Shackleton, Kenneth J.

Infoscience - École polytechnique fédérale de Lausanne

A computational acylindricity theorem for the mapping class group.Kenneth J. Shackletonkjs2006@alumni.soton.ac.ukhttp://www.is.titech.ac.jp/»kjshack5ABSTRACT: We study the usual action of the mapping class group of a surfaceon the 1-skeleton of Harvey's curve complex from a computational perspective.KEYWORDS: surface; curve complex; mapping class group.2000 MSC: 57M50 (primary); 30F60 (secondary).x0. Introduction.Let § be a compact, connected and orientable surface of genus g(§) and j@(§)jboundary components. In [Harv], Harvey associates to § a simplicial complex,C = C(§), called the curve complex. As well as encoding some of the asymptoticgeometry of the TeichmÄuller metric, the curve complex plays a central rolein the celebrated proof of Minsky and his collaborators of Thurston's endinglamination conjecture.The curve complex is constructed as follows. We shall say that a simpleloop on § is trivial only if it bounds a disc and peripheral only if it boundsan annulus containing one component of @§. Let X = X(§) be the set of allfree homotopy classes of non-trivial and non-peripheral simple loops on §. Theelements of X are referred to as curves. We take the vertex set of C to be Xand deem a family of pairwise distinct curves to span a simplex if and only ifany two of its curves can be realised disjointly.With the exception of only seven cases, namely either § is a sphere andj@§j · 4 or § is a torus and j@§j · 1, the set X(§) is non-empty and the curvecomplex is connected. For these non-exceptional cases, it can be veri¯ed thatthe dimension · = ·(§) of C(§) is equal to 3g(§) + j@§j ¡ 4. We see that § isin fact non-exceptional if and only if ·(§) is positive.When this is the case, the curve complex can be endowed with a path-metric by ¯rst declaring each edge to have length equal to 1 and then by takingeuclidean simplices. All that is important in this paper is the 1-skeleton, G, ofthe curve complex with which C is quasi-isometric and where distances betweenvertices are integers. We refer to this simplicial graph as the curve graph. Alldistances will be taken in this graph, whose own path-metric we denote by d. Itis to be noted that the curve graph is nowhere locally ¯nite and, when ·(§) ¸ 2,between any two vertices of distance at least 2 there always exist in¯nitely manygeodesic paths.The mapping class group of §, denoted Map(§), we de¯ne as the groupof all self-homeomorphisms of § modulo homotopy. This group has a naturalcocompact action on the curve graph and this has been exploited by numerousauthors in numerous ways; see [Hare] or [I] by way of example.The purpose of this paper is to prove a new theorem regarding the nature ofthis action. There have been at least two studies along these lines already, begin-ning with the work of Bestvina-Fujiwara [BeF] who, prompted by an argument1of Luo itself an adaptation of an argument of Kobayashi, establish a certainweak proper discontinuity (WPD) property. This is an important result, forBestvina-Fujiwara use this to prove that the dimension of the second boundedcohomology of the mapping class group, and any one of its non-virtually abeliansubgroups, is in¯nite.Inspired by their arguments, Bowditch [Bo] proves the acylindricity of thisaction: the number of mapping classes moving a \long" geodesic a \small"distance is uniformly bounded in terms of the topology of the surface. This ismuch stronger than WPD, where the number of such mapping classes is onlyassumed to be ¯nite and where one end of the geodesic is assumed to be theimage of the other under a pseudo-Anosov mapping class raised to a su±cientlyhigh power. Many of the interesting groups that admit an acylindrical actionon a hyperbolic metric space necessarily contain free groups of rank 2 (and thusof arbitrarily high rank); see [F].At some stage, both proofs from [BeF] and [Bo] make essential use