Unsmoothable group actions on compact one-manifolds

0.22 MB·

by  Hyungryul Baik

UNSMOOTHABLE GROUP ACTIONS ON COMPACT
ONE–MANIFOLDS
6 HYUNGRYULBAIK,SANG-HYUNKIM,ANDTHOMASKOBERDA
1
0
2 Abstract. We show that no finite index subgroup of a sufficiently complicated
mappingclassgrouporbraidgroupcanactfaithfullybyC1`bv diffeomorphisms
n
on the circle, which generalizes a result of Farb–Franks, and which parallels a
u
J resultofGhysandBurger–Monodconcerningdifferentiableactionsofhigherrank
0 latticesonthecircle. Thisanswersa questionofFarb,whichhasitsrootsinthe
1 work of Nielsen. We prove this result by showing that if a right-angled Artin
groupactsfaithfullybyC1`bvdiffeomorphismsonacompactone–manifold,then

]
its defining graph has no subpath of length three. As a corollary, we also show
T
G thatnofiniteindexsubgroupofAutpFnqandOutpFnqforně3,theTorelligroup
forgenusatleast3,andofeachtermoftheJohnsonfiltrationforgenusatleast5,
.
h canactfaithfullybyC1`bvdiffeomorphismsonacompactone–manifold.
t
a
m
[
1. Introduction

4
v 1.1. Main results. Let S “ S be an orientable surface of genus g, with n
g,n,b
0
marked points, and with b boundary components, and let ModpSq denote its map-
9
4 pingclass group. Wewillwrite
5
0 cpSq “ 3g´3`n`b
.
1 forthecomplexityofS. Throughoutthispaper,welet Mbeacompact(possiblydis-
0
connected) one–manifold, and we let Diff1`bvpMq be the group of C1 orientation–
6 `
1 preserving diffeomorphisms of M whose first derivatives have bounded variation;
:
v see Section 4 for a more detailed discussion. The following is a classical result of
Xi Nielsen:
r Theorem 1.1 (Nielsen [13]). If S “ S , the group ModpSq admits a faithful
a g,1,0
continuousactionon S1 withoutglobalfixed points.
InthisarticleweprovethefollowingresultwhichshowsthattheactioninTheo-
rem1.1,andinfactanyactionatallevenaftertakingafiniteindexsubgroup,isnot
smoothabletoC1 withboundedvariation.
Theorem 1.2. There exists a finite index subgroup G ă ModpSq and an injective
homomorphismfromG to Diff1`bvpMqif andonlyifcpSq ď 1.
`
Date:June13,2016.
1
2 H.BAIK,S.KIM,ANDT.KOBERDA
Theorem1.2generalizes aresultofFarb–Franks [11], whereinitisshownthatif
S “ S , where g ě 3 and where n P t0,1u, then there is no nontrivialC2 action
g,n,0
of ModpSq on the circle or the closed interval. As Farb and Franks remark, Ghys
establishedthisfact independentlyin thecase ofthecircle.
Note that a C2 action or a C1 action with Lipschitz derivatives on M is always
C1`bv. For a lower regularity, Parwani [31] proves that every C1 mapping class
groupaction onthecircleistrivialifthegenus ofS isat least six.
Wesayagroup H embedsintoanothergroupG ifthereexistsaninjectivehomo-
morphismfrom H intoG, and H virtuallyembeds intoG ifa finite indexsubgroup
of H embedsintoG. Theorem1.2 hasthefollowingimmediatecorollary:
Corollary 1.3. The n–strand braid group B virtually embeds into Diff1`bvpMq if
n `
andonlyifn ď 3.
Previousresultsaboutthenon–existenceofsmoothmappingclassgroup actions
on one–manifolds used particular relations which hold in mapping class groups,
especially braid relations. The difficulty with generalizing such arguments is that
mostrelationsotherthancommutation/non–commutationofelementsdonotpersist
infiniteindexsubgroups.
