Kakimizu complexes of Surfaces and 3-Manifolds

Transcription

1 Kakimizu omplexes of Surfaes and 3-Manifolds Jennifer Shultens April 29, 2016 Abstrat The Kakimizu omplex is usually defined in the ontext of knots, where it is known to be quasi-eulidean. We here generalize the definition of the Kakimizu omplex to surfaes and 3-manifolds (with or without boundary). Interestingly, in the setting of surfaes, the omplexes and the tehniques turn out to repliate those used to study the Torelli group, i.e., the nonlinear subgroup of the mapping lass group. Our main results are that the Kakimizu omplexes of a surfae are ontratible and that they need not be quasi-eulidean. It follows that there exist (produt) 3-manifolds whose Kakimizu omplexes are not quasi- Eulidean. The existene of Seifert s algorithm, disovered by Herbert Seifert, proves, among other things, that every knot admits a Seifert surfae. I.e., for every knot K, there is a ompat orientable surfae whose boundary is K. It is worth noting that the existene of a Seifert surfae for a knot K also follows from the existene of submanifolds representing homology lasses of manifolds or pairs of submanifolds, in this ase the pair (K, S 3 ). This point of view proves useful in generalizing our understanding of Seifert surfaes to other lasses of surfaes in 3-manifolds. Adding a trivial handle to a Seifert surfae produes an isotopially distint surfae. Adding additional handles produes infinitely many isotopially distint surfaes. These are not the multitudes of surfaes of primary interest here. The multitudes of surfaes of primary interest here are, for example, the infinite olletion of Seifert surfaes produed by Eisner, see [4]. Eisner realized that spinning a Seifert surfae around the deomposing annulus of a onneted sum of two non fibered knots produes homeomorphi but non isotopi Seifert surfaes. This abundane of Seifert surfaes led Kakimizu to define a omplex, now named after him, whose verties are isotopy lasses of Seifert surfaes of a given knot and whose n-simplies are (n + 1)- tuples of verties that admit pairwise disjoint representatives. Our understanding of the topology and geometry of the Kakimizu omplex ontinues to evolve. Both Kakimizu s work and, independently, a result of Sharlemann and Thompson, imply that the Kakimizu omplex is onneted. (See [15] and [24].) Sakuma and Shakleton exhibit diameter bounds in terms of the genus of a knot. 1

2 (See [23].) P. Przytyki and the author establish that the Kakimizu omplex is ontratible. (See [20].) Finally, Johnson, Pelayo and Wilson prove that the Kakimizu omplex of a knot is quasi-eulidean. (See [22].) This paper grew out of a desire to study onrete examples of Kakimizu omplexes of 3-manifolds other than knot omplements. A natural ase to onsider is produt manifolds, where relevant information is aptured by the surfae fator. The hallenge lies in adapting the idea of the Kakimizu omplex to a more general setting: odimension 1 submanifolds of n-manifolds. As it turns out, in the ase of 1-dimensional submanifolds of a surfae, the Kakimizu omplexes are related to the homology urve omplexes investigated by Hather (see [9]), Irmer (see [13]), Bestvina-Bux-Margalit (see [2]) and Hather- Margalit (see [10]) disussed in Setion 2. These omplexes are of interest in the study of the Torelli group, whih is the kernel of the ation of the mapping lass group of a manifold on the homology of the manifold. The Torelli group of a surfae, in turn, ats on the homology urve omplexes. This group ation has been used to study the topology of the Torelli group of a surfae, for instane by Bestvina-Bux- Margalit in their investigation of the dimension of the Torelli group (see [2]), by Irmer in The Chillingworth lass is a signed stable length (see [12]), by Hather-Margalit in Generating the Torelli group (see [10]), and by Putman in Small generating sets for the Torelli group (see [21]). Hather proved that the homology urve omplex is ontratible and omputed its dimension. Irmer studied geodesis of the homology urve omplex and exhibited quasi-flats. These insights guide our investigation of the Kakimizu omplex of a surfae. Speifially, we prove similar, and in some ases analogous, results in the setting of the Kakimizu omplex of a surfae. Our main results are that the Kakimizu omplexes of a surfae are ontratible and that they need not be quasi-eulidean. One example stands out: The Kakimizu omplex of a genus 2 surfae. In [2], Bestvina-Bux-Margalit reprove a theorem of Mess, that the Torelli group of a genus 2 surfae is an infinitely generated free group. They do so by showing that it ats on a tree with infinitely many edges emerging from eah vertex. As it turns out, the Kakimizu omplex of the genus 2 surfae is also a tree with infinitely many edges emerging from eah vertex. In partiular, the Kakimizu omplex of the genus 2 surfae is Gromov hyperboli. A produt manifold with the genus 2 surfae as a fator will thus also have some Gromov hyperboli Kakimizu omplexes. This is interesting as it shows that in addition to examples of 3-manifolds with quasi- Eulidean Kakimizu omplexes, as proved by Johnson-Pelayo-Wilson, there are 3- manifolds with Gromov hyperboli Kakimizu omplexes. Kakimizu omplexes exhibit more than one geometry! I wish to thank Misha Kapovih, Allen Hather, Dan Margalit and Piotr Przytyki for helpful onversations and the referee for helpful omments. This researh was supported by a grant from the NSF. Muh of the work was arried out at the Max Plank Institute for Mathematis in Bonn. I thank the Max Plank Institute for Mathematis in Bonn for its hospitality and the Hausdorff Institute for Mathematis for logistial support. 2

3 1 The Kakimizu omplex of a surfae The work here follows in the footsteps of [20]. Whereas the setting for [20] is surfaes in 3-manifolds, the setting here is 1-manifolds in 2-manifolds. It is worth pointing out that although we disuss only 1-manifolds in 2-manifolds and 2-manifolds in 3-manifolds, the definitions and arguments arry over verbatim to the setting of odimension 1 submanifolds in manifolds of any dimension. Reall that an element of a finitely generated free abelian group G is primitive if it is an element of a basis for G. In the following we will always assume: 1) S is a ompat (possibly losed) onneted oriented 2-manifold; 2) α is a primitive element of H 1 (S, S, Z). Definition 1. A Seifert urve for (S, α) is a pair (w, ), where is a union, 1 n, of pairwise disjoint oriented simple losed urves and ars in S and w is an n-tuple of natural numbers (w 1,..., w n ) suh that the homology lass w 1 [[ 1 ]] + + w n [[ n ]] equals α. Moreover, we require that S\ is onneted. We all the underlying urve of (w, ). We will denote w 1 [[ 1 ]] + + w n [[ n ]] by w. Figure 1: A Seifert urve (weights are 1) Our definition of Seifert urve disallows null homologous subsets. Indeed, a null homologous subset would bound a omponent of S\ and would hene be separating. In fat, ontains no bounding subsets. Conversely, if w d = α and d ontains no bounding subsets, then S\d is onneted. Lemma 1. If (w, ) represents α, then w is determined by the underlying urve. Proof: Suppose that (w, ) and (w, ) represent α, where w = (w 1,, w n ) and w = ((w ) 1,, (w ) n ). Then w 1 [[ 1 ]] + + w n [[ n ]] = α = (w ) 1 [[ 1 ]] + + (w ) n [[ n ]], hene (w 1 (w ) 1 )[[ 1 ]] + (w n (w ) n )[[ n ]] = 0. 3