of pass-ing from a sequence of curves to a limiting lamination to ultimately derive acontradiction and, as such, all the information found this way would appear tobe non-computable. Our main result may be viewed as a computational alter-native to the acylindricity theorem of Bowditch.Theorem 1 There is a computable function F : N4 ! N such that the followingholds. Suppose § is non-exceptional. Let r be any non-negative integer. Then,for any two curves ® and ¯ with d(®; ¯) ¸ 70r + 5, the number of mappingclasses h 2 Map(§) satisfying d(®; h®) · r and d(¯; h¯) · r is bounded aboveby F (¶(®; ¯); r; g(§); j@§j).We remark the lower bound 70r+5 can most likely be improved. Neverthe-less, this is stronger than the WPD property found by Bestvina-Fujiwara, andlogically independent of Bowditch's acylindricity theorem. While the boundingfunction, F , does depend on intersection number, whereas the bound given inBowditch's acylindricity theorem does not, in contrast it is computable and ourargument is entirely elementary. Of much interest would be to ¯nd an argu-ment that overcomes this trade-o®, yielding both computability and uniformitysimultaneously.ACKNOWLEDGEMENTS: The author wishes to thank the Centre InterfacultaireBernoulli, EPFL, for its warm and generous hospitality. He also wishes to thankBrian Bowditch for many interesting and helpful conversations.x1. Background and de¯nitions.Before starting our proof of Theorem 1 in x2 we recall a few de¯nitions, themost important of which is that of a tight multigeodesic after Masur-Minsky.x1:1 Curves and multicurves. Given any two curves ® and ¯, their intersec-tion number ¶(®; ¯) is de¯ned equal to minfja\ bj : a 2 ®; b 2 ¯g. A multicurveis a non-empty collection of pairwise distinct curves of pairwise zero intersection2number, and as such spans a simplex in the curve complex. The intersectionnumber of two multicurves is to be de¯ned additively.x1:2 Paths and multipaths. A path in the curve graph is a sequence ofvertices (°0; : : : ; °n) such that, for each index i 2 f0; : : : ; n ¡ 1g, the curves°i and °i+1 are distinct and have zero intersection number, thus spanning anedge. A geodesic in the curve graph is then a path whose length is precisely thedistance between its ends.A multipath (º0; : : : ; ºn) is a sequence of multicurves such that (°0; : : : ; °n)is a path for each curve °i 2 ºi over each index i 2 f0; : : : ; ng. We shall refer toeach multicurve ºi as a vertex of the multipath, following the language of Masur-Minsky. A multigeodesic is a multipath (º0; : : : ; ºn) such that (°0; : : : ; °n) is ageodesic, for each curve °i 2 ºi over each index i 2 f0; : : : ; ng.A multipath (º0; : : : ; ºn) is said to be k-embedded only if, for any two indicesi and j with ji ¡ jj ¸ k, we have d(°i; °j) ¸ k for each curve °i 2 ºi and eachcurve °j 2 ºj . Finally, a multipath (º0; : : : ; ºn) is said to be k-quasigeodesiconly if n · d(°0; °n) + k for each curve °0 2 º0 and each curve °n 2 ºn.x1:3 Tight multigeodesics. The notion of a tight multigeodesic was intro-duced by Masur-Minsky [MaMi] to address the lack of local ¯niteness in thecurve graph. Though there always exist in¯nitely many geodesics connectingany pair of vertices of distance at least 2, whenever ·(§) ¸ 3, Corollary 6.4of [MaMi] states that the number of tight multigeodesics connecting any givenpair of vertices is always ¯nite. A slightly more general de¯nition was latero®ered by Bowditch in [Bo], where the ¯niteness result of Masur-Minsky isstrengthened. In [S], the author o®ers computable bounds on the number oftight multigeodesics, taking either de¯nition, connecting any pair of vertices.In this paper, we shall work exclusively with Masur-Minsky's de¯nition,recalled as follows. For any two multicurves º0 and º2 connected by a multi-geodesic of length 2, we realise º0 and º2 in general position and denote by Nan open regular neighbourhood of their union. Attach to N all the discs and allthe one-holed discs, containing a component of @§, complementary to N . Thefree homotopy class of the boundary of the resulting subsurface is a well-de¯nedmulticurve associated to º0 and º2, so long as we disregard multiplicity. Wedenote this multicurve by @(º0; º2), and refer to it as the relative boundary ofº0 and º2. An example of a relative boundary is given below in Figure 1.