We are able to establish Theorem 1.2 by exhibiting a right-angled Artin group
whichcannotact faithfullybyC1`bv diffeomorphismsonacompactone–manifold.
Recall thatifΓis afinitesimplicialgraphwith thevertexset VpΓq and theedgeset
EpΓq, theright-angledArtingroup ApΓq isthefinitelypresented group
ApΓq “ xVpΓq | rv,v s “ 1ifand onlyiftv,v u P EpΓqy.
i j i j
Welet P denotethepathon nvertices (seeFigure2).
n
Thecore ofthepaper isinprovingthefollowingresult:
Theorem 1.4. The group ApP q does notvirtuallyembed intoDiff1`bvpMq.
4 `
Note that Theorem 1.4 provides a (relatively simple) purely algebraic obstruc-
tion for a group to act faithfully on a compact one-manifold by C1`bv diffeomor-
phisms. Whereas theregularityassumptioninTheorem 1.4may appearartificial at
first glance, it is actually optimal; in particular, it cannot be weakened to C1. See
Theorem 1.9below.
Recall that a graph Λ is an induced subgraph of a graph Γ if VpΛq Ă VpΓq, and
wheneverv ,v P VpΛq span an edge of Γ, they also span an edge of Λ. A graph Γ
1 2
iscalled P –freeifnoinducedsubgraph ofΓisisomorphicto P .
4 4
Corollary1.5. Supposethat ApΓq embedsin Diff1`bvpMq. Then Γ is P –free.
` 4
There are two main steps in the proof of Theorem 1.4. The first step is finding
acertainconfigurationofinfinitelymanyintervalsinthesupportsofthegenerators
ofApP q. Thisstepconstitutesmostoftheoriginalcontentofthispaper,andbuilds
4
UNSMOOTHABLEGROUPACTIONSON COMPACTONE–MANIFOLDS 3
on the classical results in one–dimensional dynamics described in Section 4. The
results of Section 5 give a detailed description of right-angled Artin group actions
on one–dimensional manifolds. Labeling the vertices of P as shown in Figure 2,
4
we prove the following proposition in Section 5, which is crucial for the proof of
Theorem 1.4:
Proposition 1.6. Suppose M is connected. If there is a faithful representation
φ: ApP q Ñ Diff1`bvpMq such that Fixφpvq ‰ ∅ for each v P VpP q, then there
4 ` 4
exist collectionsof infinitelymanyintervals
tIj: j ě 1u Ď π psuppφpaqq and tIj: j ě 1u Ď π psuppφpdqq
a 0 d 0
suchthatthefollowingtwo conditionshold:
(i) Ij ‰ Ik whenever v P ta,duand j ‰ k.
v v
(ii) Foreach j,we have Ij ‰ Ij andφpcbqIj XIj ‰ ∅.
a d a d
In Proposition 1.6 above and in what follows, π denotes the set of connected
0
components.
The second step of the proof of Theorem 1.4 consists of derivative estimates of
thediffeomorphismstφpvq | v P VpP qu; seeSection6.
4
In the last section, we will give a generalization of Theorem 1.2 to finitely gen-
erated groupsquasi–isometricto mappingclass groups.
1.2. Notes and references.
1.2.1. Higher rank phenomena for mapping class groups. Theorem 1.2 is an ex-
ample of a higher rank phenomenon for mapping class groups. The reader may
compare Theorem 1.2 to the followingresult of Ghys [14] and Burger–Monod [4],
whichbuilton workofWitte[34]:
Theorem 1.7. Let G be a lattice in a simple Lie group of higher rank. Then every
continuousaction ofG on the circle has a finite orbit, and everyC1 action ofG on
thecirclefactorsthrougha finitegroup.
LatticesinhigherranksimpleLiegroupshaveKazhdan’sproperty(T).Navas[27]
proved the following non-smoothability result for all countable groups with prop-
erty (T). Here, we use the terminology C1`τ diffeomorphism for a C1 diffeomor-
phismwhosefirst derivativeis Ho¨ldercontinuousofexponentτ ą 0:
Theorem 1.8. Let G be a countable group with property (T) and τ ą 1{2. Then
everyC1`τ actionofG onthecirclefactorsthrougha finitegroup.