4 Sine has no null homologous subsets, this ensures that w 1 (w ) 1 = 0,..., w n (w ) n = 0. Thus w 1 = (w ) 1,, w n = (w ) n. Sine the underlying urve of a Seifert urve (w, ) determines w, we will oasionally speak of a Seifert urve, when w does not feature in our disussion. Definition 2. Given a Seifert urve (w, ) we denote the urve obtained by replaing, for all i, the urve i with w i parallel omponents of i, by h(w, ). This defines a funtion from Seifert urves to unweighted urves. Conversely, let d = d 1 d m be a disjoint union of (unweighted) pairwise disjoint simple losed urves and ars suh that parallel omponents are oriented to be parallel oriented urves and ars. We denote the weighted urve obtained by replaing parallel omponents with one weighted omponent whose weight is equivalent to the number of these parallel omponents by h 1 (d). Definition 3. For eah pair (S, α), the isomorphism between H 1 (S, S) and H 1 (S) identifies an element a of H 1 (S) orresponding to α that lifts to a homomorphism h a : π 1 (S) Z. We denote the overing spae orresponding to N α = kernel(h a ) by (p α, Ŝα, S), or simply (p, Ŝ, S), and all it the infinite yli overing spae assoiated with α. We now desribe the Kakimizu omplex of (S, α). As verties we take Seifert urves (w, ) of (S, α), onsidered up to isotopy of underlying urves. We write [(w, )]. Consider a pair of verties v, v and representatives (w, ), (w, ). Here S\ and S\ are onneted, hene path-onneted. It follows that lifts of S\ and S\ to the overing spae assoiated with α are simply path omponents of p 1 (S\) and p 1 (S\ ). We obtain a graph Γ(S, α) by spanning an edge e = (v, v ) on the verties v, v if and only if the representatives (w, ), (w, ) of v, v an be hosen so that a lift of S\ to the overing spae assoiated with α intersets exatly two lifts of S\. (Note that in this ase and are neessarily disjoint.) See Figure 2. Definition 4. Let X be a simpliial omplex. If, whenever the 1-skeleton of a simplex σ is in X, the simplex σ is also in X, then X is said to be flag. Definition 5. The Kakimizu omplex of (S, α), denoted by Kak(S, α) is the flag omplex with Γ(S, α) as its 1-skeleton. Remark 6. The Kakimizu omplex is defined for a pair (S, α). For simpliity we use the expression the Kakimizu omplex of a surfae in general disussions, rather than the more umbersome the Kakimizu omplex of a pair (S, α), where S is a surfae and α is a primitive element of H 1 (S, S, Z). Note that the Kakimizu omplex of a surfae is thus unique only in onjuntion with a speified α. 4

5 Figure 2: Two Seifert urves orresponding to verties of distane 1 (weights are 1) Figure 3 provides an example of a pair (w, ), (w, ) of disjoint (disonneted) Seifert urves that do not span an edge. The ar from one side of to the other side of intersets twie with the same orientation and a lift of S\ will hene meet at least three distint lifts of S\. For a 3-dimensional analogue of Figure 3, see [1]. Figure 3: Two Seifert urves orresponding to verties of distane stritly greater than 1 (weights are 1) Example 1. The Kakimizu omplexes of the disk and sphere are empty. The annulus has a non-empty but trivial Kakimizu omplex Kak(A, α) onsisting of a single vertex. Speifially, let A = annulus, and α a generator of H 1 (A, A, Z) = Z. Then α is represented by a spanning ar with weight 1. The spanning ar is, up to isotopy, the only possible underlying urve for a representative of α. Thus Kak(A, α) onsists of a single vertex. Similarly, the torus has non-empty but trivial Kakimizu omplexes, eah onsisting of a single vertex. Speifially, let T = torus, and β a primitive element of H 1 (T, Z) = Z Z. Again, there is, up to isotopy, only one underlying urve for representatives of β. There are infinitely many hoies for β, but in eah ase, Kak(T, β) onsists of a single vertex. 5

