Hierarchical hyperbolicity of the separating curve graph

Hierarchical hyperbolicity of the separating curve graph

Kate M. Vokes Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom k.m.vokes@warwick.ac.uk
Abstract.

We show that the separating curve graph associated to a connected, compact, orientable surface of sufficient complexity is a hierarchically hyperbolic space in the sense of Behrstock, Hagen and Sisto. It also automatically has the coarse median property defined by Bowditch. Consequences for the separating curve graph include a distance formula analogous to Masur and Minsky’s distance formula for the mapping class group, an upper bound on the maximal dimension of quasiflats, and the existence of a quadratic isoperimetric inequality.

Date: 8 November 2017

1. Introduction

Let be a connected, compact, orientable surface. We shall use the notation for the surface of genus with boundary components, with . The curve graph, , of and its flag complex, the curve complex, were introduced by Harvey [18]. The curve graph has as its vertex set all isotopy classes of essential, non\hypperipheral simple closed curves in , with an edge joining two distinct vertices if they have disjoint representatives. It is equipped with the combinatorial metric defined by assigning length 1 to each edge, denoted . In addition to the curve graph, there are many other graphs which can be associated to a surface which have curves or collections of curves as vertices. Like the curve graph, these typically have a natural isometric action of the mapping class group, . One variation is the separating curve graph, , which is the full subgraph of spanned by all separating curves, again with the combinatorial metric.

The curve complex was used by Harer to study homological properties of the mapping class groups (see, for example, [16] and [17]). It was also central in the proof of the Ending Lamination Theorem [29] [12]. Moreover, the curve graph has been an important tool in the study of the large scale geometry of mapping class groups, for example in two papers by Masur and Minsky [25] [26]. The Gromov hyperbolicity of the curve graph is shown in [25]. The second paper [26] gives a distance estimate for the word metric on the mapping class group of a surface in terms of projections to the curve graphs of subsurfaces.

The separating curve graph also has applications to the study of mapping class groups. In particular, it has been used to study certain subgroups of such as the Johnson kernel (see, for example, [11] and [21] and references therein).

Unlike the curve graph, the separating curve graph of a surface is not in general Gromov hyperbolic. We shall show that it is, however, a hierarchically hyperbolic space. Hierarchically hyperbolic spaces were defined by Behrstock, Hagen and Sisto in [4] and [5]. Mapping class groups of surfaces are a motivating example, and the definition (which we shall give in Section 2.2) is inspired by the work of Masur and Minsky in [26]. Hierarchically hyperbolic spaces are equipped with projections to a family of hyperbolic spaces satisfying certain axioms. Behrstock, Hagen and Sisto prove in [5] that every hierarchically hyperbolic space satisfies a “distance formula” in terms of these projections. The specific result we shall prove is the following theorem and immediate corollary, where is the set of subsurfaces of such that every separating curve intersects non\hyptrivially. The excluded cases are those for which is not connected with the usual definition. For completeness, we give a proof of connectedness of , for as in Theorem 1.1, in Appendix A.

Theorem 1.1.

Let be a connected, compact, orientable surface. Suppose is not for , for or for . Then the separating curve graph of is a hierarchically hyperbolic space with respect to .

Corollary 1.2.

Let be as in Theorem 1.1. Then there exists a constant such that for every there exist and such that the following holds. For every pair of separating curves ,

Here is equal to if and otherwise, and refers to the distance between the subsurface projections of and to (see Section 2.1 for definitions).

The key step in proving Theorem 1.1 is to show that if the distance between the subsurface projections of two separating curves to is bounded by some for all , then there is a bound on their distance in depending only on and . We shall in fact verify this, along with the other conditions for hierarchical hyperbolicity, for a different graph (Section 3), which we shall show to be quasi\hypisometric to (Proposition 4.1). The methods of Section 3 are constructive, but we make no effort to optimise constants. In principle, the quasi\hypisometry constants are also computable.

A related notion of non-positive curvature which also provides a common framework for hyperbolic spaces and mapping class groups is the coarse median property defined by Bowditch in [7]. Every hierarchically hyperbolic space is also a coarse median space. A coarse median space is equipped with a ternary operator called a coarse median which approximates to the median operation on a finite median algebra for any finite set of points in the space. Bowditch proved in [7] that every coarse median space satisfies a quadratic isoperimetric inequality, giving us the following corollary.