Navas refined Theorem 1.8 to groups withrelativeproperty (T) in [28], and pro-
duced a finitely generated, locally indicable group with no faithful differentiable
action on the interval in [29]. In those papers, as in the present one, the main sub-
tletiesarisefromdifferentiability.
4 H.BAIK,S.KIM,ANDT.KOBERDA
Braid groups and mapping class groups of genus two surfaces (virtually) ad-
mit nontrivial homomorphisms to Z, so that these groups do admit (non–faithful)
actions on the circle without a global fixed point, which can even be taken to be
analytic.
In this context, Theorem 1.2 can be viewed as a generalization of known re-
sults to “lattices in mapping class groups”, i.e. finite index subgroups of mapping
class groups. The reader may note that Theorem 1.2 resolves a question posed by
Labourieinhis2002ICM talk, as wellas byFarb (see page47of[10]).
For C1 actions, we can contrast Theorem 1.4 with the following result of Farb–
Franks [12, Theorem1.6]:
Theorem 1.9. If X be a connected one–manifold, then every finitely generated
residually torsion–free nilpotent group embeds into the orientation–preserving C1
diffeomorphismgroupof X.
In particular, right-angled Artin groups, pure braid groups, and Torelli groups
of surfaces all admit faithful C1 actions on the circle. In particular, Theorem 1.4
cannot be weakened to cover C1 actions of right-angled Artin groups. In light of
Parwani’sTheorem [31], onemay ask thefollowing:
Question 1. Let S be a surface with genus at least two. Does ModpSq virtually
admita faithfulC1 actionon a compactone–manifold?
Ivanov conjectured that every finite index subgroup of a higher genus mapping
class group has finite abelianization [20]. Parwani [31, Conjecture 1.6] noted that
Ivanov’s conjecture anticipates a negative answer to Question 1 when the genus is
at leastfour.
1.2.2. Othergroupactions. Theorem1.4showsthatagroupwhichcontainsApP q
4
cannot act by C1`bv diffeomorphisms on the circle. There are many groups which
contain ApP q, including many subgroups of mapping class groups which do not
4
havefiniteindices.
We recall the following: let AutpF q and OutpF q denote the automorphism
n n
and the outer automorphism groups of the free group F on n generators. The
n
Torelli group IpSq of S is defined to be the kernel of the standard representation
ModpSq Ñ AutpH pS,Zqq. When S has a distinguished marked point, one can
1
write J pSq for the kth term of the Johnson filtration, i.e. J pSq is the kernel of
k k
the natural action of ModpSq on the kth universal nilpotent quotient G of π pSq.
k 1
In other words, we set γ pπ pSqq “ π pSq and γ pπ pSqq “ rπ pSq,γ pπ pSqqs.
1 1 1 k`1 1 1 k 1
Then,G “ π pSq{γ pπ pSqq. Notethat with this definition, J pSq “ ModpSq and
k 1 k 1 1
J pSq “ IpSq.
2
Corollary 1.10. No finiteindexsubgroupof thefollowinggroupsacts faithfullyby
C1`bv diffeomorphismson M:
UNSMOOTHABLEGROUPACTIONSON COMPACTONE–MANIFOLDS 5
(1) AutpF q andOutpF qfor n ě 3;
n n
(2) IpSqandJ pSq,where S has genusatleast 3;
3
(3) J pSq,where S has genusatleast 5if k ą 3.
k
It was proved in [3] that AutpF q does not admit a nontrivial action by circle
n
homeomorphisms,thoughtheBridson–Vogtmannproofreliesessentiallyontorsion
inside AutpF q and thus does not generalize to finite index subgroups of AutpF q.
n n
WewillgiveaproofofCorollary 1.10in Section 3below.