6 Having understood the above Examples, we restrit our attention to the ase where S is a ompat orientable hyperboli surfae with geodesi boundary for the remainder of this paper. Definition 7. Let (w, ) and (w, ) be Seifert urves. We say that (w, ) and (w, ) (or simply and ) are almost disjoint if for all i, j the omponent i of and the omponent j of are either disjoint or oinide. Remark 8. Let σ be a simplex in Kak(S, α) of dimension n. Denote the verties of σ by v 0,..., v n and let 0,..., n be geodesi representatives of the underlying urves of Seifert urves for v 0,..., v n suh that ar omponents of 0,..., n are perpendiular to S. It is a well known fat that losed geodesis that an be isotoped to be disjoint must be disjoint or oinide. The same is true for the geodesi ars onsidered here, and ombinations of losed geodesis and geodesi ars, beause their doubles are losed geodesis in the double of S. Hene, for all pairs i, j, the omponent i of and the omponent j of are either disjoint or oinide. Definition 9. Consider Kak(S, α). Let (p, Ŝ, S) be the infinite yli over of S assoiated with α. Let τ be the generator of the group of overing transformations of (p, Ŝ, S) (whih is Z) orresponding to 1. Note that τ is anonial up to sign. Let (w, ), (w, ) be Seifert urves in (S, α). Let S 0 denote a lift of S\ to Ŝ, i.e., a path omponent of p 1 (S\). Set S i = τ i (S 0 ), i = losure(s i ) losure(s i+1 ). Let S 0 be a lift of S\ to Ŝ. Set d K(, ) = 0 and for, set d K (, ) equal to one less than the number of translates of S 0 met by S 0. Let v, v be verties in Kak(S, α). Set d k (v, v) = 0 and for v v set d K (v, v ) equal to the minimum of d K (, ) for (w, ), (w, ) representatives of v, v. Definition 10. Let C, D be disjoint separating subsets of Ŝ. We say that D lies above C if D lies in the omponent of Ŝ\C ontaining τ(c). We say that D lies below C if D lies in the omponent of Ŝ\C ontaining τ 1 (C). Remark 11. Here d K (, ) is finite: Indeed, w = w = α and so [(w, )], [(w, )] are in the kernel, N α, of the ohomology lass dual to α. Speifially, the ohomology lass dual to α is represented by the weighted intersetion pairing with (w, ) and also the weighted intersetion pairing with (w, ). Thus, let j be a omponent of, then the value of the ohomology lass dual to α evaluated at [[ j ]] is given by the weighted intersetion pairing of (w, ) with j whih is 0. Likewise for other omponents of and. Thus eah omponent of, lies in the kernel of this homomorphism and hene in N α. Thus lifts of, are homeomorphi to,, respetively, in partiular, they are ompat 1-manifolds. It follows that d K (, ) is finite, whene for all verties v, v of Kak(S, α), d K (v, v ) is also finite. It is not hard to verify, but important to note, the following theorem (see [15, Proposition 1.4]): Proposition 1. The funtion d K is a metri on the vertex set of Kak(S, α). 6

7 2 Relation to homology urve omplexes In [9], Hather introdues the yle omplex of a surfae: By a yle in a losed oriented surfae S we mean a nonempty olletion of finitely many disjoint oriented smooth simple losed urves. A yle is redued if no subyle of is the oriented boundary of one of the omplementary regions of in S (using either orientation of the region). In partiular, a redued yle ontains no urves that bound disks in S, and no pairs of irles that are parallel but oppositely oriented. Define the yle omplex C(S) to be the simpliial omplex having as its verties the isotopy lasses of redued yles in S, where a set of k + 1 distint verties spans a k-simplex if these verties are represented by disjoint yles 0,..., k that ut S into k + 1 obordisms C 0,..., C k suh that the oriented boundary of C i is i+1 i, subsripts being taken modulo k + 1, where the orientation of C i is indued from the given orientation of S and i denotes i with the opposite orientation. The obordisms C i need not be onneted. The faes of a k-simplex are obtained by deleting a yle and ombining the two adjaent obordisms into a single obordism. One an think of a k-simplex of C(S) as a yle of yles. The ordering of the yles 0,..., k in a k-simplex is determined up to yli permutation. Cyles that span a simplex represent the same element of H 1 (S) sine they are obordant. Thus we have a well-defined map π 0 : C(S) H 1 (S). This has image the nonzero elements of H 1 (S) sine on the one hand, every yle representing a nonzero homology lass ontains a redued subyle representing the same lass (subyles of the type exluded by the definition of redued an be disarded one by one until a redued subyle remains), and on the other hand, it is an elementary fat, left as an exerise, that a yle that represents zero in H 1 (S) is not redued. For a nonzero lass x H 1 (S) let C x (S) be the subomplex of C(S) spanned by verties representing x, so C x (S) is a union of omponents of C(S). See [9, Page 1]. Lemma 2. When both are defined, i.e., when S is losed, onneted, of genus at least 2, and α is primitive, V ert(kak(s, α)) is isomorphi to a proper subset of V ert(c α (S)). Proof: Let v be a vertex of Kak(S, α). If we hoose a representative (w, ), then h(w, ) is a disjoint olletion of (unweighted) urves and ars. The requirement on the Seifert urve (w, ), that S\ be onneted implies that the multi-urve h(w, ) is redued and thus represents a vertex of C α (S). Abusing notation slightly, we denote the map from V ert(kak(s, α)) to V ert(c α ) thus obtained by h. There is an inverse, h 1, defined on the image of h, hene h is injetive. It is not hard to identify redued multi-urves that ontain bounding subsets that are not the oriented boundary of a subsurfae. Hene V ert(kak(s, α)) is a proper subset of V ert(c α (S)). Lemma 3. Suppose that S is hyperboli and let σ be an n-simplex in Kak(S, α). Denote the verties of σ by v 0,..., v n. Then there are representatives of v 0,..., v n with underlying urves 0,..., n suh that the following hold: 7

8 1. i j = i j; 2. S\( 0 n ) is partitioned into subsurfaes P 0,..., P n suh that P i = i i 1. Proof: Let (p, Ŝ, S) be the overing spae assoiated with α and let σ be a simplex in Kak(S, α). Let 0,..., n be geodesi representatives of the underlying urves of v 0,..., v n suh that ar omponents of 0,..., n are perpendiular to S. By Remark 8, i and j are almost disjoint i j. Consider a lift S 0 of S\ 0 to Ŝ. For eah j 0, j lifts to a separating olletion ĉ j of simple losed urves and simple ars. Moreover, sine S 0 is homeomorphi to S\ 0, the lifts ĉ i, ĉ j are almost disjoint as long as i j. By reindexing 0,..., n if neessary and performing small isotopies that pull apart equal omponents, we an thus ensure that ĉ i lies above ĉ j for i > j. Note that the lift of S\ 0 is homeomorphi to S\ 0. In partiular, i j = i j. Moreover, the surfae with interior below ĉ i and above ĉ i 1 projets to a subsurfae P i of S for i = 1,..., n. The subsurfaes P 1,..., P n exhibit the required properties. (2,1) (1, 2 ) (1,d 3) (2, d (2, d 1 ) ) 2 Figure 4: An edge of Kak(S, α) that does not map into C α Remark 12. When w 0,..., w n = 1, Lemma 3 ensures that 0 = h(w 0, 0 ) = h(1, 0 ),..., n = h(w n, n ) = h(1, n ) form a yle of yles. In this ase h extends over the simplex σ to produe a simplex h(σ) in C α. However, h does not extend over simplies in whih weights are not all 1. See Figure 4. Hather proves that for eah x H 1 (S), C x (S) is ontratible. (In partiular, it is therefore onneted and hene onstitutes just one omponent of C(S).) In Setion 4 we prove an analogous result for Kak(S, α), using a tehnique from the study of the Kakimizu omplex of 3-manifolds. The yli yle omplex and the Kakimizu omplex are simpliial omplexes. The omplex defined by Bestvina-Bux-Margalit (see [2]) is not simpliial, but an be subdivided to obtain a simpliial omplex. See the final omments in [10, Setion 2]. There is a subomplex of the yli yle omplex that equals this subdivision of the omplex defined by Bestvina-Bux-Margalit. This is the omplex of interest in the ontext the Torelli group. Bestvina-Bux-Margalit exploited the ation of the 8

