Pullback Invariants Of Thurston Maps

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Pullback invariants of Thurston mapsSarah Koch, Kevin M. Pilgrim, and Nikita SelingerMay 9, 2014AbstractAssociated to a Thurston map f : S 2 ! S 2 with postcritical set Pfare several di erent invariants obtained via pullback: a relation SPSP on the set SP of free homotopy classes of curves in S 2 \ P , a linearoperator f : R[SP ] ! R[SP ] on the free R-module generated by SP , avirtual endomorphism f : PMod(S 2 , P ) 99K PMod(S 2 , P ) on the puremapping class group, an analytic self-map f : T (S 2 , P ) ! T (S 2 , P ) ofan associated Teichmüller space, and an analytic self-correspondenceX Y 1 : M(S 2 , P ) M(S 2 , P ) of an associated moduli space.Viewing these associated maps as invariants of f , we investigate relationships between their properties.1

Contents1 Introduction22 Fundamental identifications2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . .773 Pullback invariants113.1 No dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Whenf: @TB ! @TA225 Whenf: TB ! TA is constant246 Noninjectivity of the virtual homomorphism286.1 Proof of Theorem 6.3 . . . . . . . . . . . . . . . . . . . . . . . 306.2 Proof of density, Theorem 6.4 . . . . . . . . . . . . . . . . . . 347 Finite global attractor for the pullback relation348 Shadowing409 Obstructed twists and repelling fixed points in @MP409.1 Obstructed twists . . . . . . . . . . . . . . . . . . . . . . . . . 409.2 Dynamical consequences . . . . . . . . . . . . . . . . . . . . . 431IntroductionThurston maps are orientation-preserving branched covers f : S 2 ! S 2 fromthe oriented topological two-sphere to itself that satisfy certain properties.They were introduced by Thurston as combinatorial invariants associatedto postcritically finite rational functions R : P1 ! P1 , regarded as dynamical systems on the Riemann sphere. A fundamental theorem of complexdynamics is Thurston’s characterization and rigidity theorem [DH], which(i) characterizes which Thurston maps f arise from rational functions, and(ii) says that apart from a well-understood family of ubiquitous exceptions,the Möbius conjugacy class of R is determined by the combinatorial class ofits associated Thurston map f . The proof of this theorem transforms thequestion of determining whether f arises from a rational function R to the2

question of whether an associated pullback map f : TP ! TP , an analyticself-map of a certain Teichmüller space, has a fixed point.As combinatorial (as opposed to analytic or algebraic) objects, Thurstonmaps, in principle, should be easier to classify than postcritically finite rational maps. For many years, the lack of suitable invariants for generalThurston maps frustrated attempts toward this goal. Kameyama [Kam]developed algebraic methods geared specifically to the classification of combinatorial classes of maps arising as twists of a given Thurston map f : theseare maps of the form h0 f h1 where h0 , h1 are homeomorphisms. In aground-breaking paper [BN1], Bartholdi and Nekrashevych introduced several new general tools. One innovation was to apply the general theory ofself-similar groups to study both the dynamics of a given Thurston map fand the classification of twists of f . Another was to exploit the existenceof an associated analytic correspondence on the moduli space MP coveredby the graph of f . These new tools have led to much better invariants forThurston maps and to a better understanding of the pullback map f , see[Nek1], [CFPP], [Koc1], [Koc2], [Pil3], [BEKP], [HP1], [HP2], [Kel], [Nek2],[BN2], [Lod].In this work, we deepen the investigations of the relationship betweendynamical, algebraic, and analytic invariants associated to Thurston maps.Fundamental invariants. Let S 2 denote a topological two-sphere, equippedb we use S 2 forwith an orientation. Fix an identification S 2 P1 C;b when formulas aretopological objects, P1 for holomorphic objects, and C2required. Let P S be a finite set with #P3 (in the case #P 3,the groups and spaces are trivial, so it is helpful to imagine at first that#P 4).The following objects are basic to our study.1. SP , the set of free homotopy classes of simple, unoriented, essential,nonperipheral, closed curves in S 2 \ P ; the symbol o denotes the unionof the sets of free homotopy classes of unoriented inessential and peripheral curves. A curve representing an element of SP we call nontrivial. A multicurve on S 2 \ P is a possibly empty subset of SPrepresented by nontrivial, pairwise disjoint, pairwise nonhomotopiccurves. Let MSP denote the set of possibly empty multicurves onS2 \ P .2. R[SP ], the free R-module generated by SP ; this arises naturally in thestatement of Thurston’s characterization theorem.3