1.2.3. Compactnessandsmoothactions. Theorem1.4alsounderlinesafundamen-
tal difference between compact and noncompact one–manifolds,as far as theirdif-
feomorphism groups are concerned. We see in this paper that many right-angled
Artin groups do not embed in Diff1`bvpMq when M is compact, though no such
`
restrictions holdwhen M is noncompact, as is illustratedby thefollowingresult of
theauthors[1]:
Theorem1.11. Everyright-angledArtingroupembedsintotheorientation–preserving
C8 diffeomorphismgroupofR.
Actions on noncompact manifolds have applications to mapping class groups of
surfaces in theirownright:
Question2. Let n ě 4. Does B virtuallyadmitafaithfulC8 actiononR?
n
Anegativeanswerwouldprovethatabraidgrouponatleastfourstrandsdoesnot
virtually embed into a right-angled Artin group, by Theorem 1.11. It has recently
been proved that such braid groups do not virtually admit cocompact actions on
CAT(0)cubecomplexes[17], [19].
1.2.4. Structureoffinitelygeneratedsubgroupsofthediffeomorphismgroup. The-
orems 1.2 and 1.4 raise natural questions about the structure of finitely generated
subgroupsofDiff1`bvpMq,whichareknowntobequitecomplicated[15]. Already,
`
we see that the possibleright-angled Artin subgroups of Diff1`bvpMq are rather re-
`
stricted whenever M is a compact one–manifold. Indeed, Corollary 1.5 states that
ifΓisagraphand ApΓqactsfaithfullybyC1`bv diffeomorphismson M thenΓmust
be P –free. The class of P –free graphs is well–understood: it is the smallestclass
4 4
offinitesimplicialgraphswhichcontainsasinglevertexgraph,andwhichisclosed
under finite disjoint unions and finite joins. See [7] and also [21] for a more de-
tailed discussion of P –free graphs and the right-angled Artin groups defined by
4
such graphs.
Let us denote by Diff1`bvpI,BIq the group of C1`bv diffeomorphisms of I which
`
aretheidentityneartheboundaryof I. By concatenatingintervals,weseethatifG
and H are finitely generated subgroups of Diff1`bvpI,BIq, then G ˆ H occurs as a
`
subgroupofDiff1`bvpI,BIq. Amuch moresubtlequestionisthefollowing:
`
6 H.BAIK,S.KIM,ANDT.KOBERDA
Question 3. LetG,H ă Diff1`bvpMq be finitelygenerated subgroups. DoesG ˚H
`
occur asa subgroupofDiff1`bvpMq?
`
Apositiveanswertothisquestionwouldcharacterizetheright-angledArtinsub-
groups of Diff1`bvpMq as exactly the ones with P –free defining graphs. Recall
` 4
that a diffeomorphismof a space is fullysupported if the fixed point set has empty
interior [11]. The referee has pointed out the following partial answer to Ques-
tion3: IfG and H consistoffullysupporteddiffeomorphisms,thenonecanembed
G ˚ H into Diff2 pMq by conjugating one of G and H by some diffeomorphism f,
`
andthechoiceofsuchan f isdenseinDiff2 pMq. Similargenericityargumentsare
`
described in[15, Proposition4.5]andin [33,Theorem 2.1].
1.2.5. Sectionsforthemappingclass group. Let
π: Homeo pSq Ñ ModpSq
`
be the projection that takes a homeomorphism of S its mapping class. A well–
known result of Markovic [26], which is closely related to mapping class group
actions on the circle, says that π does not admit a section whenever S is a closed
surface withgenusat leastsix.
Question4. LetG ă ModpSqbeafiniteindexsubgroup. Doesthereexistasection
s: G Ñ Diff1 pSqwhich splitsπ? HowabouttoHomeo pSq?