9 Torelli group on this omplex to ompute the dimension of the Torelli group. Hather- Margalit used it to identify generating sets for the Torelli group. In [13], Irmer defines the homology urve omplex of a surfae: Suppose S is a losed oriented surfae. S is not required to be onneted but every omponent is assumed to have genus g 2. Let α be a nontrivial element of H 1 (S, Z). The homology urve omplex, HC(S, α), is a simpliial omplex whose vertex set is the set of all homotopy lasses of oriented multi-urves in S in the homology lass α. A set of verties m 1,..., m k spans a simplex if there is a set of pairwise disjoint representatives of the homotopy lasses. The distane, d H (v 1, v 2 ), between two verties v 1 and v 2 is defined to be the distane in the path metri of the one-skeleton, where all edges have length one. (See [13, Page 1].) It is not hard to see the following (f, Remark 8 and Figure 3): Lemma 4. When both are defined, i.e., when S is losed, onneted, of genus at least 2 and α is primitive, Kak(S, α) is a subomplex of HC(S, α). Moreover, for verties v, v of Kak(S, α), d K (v, v ) d H (v, v ) Irmer shows that distane between verties of HC(S, α) is bounded above by a linear funtion on the intersetion number of representatives. The same is true for verties of the Kakimizu omplex. Irmer also onstruts quasi-flats in HC(S, α). Her onstrution arries over to the setting of the Kakimizu omplex. See Setion 6. 3 The projetion map, distanes and geodesis In [15], Kakimizu defined a map on the verties of the Kakimizu omplex of a knot. He used this map to prove several things, for instane that the metri, d K, on the verties of the Kakimizu omplex equals graph distane. (Quoted and reproved here as Theorem 2.) In [20], Kakimizu s map was rebranded as a projetion map. We wish to define π V ert(kak(s,α)) : V ert(kak(s, α)) V ert(kak(s, α)) on the vertex set of Kak(S, α). Let (p, Ŝ, S) be the infinite yli overing spae assoiated with α. Let v, v be verties in Kak(S, α) suh that v v. Here v = [(w, )] for some ompat oriented 1-manifold and v = [(w, )] for some ompat oriented 1-manifold. We may assume, in aordane with Definition 9 and Remark 11, that (w, ) and (w, ) are hosen so that d K (, ) = d K (v, v ). Define τ, S i, S i, i and, by analogy, i, as in Definition 9. Instead of working only with 0, we will now also work with h(w, 0). Take m = max{i S i+1 S 0 }. Consider a onneted omponent C of S m+1 S 0. Its frontier onsists of a subset of 0 and a subset of m. The subset of 0 lies above the 9

10 subset of m. In partiular, C lies above m and below 0, hene the orientations of the subset of 0 are opposite those of the subset of m. See Figure 5. Beause the subset of 0 and the subset of m obound C, they are homologous. It follows that the lowest omponents of the orresponding subset of h(w, 0) are also homologous to the subset of m. Replaing the lowest of the orresponding subsets of h(w, 0) with the subset of m and isotoping this portion of m to lie below m yields a multi-urve d 1 with the following properties: d 1 is homologous to h(w, 0) via a homology that desends to a homology in S (beause C is homeomorphi to a subset of S); d 1 has lower geometri intersetion number with m than h(w, 0); d 1 lies above h(w, 1) and an be isotoped to lie below and thus be disjoint from h(w, 0), moreover its projetion an be isotoped to be disjoint from h(w, ). For (x 1, e 1 ) = h 1 (d 1 ), we have x 1 e 1 homologous to w 0 via a homology that desends to a homology in S. See Figures 5, 6. Working with h 1 (d 1 ), d 1 instead of 0, h(w, 0) we perform suh replaements in suession to obtain a sequene of multi-urves d 1,..., d k suh that the following hold: d j is homologous to d j 1 via a homology that desends to a homology in S; d j has lower geometri intersetion number with m than d j 1 ; d j an be isotoped to lie below h(w, 0), d 1,..., d j 1, moreover its projetion an be isotoped to be disjoint from h(w, ). For (x j, e j ) = h 1 (d j ), we have x j e j homolgous to w 0 via a homology that desends to S. d k lies above h(w, 1) and below m. See Figures 7, 8 and 9. This proves the following: Lemma 5. The homology lass [[p(d k )]] = p # (x k e k ) = α. We make two observations: 1) A result of Oertel, see [19], shows that the isotopy lass of p(e k ) does not depend on the hoies made; 2) It is important to realize that (x k, p(e k )) may not be a Seifert urve, beause S\p(e k ) is not neessarily onneted. If S\p(h 1 (e k )) is onneted, set p ( ) = (x k, p(e k )). Otherwise, hoose a omponent D of S\p(e k ). If the frontier of D is null homologous, then remove the frontier of D from p(e k ). See Figures 10, 11,

11 m+1 m 0 m 1 Figure 5: The setup with m, 0 (weights are 1) m+1 m m 1 Figure 6: d 1 11

12 2 1 m 0 Figure 7: A different pair of weighted multi-urves m 1 h (d 1 ) Figure 8: h 1 (d 1 ) 12

13 m 1 h (d ) 2 Figure 9: h 1 (d 2 ) If the frontier of D is not null homologous (beause the orientations do not math up) hoose an ar a in its frontier with smallest weight. Denote the weight of a by w a. We eliminate the omponent a of p(e k ) by adding ±w a to the weights of the other omponents of p(e k ) in the frontier of D in suh a way that the resulting weighted multi-urve still has homology α. After a finite number of suh eliminations, we obtain a weighted multi-urve that is a subset of p(e k ), has homology α, and whose omplement in S is onneted. After reversing orientation on omponents with negative weights, we obtain a Seifert urve p ( ). Lemma 6. The homology lass [[p ( )]] = α. Proof: This follows from Lemma 5 and the observations above. Definition 13. We denote the isotopy lass [p ( )] by π v (v ). Lemma 7. For v v, the following hold: d K (π v (v ), v ) = 1 and d K (π v (v ), v) d K (v, v) 1. 13