3. GP : PMod(S 2 , P ), the pure mapping class group (that is, orientationpreserving homeomorphisms g : (S 2 , P ) ! (S 2 , P ) fixing P pointwise,up to isotopy through homeomorphisms fixing P pointwise). Eachnontrivial element of GP has infinite order. The group GP contains adistinguished subset Tw whose elements are multitwists; that is[Tw : Tw( )a multicurve on S 2 \Pwhere Tw( ) is the subgroup of GP generated by Dehn twists aroundcomponents of . A multitwist is positive if it is a composition ofpositive powers of right Dehn twists. Let[Tw : Tw ( )a multicurve on S 2 \Pwhere Tw ( ) is the subgroup of GP generated by positive Dehn twistsaround components of . The support of a multitwist is the smallestmulticurve about which the twists comprising it occur; this is welldefined.4. TP : T (S 2 , P ), the Teichmüller space of (S 2 , P ), as in [DH]. It comesequipped with two natural metrics, the Teichmüller Finsler metricPdTP and the Weil-Petersson (WP) inner product metric dWTP . Themetric space (TP , dTP ) is complete, and any pair of points are joinedPby a unique geodesic. In contrast, (TP , dWTP ) is incomplete. The WPcompletion T P has a rich geometric structure [Mas] (see also [HK]); itis a stratified space where each stratum TP corresponds to a multicurveand consists of noded Riemann surfaces whose nodes correspond topinching precisely those curves comprising to points. We denote by@TP the WP boundary of TP .5. MP : M(S 2 , P ) is the corresponding moduli space. It is a complexmanifold, isomorphic to a complex affine hyperplane complement. Thenatural projection : TP ! MP is a universal cover, with deck groupGP .Given a basepoint 2 TP , there is a natural identification of GP with 1 (MP , m ) where m : ( ); the isomorphism 1 (MP , m ) ! GPproceeds via isotopy extension, while the isomorphism GP ! 1 (MP , m )is induced by composing the evaluation map at those points marked by P4

with an isotopy to the identity through maps fixing three points of MP ; see[Lod] for details.A subgroup L GP is purely parabolic if L Tw. In terms of theidentification GP 1 (MP , m ), L is purely parabolic if and only if thedisplacements of each of its elements satisfies inf 2TP dTP ( , g. ) 0 for allg 2 L. A purely parabolic subgroup is complete if for each multitwist M 2 Land each 2 SP , we have 2 supp(M ) ) T 2 L where T is the rightDehn twist about . A complete parabolic subgroup is necessarily of theform Tw( ), the set of all multitwists about elements of , where is somenonempty multicurve.Let f : S 2 ! S 2 be an orientation-preserving branched covering mapwith critical set f . Its postcritical set is[Pf : f n ( f );n 0we assume throughout this work that #Pf 1. Suppose P S 2 is finite,Pf P , and f (P ) P ; we then call the map of pairs f : (S 2 , P ) ! (S 2 , P )a Thurston map. Throughout this work, we refer to a Thurston map by thesymbol f , and often suppress mention of the non-canonical subset P .The following objects are associated to a Thurston map f : (S 2 , P ) !2(S , P ) via pullback:1. a relationf: SP [ {o} ! SP [ {o},2. a non-negative linear operator3. a virtual endomorphismff: R[SP ] ! R[SP ],: GP 99K GP ,4. an analytic map f : TP ! TP ; this extends continuously toT P ! T P by [Sel1, §4],f:5. an analytic correspondence X Y 1 : MP MP ; the double-arrownotation reflects our view that this is a one-to-many “function”.Precise definitions and a summary of basic properties of these associatedmaps will be given in the next two sections.Main results. The goal of this work is to examine how properties ofthe objects in (1) through (5) are related. That such relationships shouldexist is expected since there are fundamental identifications between variouselements associated to the domains of these maps; see §2. Our main resultsinclude the following.5