` `
2. Acknowledgements
The authors thank N. Kang, S. Lee, J. McCammond, C. McMullen, A. Navas,
D. Rolfsen, and M. Sapir for helpful discussions and comments on an earlier draft
of this paper. The authors are particularly grateful to B. Farb and A. Wilkinson
for helpful comments. The authors thank the hospitality of the Mathematisches
Forschungsinstitut Oberwolfach workshop “New Perspectives on the Interplay be-
tween Discrete Groups in Low–Dimensional Topology and Arithmetic Lattices”,
where part of this research was carried out, and the hospitality of the Tsinghua
SanyaInternationalMathematicsForum,where thisresearch was completed.
The authors are grateful to an anonymous referee who provided many useful
commentswhich significantlyimprovedtheexposition.
Thefirst-namedauthorwaspartiallysupportedbytheERCGrantNb. 10160104.
Thesecond-namedauthorwassupportedbyBasicScienceResearchProgramthrough
the National Research Foundation of Korea (NRF) funded by the Ministry of Sci-
ence, ICT & FuturePlanning(NRF-2013R1A1A1058646). Thesecond-named au-
thor is also supported by Samsung Science and Technology Foundation (SSTF-
BA1301-06).
UNSMOOTHABLEGROUPACTIONSON COMPACTONE–MANIFOLDS 7
3. Right-angled Artin groupsandfinite index subgroupsofmappingclassgroups
Forour purposes, right-angledArtin groups are an essential tool for understand-
ing all finite index subgroups of a given mapping class group. In this section, we
summarizetherelevantfacts andreduce theresultsofthispaperto Theorem1.4.
Lemma3.1. Ifaright-angledArtingroupAembedsintoagroupG,thenAembeds
intoeachfiniteindexsubgroupofG.
Proof. Let H be a finite index subgroup of G. By taking a smaller finite index
subgroup if necessary, we may assume H is normal in G. Let φ: A Ñ G be the
giveninjectivehomomorphismandN “ rG : Hs. WriteA “ ApΓq. ThenK “ xvN |
v P VpΓqy ď A isisomorphicto A and φpKq ď H. (cid:3)
LetS beasurface. RecallthatthecurvegraphCpSqisagraphwhoseverticesare
isotopy classes of non–peripheral simpleclosed curves on S, and the edge relation
is given by disjoint realization on S. A graph Γ is called an induced subgraph
of CpSq if there exists an injective map of graphs ι: Γ Ñ CpSq which preserves
adjacency andnon–adjacency.
Theorem 3.2 (See [23]). Let Γ ă CpSq be a finite induced subgraph. Then there
existsan injectivehomomorphism ApΓq Ñ ModpSq.
Explicitly, let us suppose a graph map ι: Γ Ñ CpSq preserves adjacency and
non–adjacency. Foreach N P N, weget an inducedmap
ι : ApΓq Ñ ModpSq
˚,N
by sending v ÞÑ TN , where T denotes the Dehn twist about the curve ιpvq. A
ιpvq ιpvq
more detailed version of Theorem 3.2 as proved in [23] is that for N " 0, the
homomorphismι isinjective.
˚,N
Corollary3.3. Thegroup ApP q embedsintoModpSqwhenever cpSq ě 2.
4
Proofof Theorem 1.2. For surfaces S with cpSq ě 2, the conclusion of the theo-
rem follows from Theorem 1.4, combined with Lemma 3.1 and Corollary 3.3. For
surfaces withcpSq ă 2, wehave thatModpSqis virtually aproduct of afree group
and a cyclic group [5] and thus virtually admits aC1`bv action on M; for example,
see[16]. (cid:3)
Proofof Corollary1.10. ByTheorem1.4,itsufficestofindacopyofApP qineach
4
ofthesegroups.
(1)NotethatthesurfaceS ofgenusonewithtwoboundarycomponentscon-
1,0,2
tains a chain of four simple closed curves, by which we mean a collection of four
pairwise non–isotopicessential simple closed curves tγ ,...,γ u in minimal posi-
1 4
tionwiththepropertythatγ Xγ ‰ ∅,butγ Xγ “ ∅otherwise. SeeFigure1(a)
i i`1 i j
8 H.BAIK,S.KIM,ANDT.KOBERDA
below. By Theorem 3.2, this realizes ApP q in ModpS q ă OutpF q. Equipping
4 1,0,2 3
S withamarked pointrealizes ModpS q as asubgroupofAutpF q.