14 m m 1 m 2 0 Figure 10: The setup with m, 0 (weights are 1) It will follow from Theorem 2 below that the inequality is in fat an equality. Proof: By onstrution, e k lies stritly between 0 and 1. So τ(e k ) lies stritly between 1 and 0. Thus the lift of S\p ( ) with frontier in e k τ(e k ) meets S 0 and S 1 and is disjoint from S i for i 0, 1. It follows that the lift of S\p ( ) with frontier ontained in e k τ(e k ) also meets S 0 and S 1 and is disjoint from S i for i 0, 1. Hene d K (π v (v ), v ) = 1. In addition, suppose that 0 S i if and only if i {n,..., m + 1}. Then 1 S i if and only if i {n+1,..., m+2}. Hene the lift of S\ that lies stritly between 0 and 1 meets exatly S n,..., S m+2. By onstrution, e k S i an be non empty only if i {n,..., m} and thus τ(e k ) S i an be non empty only if i {n+1,..., m+1}. Hene the lift of S\p ( ) with frontier in e k τ(e k ) an meet S i only if i {n,..., m + 1}. It follows that the lift of S\p ( ) with frontier ontained in e k τ(e k ) an meet S i only if i {n,..., m + 1}. Whene d K (π v (v ), v) m + 1 n 1 = d K (v, v) 1. Definition 14. The graph distane on a omplex C is a funtion that assigns to eah pair of verties v, v the least possible number of edges in an edge path in C from v to v. 14

15 m m 1 0 m 2 Figure 11: d k m m 1 0 m 2 Figure 12: A subset of d k 15

16 Theorem 2. (Kakimizu) The funtion d K equals graph distane. Proof: Denote the graph distane between v and v by d(v, v). If d K (v, v) = 1, then d(v, v) = 1 and vie versa by definition. So suppose d K (v, v) = m > 1 and onsider the path with verties v, π v (v ), π 2 v(v ),..., π m 1 v (v ), π m v (v ) = v. By Remark 7, d K (π v (v ), v ) = 1 and d K (πv(v i ), πv i 1 (v )) = 1. Thus the existene of this path guarantees that d(v, v) m. Hene d(v, v) d K (v, v). Let v = v 0, v 1,..., v n = v be the verties of a path realizing d(v, v). By the triangle inequality and the fat that d(v i 1, v i ) = 1 = d K (v i 1, v i ): d K (v, v) d K (v 0, v 1 ) + + d K (v n 1, v n ) = = d(v 0, v 1 ) + + d(v n 1, v n ) = d(v, v) The following theorem is a reinterpretation of a theorem of Sharlemann and Thompson, see [24], that was proved using different methods: Theorem 3. The Kakimizu omplex is onneted. Proof: Let v, v be verties in Kak(S, α). By Remark 11, d K (v, v ) is finite. By Theorem 2, d(v, v ) is finite. In partiular, there is a path between v and v. Definition 15. A geodesi between verties v, v in a Kakimizu omplex is an edgepath that realizes d(v, v ). Theorem 4. The path with verties v, π v (v ), π 2 v(v ),..., π v (v ) m 1, π m v (v ) = v is a geodesi. Proof: This follows from Theorem 2 beause the path realizes d(v, v). v, π v (v ), π 2 v(v ),..., π v (v ) m 1, π m v (v ) = v Remark 16. Theorem 4 tells us that geodesis in the Kakimizu omplex joining two given verties are, at least theoretially, onstrutible. Note that, typially, π v (v ) π v (v). See Figure 13 for a step in the onstrution of π v (v). 16

17 2 1 m+1 0 Figure 13: u( m+1, 0 ) 4 Contratibility The proof of ontratibility presented here is a streamlined version of the proof given in the 3-dimensional ase in [20]. Those familiar with Hather s work in [9], will note ertain similarities with his first proof of ontratibility of C α (S) in the ase that α is primitive. Lemma 8. Suppose that v, v 1, v 2 are verties in Kak(S, α). Then there are representatives, 1, 2 with v = [(w, )], v 1 = [(w 1, 1 )], and v 2 = [(w 2, 2 )] that realize d K (v, v 1 ), d K (v, v 2 ), and d K (v 1, v 2 ). Proof: Let, 1, 2 be geodesi representatives of the underlying urves of representatives of v, v 1, v 2 suh that ar omponents of, 1, 2 are perpendiular to S. Lifts of, 1, and 2 to (p, Ŝ, S), the infinite yli overing of S assoiated with α, are also geodesis. Points of intersetion lift to points of intersetion. Geodesis that interset an t be isotoped to be disjoint. Hene, 1, 2, with appropriate weights, realize d K (v, v 1 ), d K (v, v 2 ), and d K (v 1, v 2 ). Lemma 9. Suppose that v, v 1, v 2 are verties in Kak(S, α) suh that d K (v, v i ) > 1 and d K (v 1, v 2 ) = 1. Then d K (π v (v 1 ), π v (v 2 )) 1. Proof: In the ase that, say, v 1 = v, note that d K (v 1, v 2 ) = 1 means that d K (v, v 2 ) = 1. Thus π v (v 1 ) = π v (v) = v and π v (v 2 ) = v. Thus d K (π v (v 1 ), π v (v 2 )) = 0. In the ase 17