We characterize when the map on Teichmüller space f : TP ! TPsends the Weil-Petersson boundary to itself (Theorem 4.1). We characterize when the map on Teichmüller space f : TP ! TP isconstant (Theorem 5.1); the proof we give corrects an error in [BEKP]. We show that the virtual endomorphism on the pure mapping classgroup f : GP 99K GP is essentially never injective (Theorem 6.3). It is natural to investigate whether a Thurston map f induces an actionon projectivized measured foliations, i.e. on the Thurston boundaryof TP . Using Theorem 6.3, however, we show (Theorem 6.4) thateach nonempty fiber of the pullback map f accumulates on the entireThurston boundary. This substantially strengthens the conclusion of[Sel1, Theorem 9.4]. We give sufficient analytic criteria (Theorem 7.2) for the existence of afinite global attractor for the pullback relation SPequivalently, for the action of the pullback mapstrata.ffSP on curves—: T P ! T P on We prove an orbit lifting result (Proposition 8.1) which asserts thatunder the hypothesis that the virtual endomorphism f is surjective,finite orbit segments of the pullback correspondence X Y 1 on modulispace can be lifted to finite orbit segments of f . In the case when #P 4, we relate fixed points of the various associated maps (Theorem 9.1). If in addition the inverse of the pullbackcorrespondence is actually a function (i.e. the map X is injective),sharper statements are possible (Theorem 9.2). Our results generalize, clarify, and put into context the algebraic, analytic, and dynamical findings in the analysis of twists of z 7! z 2 i given in [BN1,§6]. Using the shadowing result from Proposition 8.1, these resultsalso demonstrate that for certain unobstructed Thurston maps, onecan build finite orbits of the pullback map whose underlying surfacesbehave in prescribed ways. For example, one can arrange so that thelength of the systole can become shorter and shorter for a while beforestabilizing.While motivated by the attempt at a combinatorial classification of dynamical systems, many of the results we obtain are actually more naturallyphrased for a nondynamical branched covering map f : (S 2 , A) ! (S 2 , B).6

Where possible, we first phrase and prove more general results, (Proposition3.1, Theorem 4.1, Theorem 5.1, Proposition 6.2, Theorem 6.3, and Theorem6.4); the aforementioned theorems then become corollaries of these moregeneral results.Conventions. All branched covers and homeomorphisms are orientationpreserving. Throughout, f denotes a branched covering of S 2 ! S 2 of degreed2. The symbols A, B, P denote finite subsets of S 2 , which contain atleast three points.Acknowledgements. We thank Indiana University for supporting the visits of S. Koch and N. Selinger. S. Koch was supported by an NSF postdoctoral fellowship and K. Pilgrim by a Simons collaboration grant. We thankA. Edmonds, D. Margalit, and V. Turaev for useful conversations.2Fundamental identifications2.1PreliminariesTeichmüller and moduli spaces. The Teichmüller space TP T (S 2 , P )is the space of equivalence classes of orientation-preserving homeomorphisms' : S 2 ! P1 , whereby '1 '2 if there is a Möbius transformation µ : P1 !P1 so that '1 µ '2 on the set P , and '1 is isotopic to µ '2 relative to the set P .The moduli space MP M(S 2 , P ) is the set of all injective maps ' : P ,!P1 modulo postcomposition by Möbius transformations. The Teichmüllerspace and the moduli space are complex manifolds of dimension #P 3,and the map P : TP ! MP given by P : ['] 7! [' P ], is a holomorphicuniversal covering map.Note that since we have identified S 2 P1 , both MP and TP havenatural basepoints represented by the classes of the inclusion and identitymaps, respectively.Teichmüller metric. Equipped with the Teichmüller Finsler metric dTP ,the space TP becomes a complete uniquely geodesic metric space, homeomorphic to the open ball B #P 3 ; it is not, however, nonpositively curved.PWP metric. In contrast, when equipped with the WP metric dWTP , thespace TP is negatively curved but incomplete. The completion T P is a7