Corollary 1.3.

Let be as in Theorem 1.1. Then satisfies a quadratic isoperimetric inequality.

Also implied by the hierarchical hyperbolicity of is the following result on a notion of “rank”. Statements of the specific results used are given in Section 2.3.

Corollary 1.4.

Let be as in Theorem 1.1. Then there is no quasi\hypisometric embedding of the \hypdimensional Euclidean space or half\hypspace into , where if and otherwise. In fact, for the same , the radius of an \hypdimensional Euclidean ball which can be quasi\hypisometrically embedded into is bounded above in terms of and the quasi\hypisometry constants.

In other words, when , can have quasiflats of dimension 2 but not of any higher dimension. Such quasiflats correspond to pairs of disjoint subsurfaces in ; see Section 2.4 for a description of these. When , has no quasiflats of any dimension greater than 1. More detail on how quasiflats can behave in a hierarchically hyperbolic space is given by Behrstock, Hagen and Sisto in [6]. Moreover, the coarse median property implies that in the rank 1 cases, the separating curve graph is in fact hyperbolic.

Corollary 1.5.

Let be as in Theorem 1.1, with . Then is Gromov hyperbolic.

Acknowledgements.

I am grateful to my supervisor, Brian Bowditch, for many valuable suggestions and interesting conversations, and for reading drafts of this paper. I would also like to thank Saul Schleimer and Alex Wendland for helpful discussions. This work was supported by an Engineering and Physical Sciences Research Council Doctoral Award.

2. Preliminaries

In this section, we give some background and state some standard results which are used later on, or to obtain the corollaries to Theorem 1.1.

2.1. Curves and subsurface projection

We say that a simple closed curve in a surface is essential if it is not homotopic to a point and non\hypperipheral if it is not homotopic to a boundary component of . Unless otherwise stated, the word “curve” will always refer to an isotopy class of essential, non\hypperipheral simple closed curves. Similarly, a “multicurve” will be an isotopy class of multicurves, and the curves of the multicurve will be essential, non\hypperipheral and pairwise non\hypisotopic. Two multicurves and are in minimal position if the number of intersections between and is minimal among all pairs of multicurves , isotopic to , respectively. The intersection number, , of two multicurves and is the number of intersections between and when they are realised in minimal position.

We shall be considering several graphs associated to a surface which have curves or multicurves as vertices. In particular, the curve graph, , and the separating curve graph, , were defined in the introduction. For notational convenience, we shall usually consider these as discrete sets of vertices with the combinatorial metric induced from the graphs. Maps between the graphs should be considered as maps between their vertex sets and will not necessarily be graph homomorphisms. The importance of connectedness for the graphs we will be considering is the consequence that the distance between any two vertices is finite.

The complexity, , of a surface is defined by . This is the maximal number of curves in a multicurve of , and is strictly decreasing under taking proper subsurfaces. An essential subsurface of a surface is a connected subsurface so that every boundary component of is either a boundary component of or an essential, non\hypperipheral curve of . From now on, the word “subsurface” will always refer to an isotopy class of essential subsurfaces. Given a subsurface of , we denote by the multicurve of made up of the boundary components of which are not in .

Given a surface and a subsurface of , we have a subsurface projection map from to the power set of . The image of a point under this map may be empty, and always has uniformly bounded diameter (see Proposition 2.1 below). We define this subsurface projection for subsurfaces with positive complexity following [26]. A subsurface projection to annuli can also be defined but we will not need it here. Let be a subsurface of and a curve of intersecting minimally. That is, and are in minimal position, and if is isotopic to a boundary component of then it is isotoped to be disjoint from . If is contained in then . If is disjoint from then . Otherwise, the intersection of and is a collection of properly embedded arcs in . Then is the set containing each essential, non-peripheral curve in which arises as a boundary component of a regular closed neighbourhood of the union of some in and the components of it meets. We may similarly consider a subsurface projection for any complex whose vertices are curves or multicurves in , and any subsurface of . If is a collection of curves, then . We define the distance between two sets , of curves in by , where denotes the diameter in . We will usually abbreviate by . The following result is included in Lemma 2.3 of [26].

Proposition 2.1.