18 that, say, d K (v 1, v) = 1 and v 2 v, note that π v (v 1 ) = v and thus d K (v, v 2 ) d K (v, v 1 ) + d K (v 1, v 2 ) = d K (v, π v (v 2 ) 1, by Lemma 7, and d K (π v (v 1 ), π v (v 2 )) 1. Hene we will assume, for the rest of this proof, that d K (v, v i ) > 1. By Lemma 8, there are representatives (w, ), (w 1, 1 ), and (w 2, 2 ) of v, v 1 and v 2 that realize d K (v, v 1 ), d K (v, v 2 ), and d K (v 1, v 2 ). Let (p, Ŝ, S) be the infinite yli over of S assoiated with α. Define τ, S i, Si 1, Si 2, i, 1 i, 2 i as in Definition 9 but with a aveat: Label Si 1, Si 2 so that S0, 1 S0 2 meet S 1 and meet S j only if j 1. Sine d K ( 1, 2 ) = d K (v 1, v 2 ), 1 and 2 must be disjoint. Sine 1 0 is separating, 2 0 lies either above or below 1 0. Without loss of generality, we will assume that 2 0 lies above 1 0 (and below τ( 1 0)). See Figures 14 and 15. Note that h(w 2, 2 0) also lies above h(w 1, 1 0). Proeeding as in the disussion preeding Lemma 6, onstrut e 1 k whose projetion ontains p ( 1 ) and then e 2 l whose projetion ontains p ( 2 ), noting that this onstrution an be undertaken so that e 2 l lies above e 1 k (and below τ(e1 k )). Consider the lift of S\p ( 2 ) with frontier in e 2 l τ(e2 l ). This lift of S\p ( 2 ) meets at most the two lifts of S\p ( 1 ) whose frontiers lie in e 1 k τ(e1 k ) and τ(e1 k ) τ 2 (e 1 k ). Whene d K (π v (v 1 ), π v (v 2 )) 1. Lemma 10. If d K (v 1, v 2 ) = m, then d K (π v (v 1 ), π v (v 2 )) m. Proof: Let v 1 = v 0, v 1,..., v m 1, v m = v 2 be the verties of a path from v 1 to v 2 that realizes d K (v 1, v 2 ). By Lemma 9, d K (π v (v i ), π v (v i+1 )) d K (v i, v i+1 ) = 1 for i = 0,..., m 1. Hene d K (π v (v 1 ), π v (v 2 )) d K (π v (v 0 ), π v (v 1 )) + + d K (π v (v m 1 ), π v (v m )) d K (v 0, v 1 ) + + d K (v m 1, v m ) m. Theorem 5. The Kakimizu omplex of a surfae is ontratible. Proof: Let Kak(S, α) be a Kakimizu omplex of a surfae. It is well known (see [8, Exerise 11, page 358]) that it suffies to show that every finite subomplex of Kak(S, α) is ontained in a ontratible subomplex of Kak(S, α). Let C be a finite subomplex of Kak(S, α). Choose a vertex v in C and denote by C the smallest flag omplex ontaining every geodesi of the form given in Theorem 4 for v a vertex in C. Sine C is finite, it follows that C is finite. Define : V ert(c ) V ert(c ) on verties by (v ) = π v (v ). By Lemma 9, this map extends to edges. Sine C is flag, the map extends to simplies and thus to all of C. By Lemma 10 this map is ontinuous. It is not hard to see that is homotopi to the identity map. In partiular, is a ontration map. (Speifially, d, where d is the diameter of C, has the set {v} as its image.) 18

19 1 0 1 Figure 14: 1 0 and Figure 15: d 1 k and d2 l 19

20 5 Dimension In [9], Hather proves that the dimension of C α (S) is 2g(S) 3, where g(s) is the genus of the losed oriented surfae S. An analogous argument derives the same result in the ontext of Kak(S, α). Lemma 11. Let S be a losed onneted orientable surfae with genus(s) 2 and α a primitive lass in H 1 (S, S). The dimension of Kak(S, α) is χ(s) 1 = 2genus(S) 3. Proof: It is not hard to build a simplex of Kak(S, α) of dimension 2genus(S) 3. See for example Figure 16, where 0, 1, 2, and 3 are multi-urves (eah of weight 1) representing the verties of a simplex. Thus the dimension of Kak(S, α) is greater than or equal to 2genus(S) 3. Conversely, let σ be a simplex of maximal dimension in Kak(S, α). Label the verties of σ by v 0,..., v n and let 0,..., n be geodesi representatives of the underlying urves of representatives of v 0,..., v n. By Lemma 3, S\( 0 n ) onsists of subsurfaes P 0,..., P n with frontiers 0 n, 1 0,..., n n 1. Sine i and i 1 are not isotopi, no P i an onsist of annuli. In addition, no P i an be a sphere, hene eah must have negative Euler harateristi. Thus the number of P i s is at most χ(s). I.e., n χ(s) = 2genus(S) 2. In other words, the dimension of σ and hene the dimension of Kak(S, α) is less than or equal to 2genus(S) Figure 16: A simplex in a genus 3 surfae We an extend this argument to ompat surfaes, by introduing the following notion of omplexity: Definition 17. Let S be a ompat surfae and let P be an open subset of S whose boundary onsists of open subars of S and, possibly, omponents of S. Define (P, S) = 2χ(P ) + number of open subars in P 20

21 The following lemma is immediate: Lemma 12. Let C be a union of simple losed urves and simple ars in S. Then (S, S) = (S\C, S) Theorem 6. Let S be a ompat onneted orientable surfae with χ(s) 1 and α a primitive lass in H 1 (S, S). The dimension of Kak(S, α) is 2χ(S) 1 = 4genus(S) + 2b 5, where b is the number of boundary omponents of S. a 0 d 1 0 b a 4 5 d b Figure 17: A simplex in a puntured genus 2 surfae 3 Proof: To build a simplex of Kak(S, α) of dimension 4genus(S) + 2b 5, see for example Figure 17, where 0, 1, 2, 3, 4 and 5 are multi-urves (eah of weight 1) representing the verties of a simplex. Thus the dimension of Kak(S, α) is greater than or equal to 4g(S) + 2b 5. Conversely, let σ be a simplex of maximal dimension in Kak(S, α). Label the verties of σ by v 0,..., v n and let 0,..., n be geodesi representatives of the underlying urves of representatives of v 0,..., v n suh that ar omponents of 0,..., n are perpendiular to S. By Lemma 3, S\( 0 n ) onsists of subsurfaes P 0,..., P n with frontiers ontaining 0 n, 1 0,..., n n 1. Sine i and i 1 are not isotopi, P i an t onsist of annuli or disks with exatly two open subars of S in their boundary. In addition, no P i an be a sphere or a disk with exatly one open 21