º0º1 º1º1º1º2Figure 1: The multicurve º1 is the relative boundary of the curves º0 and º2.3We are now ready to recall the relevant de¯nition of a tight multigeodesic.De¯nition 2 (MaMi) A multigeodesic (º0; : : : ; ºn) is said to be tight at in-dex j, for j 2 f1; : : : ; n ¡ 1g, only if ºj = @(ºj¡1; ºj+1). We shall say that(º0; : : : ; ºn) is tight only if tight at each such index.Let us now suppose that (º0; º1; º2; º3) is a multigeodesic tight at º2. Wereplace º1 with the relative boundary of º0 and º2. It is veri¯ed in the proofof Lemma 4.5 from [MaMi] that this does not a®ect tightness at º2, so that(@(º0; º2); º2; º3) is a tight multigeodesic. Moreover, we are free to replace º0with a second multicurve º00 without a®ecting tightness at º2 in any way, solong as (º00; º1; º2; º3) is a multigeodesic. This is a fundamental observation, aproof of which is implicit in the fourth paragraph of the proof of Lemma 4.5from [MaMi].As we shall be extensively referring to this tightening procedure let ussummarise the above discussion with two statements, the second of which impliesthe ¯rst.Lemma 3 (MaMi) Let (º0; º1; º2; º3) be a multigeodesic tight at º2, and letº01 denote the relative boundary @(º0; º2) of º0 and º2. Then, (º0; º01; º2; º3) isa tight multigeodesic.Lemma 4 (MaMi) Let (º0; º1; º2; º3) be a multigeodesic tight at º2, and letº00 denote any multicurve such that (º00; º1; º2; º3) is a multigeodesic. Let º01denote the relative boundary @(º00; º2) of º00 and º2. Then, (º00; º01; º2; º3) is atight multigeodesic.The tightening procedure is therefore very robust, and can be applied to thevertices of a given multigeodesic in any order to produce a tight multigeodesicwith the same ends. As observed in [MaMi] it is conceivable that tightening ina di®erent order may produce distinct tight multigeodesics of the same ends,though this author is not aware of any such examples existing in the literature.As we shall see in this paper, the tightening procedure can generalised toaccept a larger collection of multipaths. While we also have to keep in mindthe technical di±culties of working with multicurves, and not just single curves,the only real catch is the multipath that results from tightening a 3-embeddedmultipath at a single vertex might not be 3-embedded. This, however, presentsan obvious place to locally shortcut the resulting multipath. Combining the twooperations of tightening and shortcutting, we have a ¯nite-time algorithm whichreturns a tight and 3-embedded multipath sharing the same ending vertices asthe input multipath. What is more, \much" of the original multipath can remainintact if it happens to be geodesic over much of its length. Pertinent examplesinclude multipaths obtained as the concatenation of a \short" multigeodesic, a\long" multigeodesic, and then another short multigeodesic. The full details ofthis procedure amount to the proof of Lemma 5 in x2.4x2. Proof of Theorem 1.Let us introduce the notation T (®; ¯) for the set of all curves belonging to atight multigeodesic connecting ® to ¯. For r a non-negative integer, we de¯neT (®; ¯; r) to be equal to the union of all sets T (±; °) where ± 2 B(®; r) and° 2 B(¯; r). For a second non-negative integer s, we de¯ne T (®; ¯; r; s) to beequal to the setf´ 2 T (®; ¯; r) : d(®; ´) ¸ s; d(´; ¯) ¸ sg:The supporting lemma, which may hold independent interest, o®ers com-putable bounds on the number of curves that can lie on a tight multigeodesic,su±ciently far from its ends, that connects two bounded subsets su±ciently farapart in the