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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 diffeomorphisms. 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 generated groupsquasi–isometricto mappingclass groups. 1.2. Notes and references. 1.2.1. Higher rank phenomena for mapping class groups. Theorem 1.2 is an example 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 property (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) admit 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 results 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 detailed 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 between 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 theorem 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 contains 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 fñpxq´ 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 diffeomorphisms. Theorem 4.4 (Denjoy’s Theorem [9], [30, Theorem 3.1.1]). If f P Diff1`bvpS1q ` andPer f “ H, then f istopologicallyconjugatetoan irrationalrotation.