22 subar of S in its boundary, hene eah must have positive omplexity. Thus the number of P i s is at most (S, S\( 0 n )). I.e., n (S, S\( 0 n )) = (S, S) = 2χ(S). In other words, the dimension of σ and hene the dimension of Kak(S, α) is less than or equal to 2χ(S) 1 = 4genus(S) + 2b 5. 6 Quasi-flats In this setion we explore an idea of Irmer. See [13, Setion 7]. t 2 t 4 t 1 t 3 Figure 18: Building a quasi-flat by Dehn twists Consider Figure 18. Denote the surfae depited by S and the homology lass of by α. The urves t 1 and t 2 are homologous as are t 3 and t 4. Denote by v the vertex (1, ) of Kak(S, α), by v 1 the vertex orresponding to the result, d 1, obtained from by Dehn twisting n times around t 1 and n times around t 2, and by v 2 the vertex orresponding to the result, d 2, obtained from by Dehn twisting n times around t 3 and n times around t 4. Then d 1, d 2 are homologous to, so we obtain three verties v, v 1, v 2 in Kak(S, α) (all weights are 1). Note the following: d(v, v i ) = d K (v, v i ) = n d(v 1, v 2 ) = d K (v 1, v 2 ) = n For i = 1, 2, we onsider the geodesis g i with verties v i, π v (v i ),..., π n v (v i ) = v. In addition, onsider the geodesi g 3 with verties v 2, π v1 (v 2 ),..., π n v 1 (v 2 ) = v 1 and note that π i v 1 (v 2 ) is represented by a urve obtained from by Dehn twisting i times around t 1, i times around t 2, n i times around t 3 and (n i) times around t 4. Definition 18. Let (X, d) be a metri spae. A triangle is a 6-tuple (v 1, v 2, v 3, g 1, g 2, g 3 ), where v 1, v 2, v 3 are verties and the edges g 1, g 2, g 3 satisfy the following: g 1 is a distane minimizing path between v 1 and v 2, g 2 is a distane minimizing path between v 2 and v 3, g 3 is a distane minimizing path between v 3 and v 1. A triangle (v 1, v 2, v 3, g 1, g 2, g 3 ) is δ-thin if eah g i lies in a δ-neighborhood of the other two edges. A metri spae (X, d) is δ-hyperboli if every triangle in (X, d) is δ-thin. It is hyperboli if there is a δ > 0 suh that (X, d) is δ-hyperboli. 22

23 For n even, the midpoint, m 1, of the geodesi g 1 is the vertex orresponding to the result, d 1, obtained from by Dehn twisting n 2 times around t 1 and n 2 times around t 2. Likewise, the midpoint, m 2, of the geodesi g 2 is the vertex orresponding to the result, d 2, obtained from by Dehn twisting n 2 times around t 3 and n 2 times around t 4. The midpoint, m 3, of g 3 is represented by a urve obtained from by Dehn twisting n 2 times around t 1 and around t 3 and n 2 times around t 2 and t 4. Lemma 13. Let S be the losed oriented surfae of genus 4. Then Kak(S, α) is not hyperboli. Proof: For S the losed genus 4 surfae, the triangle (v, v 1, v 2, g 1, g 2, g 3 ) desribed depends on n, so we will denote it by T n. In T n we have the following: d(v, m 3 ) = d K (v, m 3 ) = n d(v 1, m 2 ) = d K (v 1, m 2 ) = n d(v 2, m 1 ) = d K (v 2, m 1 ) = n In partiular, g 3 is ontained in a δ-neighborhood of the two geodesis g 1, g 2 only if n is less than δ. Thus the triangle T n in Kak(S, α) is not δ-thin for n δ. It follows that Kak(S, α) is not hyperboli. Definition 19. Let (X, d) be a metri spae. A quasi-flat in (X, d) is a quasi-isometry from R n to (X, d), for n 2. Note the following: d(m 1, m 2 ) = d K (m 1, m 2 ) = n 2 d(m 1, m 3 ) = d K (m 1, m 3 ) = n 2 d(m 2, m 3 ) = d K (m 2, m 3 ) = n 2 Thus the triangle T n sales like a Eulidean triangle. It is not too hard to see that a triangle with this property an be used to onstrut a quasi-isometry between R 2 and an infinite union of suh triangles lying in Kak(S, α). Thus Kak(S, α) ontains quasi-flats. It is also not hard to adapt this onstrution to show that, for S an oriented surfae, Kak(S, α) is not hyperboli and ontains quasi-flats if the genus of S is greater than or equal to 4, or the genus of S is greater than or equal to 2 and χ(s) 6. 23

24 7 Genus 2 We onsider the example of a losed orientable surfae S of genus 2. A non trivial primitive homology lass α an always be represented by a non separating simple losed urve with weight 1. Moreover, a Seifert urve in a losed orientable surfae of genus 2, sine its underlying urve is non separating, an have at most two omponents. Figure 19 depits multi-urves and d 1 d 2 suh that [[]] = [[d 1 ]] + [[d 2 ]]. We refer to a Seifert urve with one omponent as type 1 and a Seifert urve with two omponents as type 2. Sine α is primitive, a Seifert urve of type 1 must have weight 1. It follows that distint Seifert urves of type 1 must interset. A Seifert urve of type 1 an be disjoint from a Seifert urve of type 2, see Figure 19 and distint Seifert urves of type 2 an be disjoint, see Figure 20. Let be the underlying urve of a Seifert urve of type 1 and d = d 1 d 2 the underlying urve of a Seifert urve of type 2 that are disjoint. Then the three disjoint simple losed urves d ut S into pairs of pants. Any Seifert urve that is disjoint from d must have underlying urve parallel to either or d. Note that, sine the weight of is 1, the weights for d 1, d 2 must also be 1. Consider the link of [(1, )] in Kak(S, α). It onsists of equivalene lasses of Seifert urves of type 2. The Seifert urves of type 2 have underlying urves that are pairs of urves lying in S\, aren t parallel to, and are separating in S\ but not in S. There are infinitely many suh pairs of urves. More speifially, S\ is a twie puntured torus, so the urves are parallel urves that separate the two puntures and an be parametrized by Q. Distint suh urves an t be isotoped to be disjoint and hene orrespond to distane two verties of Kak(S, α). This onfirms that Kak(S, α) has dimension 1 = (2)(2) 3 near [(1, )], as presribed by Theorem 11. For d = d 1 d 2 as in Figure 19 or 20, we onsider S\(d 1 d 2 ), a sphere with four puntures. The link of [(w 1, w 2, d 1 d 2 )] ontains isotopy lasses of Seifert urves of type 1. These are essential urves that are separating in S\(d 1 d 2 ) but not in S and that partition the puntures of S\(d 1 d 2 ) appropriately. There are infinitely many suh urves. They too an be parametrized by Q. Note that distint Seifert urves of type 1 an t be isotoped to be disjoint and hene orrespond to verties of Kak(S, α) of distane two or more. In addition, the link of [(w 1, w 2, d 1 d 2 )] ontains verties [(u 1, u 2, e 1 e 2 )] suh that one omponent of e 1 e 2, say e 1, is parallel to a omponent of d 1 d 2, say d 1 and S\(d 1 d 2 e 1 e 2 ) onsists of two pairs of pants and one annulus. Seifert urves of this type an also be parametrized by Q, sine e 2 is a urve in a twie puntured torus that partitions the puntures appropriately and e 1 is parallel to d 1. Note that, sine the weights of d 1, d 2 are w 1, w 2, we must have w 1 = u 1 ± u 2 w 2 = u 2 In summary, Kak(S, α) is a tree eah of whose verties has a ountably infinite disrete (i.e., 0-dimensional) link. 24