curve graph. We momentarily postpone its proof.Lemma 5 There exists a computable function F1 : N4 ! N such that the fol-lowing holds. Let r be any non-negative integer and let ® and ¯ be any twocurves. Then,jT (®; ¯; r; 30r)j · F1(¶(®; ¯); r; g(§); j@§j):Given this, a proof of Theorem 1 is soon completed.Proof: [of Theorem 1] We start by connecting ® to ¯ by a tight multigeodesicz. For any mapping class h 2 Map(§) such that d(®; h®) · r and d(¯; h¯) · r,the h-translate of z is a tight multigeodesic whose curves all belong to theset T (®; ¯; r). In particular, our choice of lower bound for d(®; ¯) implies theexistence of a pair of curves f±; ´g contained in the vertices of z and of distance3 such that each of their translates by each such mapping class h is contained inT (®; ¯; r; 30r). According to Lemma 5, the number of such translates of either± or ´ is at most F1(¶(®; ¯); g(§); j@§j). Combining this fact with Lemma 7.4from [Bo], asserting that the stabiliser in Map(§) of a pair of curves at leastdistance 3 apart in the curve graph is uniformly and explicitly bounded in termsof ·(§), we complete a proof of Theorem 1. }To prove Lemma 5 we shall rely on the following theorem, a proof of whichis implicit in the proof of Theorem 6 from [S]. The argument found therein isby contradiction, and constructs a path of length at most 2 between two curvesof distance at least 3 and as such needs only the 3-embedded property of tightmultigeodesics to apply.Theorem 6 (S) There exists a computable function F2 : N4 ¡! N such thatthe following holds. Suppose § is non-exceptional, and let k denote any non-negative integer. Then, for any k-quasigeodesic, 3-embedded and tight multipath(º0; : : : ; ºn) we have, for each index j,¶(º0; ºj) + ¶(ºj ; ºn) · F2(¶(º0; ºn); k; g(§); j@§j):5Proof: [of Lemma 5] For any multigeodesic z beginning in B(®; r) and ending inB(¯; r), suppose there exists a tight 2r-quasigeodesic multipath q connecting ®to ¯, containing every curve from z at least distance 30r from both ® and ¯, andwhich is 3-embedded. According to Theorem 6, each vertex of q has intersectionnumber with ® and intersection number with ¯ uniformly bounded in terms ofthe quasigeodesic parameter of q, and hence in terms of r, and ¶(®; ¯), g(§)and j@§j. Each vertex of z at least distance 30r from both ® and ¯ thus hassimilarly bounded intersection number with ® and with ¯. This is enough toexplicitly bound the cardinality of T (®; ¯; r; 30r).All that remains is to establish the existence of such a multipath, whoselength is at most d(®; ¯) + 2r, and we do so by a careful surgery argument.To this end we will introduce two new and non-symmetric relations betweenmultipaths, denoted ! and ).The relation ! will describe the shortening of a given multipath whenit fails to be 3-embedded. The relation ) will describe the tightening of a 3-embedded multipath at a single vertex. When the original multipath is geodesicover much of its length these operations are localised and much of the originalmultipath remains intact. In particular, this is the case when our multipathis the concatenation of a short multigeodesic, a long multigeodesic, and thenanother short multigeodesic each ending on single curves.The interaction of! and) need not be entirely well-behaved for the goodwork done by a sequence of! might be undone by a single). That is the mul-tipath that results from tightening a 3-embedded multipath once might not be3-embedded and will be in need of further shortening. However, such complica-tions only occur a bounded number of times, for path length strictly decreasesin the direction of !. In addition, this behaviour is prevented from seepinginto the middle of our multipaths by Lemma 4. Thus our tightening procedurewill stabilise, at which point we have found a tight and 3-embedded multipathconnecting ® to ¯ and containing much of the original tight multigeodesic z, asrequired.To be more precise, for two