stratified space whose strata are indexed by (possibly empty) multicurves.Each stratum is homeomorphic to the product of the Teichmüller spacesof the components of the noded surface obtained by collapsing each curveofto a point. Indeed, this completion coincides with the augmentedTeichmüller space parametrizing noded Riemann surfaces marked by P (see[HK]). It is noncompact, coarsely negatively curved, and quasi-isometric tothe pants complex; for an extensive overview, see [Wol].Thurston compactification. For 2 SP and 2 TP let ( ) denotethe length of the unique geodesic in the hyperbolic surface associated to .The map 7! ( ( )) 2SP defines an embedding TP ! RSP0 which projectsto an embedding TP ! PRSP0 . Thurston showed that the closure of the#P 3image is homeomorphic to the closed ball B, and that the boundarypoints may be identified with projective measured foliations on (S 2 , P ). Acomprehensive reference is [FLP]; cf. also the book by Ivanov [Iva1].Fix a basepoint 2 TP ; this gives rise to a basepoint m : ( ) 2MP ; recall that we then have an identification GP 1 (MP , m ). Thefollowing folklore theorem is well-known.Theorem 2.1 (Fivefold way) There are natural bijections between the following sets of objects:1. multicurveson S 2 \ PP2. “purely atomic” measured foliations F( ) : 2 , where is thedelta-mass at , which counts the number of intersections of a curvewith .3. complete purely parabolic subgroups L of GP ;4. strata TP T P ;5. certain subgroups of loops in moduli space (thought of as generatedby certain pure braids) and corresponding via “pushing” to completepurely parabolic subgroups L of 1 (MP , m ).The subgroups arising in (5) will be described shortly. The bijections aregiven as follows:(1) ! (2) Take the zero measured foliation if is empty. Otherwise:take a foliation of a regular neighborhood of with the width (transversemeasure) of each neighborhood equal to 1—this gives the so-called normalform of a measured foliation F obtained by collapsing the unfoliated regions8

onto their spines ([FLP, §6.5]); the result is unique, up to the natural Whitehead equivalence on foliations. Alternatively: choosing a complex structureon S 2 \ P , one may apply a well-known result of Jenkins and Strebel (seee.g. [DH, Prop. 4.2]) to obtain a corresponding quadratic di erential whoseclosed horizontal trajectories provide the associated measured foliation.(2) ! (1) If the foliation is the zero foliation, take o, the emptymulticurve; otherwise, take one core curve from each cylinder in the normalform of the foliation.(1) ! (3) Take the subgroup L : Tw( ) generated by Dehn twistsabout elements of .(3) ! (1) Take the union of the core curves of the representing twists;this is well-defined.(1) ! (4) We take those marked noded spheres in which precisely all thecurves comprising correspond to nodes, and nothing else.(3) ! (4) The stratum TP is the unique stratum in Fix(L) of maximumdimension.(4) ! (3) Given the stratum TP , take the pointwise stabilizer L : Tw( ) StabGP (TP ).(3) ! (5) We now describe this correspondence.Let P : {p1 , . . . , pn } S 2 , n : #P 4. Recall that the configurationspace Config(S 2 , P ) is the space of injections P ,! S 2 ; the inclusion P S 2gives a natural basepoint. Let : Config(S 2 , P ) ! MP be the naturalprojection and the induced map on fundamental groups. Letbe a2nonempty multicurve represented by curves S \ P . Below, we describea procedure for producing, for each 2 , a loop in Config(S 2 , P ) basedat P which projects to the right Dehn twist Tw( ). This procedure willhave the additional property that as elements of 1 (Config(S 2 , P )), given1 , 2 2 , the elements represented by 1 , 2 will commute.The idea is based on simultaneous “pushing” (or Birman spin) of points;see Figure 1. Suppose z1 , . . . , zm 2 D : { z 1} are nonzero complexnumbers, pick r with maxi { zi } r 1, set z0 0, and consider the motiont 7! zi (t) : exp(2 it)zi , t 2 [0, 1], i 0, . . . , m. This motion extends to anisotopy of D fixing the origin and the boundary. Taking the result of thisextension when t 1, the resulting “multi-spin” homeomorphism of theplane is homotopic, through homeomorphisms fixing z0 , z1 , . . . , zm , to theright-hand Dehn twist about the circle : { z r} surrounding the zi ’s.To see this, note that the extension of each individual motion zi (t) yieldsa “spin” which is the composition of a left Dehn twist about the left-handcomponent of a regular neighborhood of the circle traced by zi (t) with the9