25 d1 d 2 Figure 19: Underlying urves and d 1 d 2 in a genus 2 surfae (all weights are 1),e ( 1,e 2 ) ( 1 1 ) ( 2,d 1) ( 1,d 2) Figure 20: Seifert urves (2, 1, d) and (1, 1, e) ( 1,e 2 ) ( 1,e ) 1, (1 ) Figure 21: Seifert urves (1, 1, e) and (1, ) 25

26 Reall that Johnson, Pelayo and Wilson showed that the Kakimizu omplex of a knot in the 3-sphere is quasi-eulidean. The Kakimizu omplex of the genus 2 surfae is an infinite graph, thus Gromov hyperboli. In partiular, it is not quasi-eulidean. 8 3-manifolds The definitions given for Seifert urve, infinite yli over, Kakimizu omplex and so forth arry over to odimension 1 submanifolds in manifolds of any dimension. In partiular, they arry over to Seifert surfaes and Kakimizu omplexes in the ontext of ompat (possibly losed) 3-manifolds. One need merely replae ones by twos and twos by threes. Instead of Seifert urves, one onsiders Seifert surfaes. Seifert surfaes are weighted essential surfaes that represent a given relative seond homology lass and have onneted omplement. This ties into and generalizes some of the work in [20]. Let S be a ompat oriented surfae. Take M = S I. Inompressible surfaes in a produt manifold are either horizontal or vertial. Vertial surfaes have the form I, where is a multi-urve in S. It follows that Kak(M, [[ I]]) = Kak(S, [[]]), where [[ ]] denotes the homology lass of. Theorem 7. There exist 3-manifolds with Gromov hyperboli Kakimizu omplex. Proof: Let S be the losed oriented surfae of genus 2, α a primitive homology lass in H 1 (S) and a ompat 1-manifold representating α. Then Kak(S, α) is the graph disussed in Setion 7. In partiular, Kak(S, α) is quasi-hyperboli. Take M = S I. Then Kak(M, [[ I]]) = Kak(S, α) is also quasi-hyperboli. 26

27 Referenes [1] Banks, Jessia, The Kakimizu omplex of a onneted sum of links, Trans. Amer. Math. So. 365 (2013), [2] Bestvina, Mladen and Bux, Kai-Uwe and Margalit, Dan, The dimension of the Torelli group, J. Amer. Math. So. 23, no. 1 (2010) [3] Bridson, Martin R. and Haefliger, André, Metri spaes of non-positive urvature, Grundlehren der Mathematishen Wissenshaften [Fundamental Priniples of Mathematial Sienes], Volume 319, Springer-Verlag, Berlin, 1999, ISBN [4] J. R. Eisner: Knots with infinitely many minimal spanning surfaes, Trans. Amer. Math. So. 229 (1977) [5] U. Hamenstädt, Geometry of the omplex of urves and of Teihmüller spae in Handbook of Teihmüller theory, IRMA Let. Math. Theor. Phys., Volume 11, Eur. Math. So., Zürih, [6] J.L. Harer, Stability of the homology of the mapping lass groups of orientable surfaes, Ann. of Math. (2) 121 (1985), no. 2, [7] W. J. Harvey, W. J., Boundary struture of the modular group, In Riemann surfaes and related topis: Proeedings of the 1978 Stony Brook Conferene (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, , Prineton Univ. Press, Prineton, N.J., [8] A. Hather, Algebrai Topology, Cambridge University Press, Cambridge, 2002, ISBN: X; [9] A. Hather, The Cyli Cyle Complex of a Surfae, unpublished, hather/papers/yles.pdf [10] A. Hather, D. Margalit, Generating the Torelli Group, Enseign. Math. (2) 58 (2012), no. 1-2, [11] Hempel, John, 3-manifolds as viewed from the urve omplex Topology 40 (2001), no. 3, [12] I. Irmer, The Chillingworth lass is a signed stable length, Algebr. Geom. Topol. 15 (2015), no. 4, [13] I. Irmer, Geometry of the Homology Curve omplex, J. Topol. Anal. 4 (2012), no. 3, [14] N.V. Ivanov, Complexes of urves and the Teihmüller modular groups, Uspekhi Mat. Nauk 42 (1987) no. 3 (255)

28 [15] O. Kakimizu, Finding disjoint inompressible spanning surfaes for a link, Hiroshima Math. J. 22 (1992) [16] H. A. Masur, Y. N. Minsky, Geometry of the omplex of urves. I. Hyperboliity. Invent. Math. 138 (1999), no. 1, [17] H. A. Masur, Y. N. Minsky, Geometry of the omplex of urves. II. Hierarhial struture. Geom. Funt. Anal. 10 (2000), no. 4, [18] U. Oertel, On the existene of infinitely many essential surfaes of bounded genus, Paifi J. Math. 202, no. 2 (2002) [19] U. Oertel, Sums of inompressible surfaes, Pro. Amer. Math. So. 102 (1988), no. 3, [20] P. Przytyki, J. Shultens, Contratibility of the Kakimizu omplex and symmetri Seifert surfaes, Trans. Amer. Math. So. 364, no. 3 (2012) [21] A. Putman, Small generating sets for the Torelli group Geom. Topol. 16 (2012), no. 1, [22] J. Johnson, R. Pelayo, R. Wilson, The oarse geometry of the Kakimizu omplex, Algebr. Geom. Topol. 14 (2014), no. 5, [23] M. Sakuma, K. J. Shakleton, K. J., On the distane between two Seifert surfaes of a knot, Osaka J. Math. 46 (2009), no. 1, [24] M. Sharlemann, A. Thompson, Finding disjoint Seifert surfaes, Bull. LMS 20 (1988), no. 1, [25] S. Shleimer, The end of the urve omplex, Groups, Geometry, and Dynamis 5, (2011), no. 1, Department of Mathematis 1 Shields Avenue University of California, Davis Davis, CA USA 28