multipaths p and p0 with common ends we shallwrite p! p0 only if there exist consecutive vertices (¹0; : : : ; ¹i) of p, with i ¸ 2,and consecutive vertices (!0; : : : ; !j) of p0, with j · minf2; i¡ 1g, such that:² (!0; : : : ; !j) is a tight multigeodesic;² p¡ f¹0; : : : ; ¹ig = p0 ¡ f!0; : : : ; !jg;² !0 µ ¹0, and² !j µ ¹i.Note that the length of p is strictly greater than the length of p0, andthat p0 may fail to be tight at the two vertices of p0 either side of the shortcut(!0; : : : ; !j) and its ends. An instance of the relation ! is depicted below inFigure 2.For a 3-embedded multipath q and a multipath q0 with the same ends butdistinct from q, we shall write q ) q0 only if q0 results from tightening q at asingle vertex. Notice even if q is 3-embedded and q ) q0 we do not presume q0to be 3-embedded. If q ) q0, then q and q0 have the same length and do agreeon all but one vertex.6pp0!0 µ ¹0 !1¹1¹9 ¶ !2Figure 2: The multipath p0 is formed by shortcutting the multipath p.Choose any tight multigeodesic z connecting a curve in B(®; r) to a curvein B(¯; r) and form a new multipath p by concatenating a tight multigeodesicconnecting ® to the end of z in B(®; r), the tight multigeodesic z, and a tightmultigeodesic connecting the end of z in B(¯; r) to ¯. Now take any sequencep = p00 ! p01 ! ¢ ¢ ¢ ! p0k0 ) p10 ! p11 ! ¢ ¢ ¢ ! ptkt ) pt+10 ;or ¢ ¢ ¢ ! ptkt if ptkt is already tight, that is of maximal length and such thatpiki is 3-embedded for each i. Note the number of ! relations appearing in thesequence is at most 2r, for path length strictly decreases in the direction of !.Moreover, the number of ) in the sequence is at most 8r since, by Lemma 4,for each ! we are only ever required to tighten at the ends of the shortcuttingmultipath (contributing at most 4r) and then at the two vertices either sideof the shortcutting multipath (contributing at most 4r). We conclude that thelength of the sequence is at most 2r + 8r, or 10r.Let us denote the ¯nal multipath in our sequence by q. By the maximalityof our sequence, q is tight and 3-embedded. Since p can only fail to be 3-embedded inside the two balls of radius r + 2 about either end of z, so everycurve belonging to z and at distance at least 30r from both ® and ¯ also belongsto q. With this, we conclude a proof of Lemma 5. }7References.[BeF] M. Bestvina, K. Fujiwara, Bounded cohomology of subgroups of the map-ping class groups : Geometry & Topology 6 (2002) 69 - 89.[Bo] B. H. Bowditch, Tight geodesics in the curve complex : to appear in Invent.Math.[F] K. Fujiwara, Subgroups generated by two elements : preprint (2007).[Hare] J. L. Harer, The virtual cohomological dimension of the mapping classgroups of an orientable surface : Invent. Math. 84 (1986) 157-176.[Harv] W. J. Harvey, Boundary structure of the modular group : in \Riemannsurfaces and related topics: Proceedings of the 1978 Stony Brook Conference"(ed. I. Kra, B. Maskit), Annals of Mathematical Studies No. 97, PrincetonUniversity Press (1981) 245-251.[I] N. V. Ivanov, Mapping class groups : in \Handbook of geometric topology"(ed. R. Daverman and R. Sher), Elsevier (2001) 523-633.[MaMi] H. A. Masur, Y. N. Minsky, Geometry of the complex of curves II:Hierarchical structure : Geom. Funct. Anal. 10 (2000) 902-974.[S] K. J. Shackleton, Tightness and computing distances in the curve complex :preprint, Southampton (2004).Kenneth J. ShackletonDepartment of Mathematical and Computing SciencesTokyo Institute of Technology2-12-1 O-okayamaMeguro-kuTokyo 152-8552Japane-mail: shackleton.k.aa@m.titech.ac.jpe-mail: kjs2006@alumni.soton.ac.ukURL: http://www.is.titech.ac.jp/»kjshack58

A Computational Acylindricity Theorem for the Mapping Class Group

http://infoscience.epfl.ch/record/126354/files/ShackletonKenneth.pdf

A Computational Acylindricity Theorem for the Mapping Class Group

Abstract

Similar works

Full text

Available Versions

Infoscience - École polytechnique fédérale de Lausanne