1,0,2 1,1,2 3
(2) A closed surface S of genus at least 3 contains a chain of four separating
closed curves as shown in Figure 1 (b). The curves are labeled by a,b,c,d so that
the intersection graph coincides with the graph shown in Figure 2 which is a copy
of P . By Theorem 3.2 this furnishes a copy of ApP q in IpSq. The reader may
4 4
notethat thisactuallyfurnishes ApP q inJ pSq.
4 3
(3)(Sketch)IfS hasgenus5ormorethenthereexistsachainoffoursubsurfaces
tS ,...,S uinS,whereeach S ishomeomorphictoagenustwosurfacewithone
1 4 i
boundary component. A chain of subsurfaces is defined analogously to a chain of
simple closed curves, and a chain of two subsurfaces is illustrated in Figure 1 (c)
below. Let k ą 3.
For each i “ 1,2,3,4, we claim the surface S supports a pseudo-Anosov map-
i
ping classes ψ which lie in the Johnson kernel J pSq. This follows from a well–
i k
known fact that every non–central normal subgroup of a mapping class group con-
tains pseudo-Anosov elements (see [20]). One way to see this fact is to appeal to
a result of Dahmani–Guirardel–Osin in [8], which shows that the mapping class
groupcontainsan infinitelygenerated, purelypseudo-Anosov,normalsubgroupN.
If 1 ‰ ψ P J pSq is pseudo-Anosov then there is nothing to show. If ψ fails to be
k
pseudo-Anosovthenwetake1 ‰ n P N,andconsiderrn,ψs. Sinceψandnwillnot
commute as the centralizer of a pseudo-Anosov mapping class is virtually cyclic,
we havethat φ “ rn,ψs is nontrivial. Since J pSq is normal, φ P J pSq. Since N is
k k
normal,wealso haveφ P Nand is thereforepseudo-Anosov.
Finally,sufficientlylargepowersoftψ ,...,ψ uwillgenerateacopyofApP q ă
1 4 4
J pSq,by [6,Theorem 1.1]or[23, Theorem 1.1]. (cid:3)
k
We also record the following fact, which allows us to reduce to the case where
M isaconnected manifold:
Lemma 3.4. Let φ: ApP q Ñ G ˆ H be an injective homomorphism, and let φ
4 G
and φ be the compositionof φ with theprojectionstoG and H respectively. Then
H
oneofφ orφ isinjective.
G H
Proof. Let K and K be the kernels of φ and φ respectively, which we assume
G H G H
are nontrivial. On the one hand, note that K K – K ˆ K , so that if g P K
G H G H G
and h P K are nontrivial, then Z2 – xg,hy ă ApP q. On the other hand, we
H 4
havethat K and K bothcontainloxodromicelements,each ofwhichhas acyclic
G H
centralizer in ApP q (see Lemma 52 of [22] and Theorem 2.4 of [24]). This is a
4
contradiction,sothatat least oneof K and K istrivial. (cid:3)
G H
UNSMOOTHABLEGROUPACTIONSON COMPACTONE–MANIFOLDS 9
γ γ
3 2
S
2
c
a
γ γ
1 4
d
b S
1
(a)Achainofcurves (b)A chainoffour (c) A chain of
separatingcurves subsurfaces
Figure 1. Corollary 1.10.
4. Classical one–dimensional dynamics
From Sections 4 through 6, we will assume M is connected. In other words, we
let M “ I “ r0,1s or M “ S1 “ R{Z. For x,y P S1, the open interval px,yq “
tz P M: x ă z ă yu is naturallydefined as a subsetoftheimageofpx,x`1q Ď R.