Figure 1: Simultanteously pushing the points labelled 4 and 5 around thepoint labelled 3 yields the right Dehn twist about the bold curve on theouter side of the thin curve passing through point 5.right Dehn twist about the corresponding right-hand component. The lefttwist resulting from the motion of z1 (t) is trivial, since this left boundarycomponent is peripheral, and for i 1, . . . , m 1 the right twist from themotion of zi cancels the left twist from the motion of zi 1 .Now suppose is a nonempty multicurve. Choose an element 2among the possibly several components of such that does not separate apair of elements of . Let V be a component of S 2 \ whose closure contains(the component V is unique if # 1). Pick 2 . Then bounds adistinguished Jordan domain D V . Let {pj0 , pj1 , . . . , pjm } D \ Pwhere 1 j0 j1 . . . jm n, so that j0 is the smallest indexof an element of P in D . Up to postcomposition with rotations aboutthe origin, there is a unique Riemann map: (D , pj0 ) ! (D, 0). Setzi (pji ), i 0, . . . , m, and transport the motion of the zi constructed inthe previous paragraph to a motion of {pi0 , pi1 , . . . , pim } in S 2 ; it is supportedin the interior of D . This motion gives a loop in the space Config(S 2 , P ).By construction, projects to the right Dehn twist aboutin thepure mapping class group. Suppose 1 , 2 are distinct elements of . IfD 1 \D 2 ; then clearly the elements represented by 1 and 2 commute.Otherwise, we may assume D 1 D 2 . By construction, the loop 2 ,10

thought of as a motion of points in the disk D 2 represents the centralelement in the braid group on #P \ D 2 strands, so it commutes with theelement represented by 1 .Remark: By choosing arbitrarily an element p1 2 P \ (S 2 D ), andupon identifying S 2 {p1 } with the plane R2 , we may view the pathsconstructed above as loops in the configuration space Config(R2 , P {p1 })whose fundamental group is then isomorphic to the pure braid group P Bn 1 .3Pullback invariantsMany of the objects we are concerned with arise nondynamically. We firstdefine them in the nondynamical setting (§3.1), and then consider the sameobjects in the dynamical case (§3.2).3.1No dynamicsAdmissible covers. Suppose A, B S 2 are finite sets, each containing atleast three points. A branched covering f : (S 2 , A) ! (S 2 , B) is admissibleif (i) B Vf , the set of branch values of f , and (ii) f (A) B; we do notrequire A f 1 (B).For the remainder of this subsection, we fix an admissible cover f :(S 2 , A) ! (S 2 , B). To (f, A, B) we associate the following objects

Pullback invariants of Thurston maps Sarah Koch, Kevin M. Pilgrim, and Nikita Selinger May 9, 2014 Abstract Associated to a Thurston map f : S2! S2 with postcritical set P are several di erent invariants obtained via pullback: a relation S P f S P on the set S P of free homotopy clas

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