslimit,andsobehomeomorphictoitby
Perelman’sStabilityTheorem[14].
There are only three types of Alexandrov spaces of maximal volume.
More thorough descriptions of these types are given in [10]. Two of these
typesachievethevolumevn(r). Thesearethe“crosscap”(thediskofradius
k
rwithantipodalpointsontheboundaryidentified),whichishomeomorphic
to RPn, and isometricin case r = π , and the “purse” (the disk of radius
2√k
CONVERGENCEOFISOMETRIES 9
r,butwithboundarypointsidentifiedbyareflectionthroughahyperplane),
whichishomeomorphicto Sn.
In case k > 0 and r > π , the maximal volume w(k,r,n) is attained
2√k
by the “lemon”. The lemon is obtained by removing a wedge (that is, a
connected component ofthe complementoftwo totallygeodesichypersur-
faces)from theroundsphereSn ofcurvaturek, and then identifyingpoints
ontheboundaryviaareflection. Thelemonishomeomorphictothesphere,
andwhen r = π itis isometricto aroundsphere.
√k
The isometry groups of each of these spaces are easily calculated. Let
the dimension of the space be n. In case the space is smooth (k > 0 and
r = π or π ), it is homogeneous with isometry group O(n + 1) (with
2√k √k
ineffectivekernel oforder2 intheRPn case).
Otherwise,thecrosscaphasisometrygroupO(n). Galaz-GarciaandGui-
jarrohaveshownthat,justaswithRiemannianmanifolds,thisisthelargest
possibledimension for the isometry group of a non-homogeneous Alexan-
drov space [6]. The purse and the lemon, however, have as isometry group
thedirect productZ O(n 1).
2
× −
Combining the information on the isometry groups of these spaces with
Proposition4.1, oneobtains the followingresult on the symmetriesof Rie-
mannianmanifoldswithalmostmaximalvolume.
Proposition 4.5. Fix a real number k, a positive r (in case k > 0, requir-
ing r < π/√k and = π/2√k), and an integer n 2. Any Riemann-
6 ≥
ian n-manifold M with sec(M,g) k and radM r and volM suffi-
ciently close to w(k,r,n) is either R≥Pn with IsomM≤ O(n) or Sn with
IsomM Z O(n 1). ⊂
2
⊂ × −
The largest possible isometry group can always be achieved, simply by
carryingoutthesmoothingconstructiongivenin[10]inanequivariantway.
In the RPn case, the Proposition states merely that M is not homoge-
neous,by[6].
It is thesymmetry gap between the two topologicaltypes that is of more
interest. A round RPn has a larger volume than a round Sn of the same
radius. The gap might be read as an expression of this fact, quantitatively
expressing the idea that volume can be maximized in RPn more naturally,
whereas maximizing it in Sn involves forcing volume into the space in a
waywhich destroysmoresymmetries.
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MATHEMATISCHES INSTITUT, UNIVERSITÄT MÜNSTER, EINSTEINSTR. 62, 48149
MÜNSTER,GERMANY.
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