We let Homeo pMq denote the group of orientation–preserving homeomorphisms
`
of M. For f P Homeo pMq, we use the notations Fix f “ tx P M: fx “ xu
`
and supp f “ MzFix f. The set supp f is called the support (or, open support)
of f. We will broadly appeal to the following equality, which is trivial to prove:
supp f “ supp f´1 foran arbitrary automorphismofan arbitrary set.
Wedefinethesetofperiodicpoints:
8
Per f “ tx P M: fpx “ x forsome0 ‰ p P Zu “ Fixpfpq.
p“1
ď
ForY Ď M, thesetoftheconnected componentsofY isdenoted asπ pYq. Fortwo
0
groupelementsa and b, weletra,bs “ a´1b´1ab.
Lemma 4.1. For f,g P Homeo pMq, we havethefollowing.
`
(1) Fixpgfg´1q “ gFix f,supppgfg´1q “ gsupp f andPerpgfg´1q “ gPer f.
10 H.BAIK,S.KIM,ANDT.KOBERDA
(2) Let Y Ď Z be intervals in M such that each component of supp f is ei-
ther contained in Y or disjoint from Z. If gpYq Ď Y and gpZq “ Z, then
gf˘1g´1pYq “ Y.
(3) If f andg commute, then f permutestheelementsof π psuppgq.
0
Proof. (1)For x P M,wehave
x P Fixpgfg´1q ô gfg´1x “ x ô fpg´1xq “ g´1x ô g´1x P Fix f.
Andtheothertwoassertionsfollowsimilarly.
(2)Theassertionisimmediatefrom
supppgf˘1g´1qXZ “ gsupp f XZ Ď gpYq Ď Y.
(3) Let Y P π psuppgq. Since fY is connected and contained in f suppg “
0
suppg, there exists Z P π psuppgq containing fY. Since f´1Z is connected and
0
containedin f´1psuppgq “ suppg,wehaveY “ f´1Z and fY “ Z. (cid:3)
Let us say f is grounded if Fix f ‰ ∅. For example, every map in Homeo pIq
`
is grounded. We say a group action of G on M is free (or, G acts freely on M) if
Fixg “ ∅foreach g P Gzt1u.
For f P Homeo pS1q, choose an arbitrary lift f˜: R Ñ R and x P R. Then
`
we define the rotation number of f as rot f “ lim f˜npxq´ x {n, which is
nÑ8
uniquely defined in R{Z regardless of the choice of the lift f˜ and x. We have
` ˘
rotpfnq “ nrotpfqforeach n P Z. It is astandard fact that rotpfq “ 0 ifand onlyif
f isgrounded.
Theorem 4.2 (Ho¨lder’s Theorem [30]). If G is a subgroup of Homeo pMq such
`
that G acts freely on MzBM, then G is abelian. In this case, if M “ S1 then the
rotationnumberisan embeddingfromG to themultiplicativegroupS1.
Theorem 4.3 ([11, Proposition 2.10]). The centralizer of an irrational rotation in
Homeo pS1q isSOp2,Rq.
`
Wedenoteby varpg;Mqthetotalvariationofamapg: M Ñ R:
n´1
varpg;Mq “ sup |gpa q´gpaq| : pa : 0 ď i ď nqis apartitionof M .
i`1 i i
# +
i“0
ÿ
Inthecase M “ S1,werequirea “ a intheabovedefinition. Following[30],we
n 0
say a C1 diffeomorphism f on M is C1`bv if varpf1;Mq ă 8. We let Diff1`bvpMq
`
denote the group of orientation–preserving C1`bv diffeomorphisms of M. Our ap-
proachessentiallybuildsonthefollowingtwofundamentalresultsonC1`bv diffeo-
morphisms.
Theorem 4.4 (Denjoy’s Theorem [9], [30, Theorem 3.1.1]). If f P Diff1`bvpS1q
`
andPer f “ H, then f istopologicallyconjugatetoan irrationalrotation.