Let be a subsurface of of positive complexity and let be a multicurve in . Then either or .

This implies that if is a path in such that every intersects , then .

Given a complex , the subsurfaces of which every vertex of must intersect are of particular interest. These are called “holes” in [28], and “witnesses” in some more recent papers (see, for example, [2] [14]).

2.2. Hierarchically hyperbolic spaces

Hierarchically hyperbolic spaces were defined by Behrstock, Hagen and Sisto in [4]. Hierarchical hyperbolicity of a space is always with respect to some family of uniformly hyperbolic spaces with projections from to these spaces. The same authors give an equivalent definition of hierarchically hyperbolic spaces in [5], and that is the definition we shall use here. For an exposition of the topic of hierarchically hyperbolic spaces, see [33]. The space is assumed to be a quasigeodesic space, that is, any two points in the space can be connected by a quasigeodesic with uniform constants. All of the spaces we will deal with in this paper will in fact be geodesic spaces.

We say that is a hierarchically hyperbolic space if there exist a constant , an indexing set and, for each , a \hyphyperbolic space such that the following axioms are satisfied.

1. Projections. There exist constants and such that for each , there is a \hypcoarsely Lipschitz projection such that the image of each point of has diameter at most in .

2. Nesting. The set has a partial order , and if is non-empty then it contains a unique \hypmaximal element. If then we say that is nested in . For all , . For all such that (that is, and ) there is an associated subset with diameter at most , and a projection map .

3. Orthogonality. There is a symmetric and anti-reflexive relation on called orthogonality. Whenever and , . For every and , either there is no such that , or there exists such that whenever and , . If then and are not \hypcomparable, that is, neither is nested in the other.

4. Transversality and consistency. If and are not orthogonal and neither is nested in the other, then we say and are transverse, . There exists such that whenever there are sets and , each of diameter at most , satisfying, for all :

If and then:

These are called the consistency inequalities. If then for any such that either or and then .

5. Finite complexity. There exists , called the complexity of with respect to , such that any set of pairwise \hypcomparable elements of contains at most elements.

6. Large links. There exist and such that the following holds. Let , and . Then either for every , or there exist such that for each , , and such that for all , either for some , or . Also, for each .

7. Bounded geodesic image. For all , and , and for all geodesics of , either or .

8. Partial realisation. There exists a constant with the following property. Let be a set of pairwise orthogonal elements of and let for each . Then there exists such that:

  • for all ,

  • for each and each such that , ,

  • if for some , then .

9. Uniqueness. For all , there exists such that if satisfy for all , then .

2.3. Properties of hierarchically hyperbolic spaces

An important basic property is that hierarchical hyperbolicity is a quasi\hypisometry invariant (Proposition 1.7 of [5]).

Proposition 2.2.

If is hierarchically hyperbolic with respect to and is a quasigeodesic space quasi\hypisometric to , then is hierarchically hyperbolic with respect to .

It is shown in [5] (Theorem 5.5) that hierarchically hyperbolic spaces satisfy the following distance estimate, which is analogous to the result for mapping class groups given by Masur and Minsky in [26]. This was one of the axioms for the original definition of hierarchical hyperbolicity in [4] but is a consequence of the modified axioms in [5].

Theorem 2.3.

Let be hierarchically hyperbolic with respect to a set . Then there exists a constant such that for all there exist and such that the following holds. For every ,

Here denotes the cut\hypoff function as for Corollary 1.2.

Theorem J of [4] gives an upper bound on the dimension of a Euclidean space which can be quasi\hypisometrically embedded in a hierarchically hyperbolic space in terms of the maximal cardinality of a set of pairwise orthogonal elements of . We may obtain a stronger result by combining the following two results.

Theorem 2.4.

Let be a coarse median space of rank , and fix some quasi\hypisometry constants. Then there exists , depending only on and the quasi\hypisometry constants, such that there is no quasi\hypisometric embedding of the \hypdimensional Euclidean ball of radius into .

Theorem 2.5.

Let be hierarchically hyperbolic with respect to and let be the maximal cardinality of a set of pairwise orthogonal elements of . Then is a coarse median space of rank at most .

Theorem 2.4 is Lemma 6.10 of [9]. Theorem 2.5 is observed in [9], without the specific bound on rank. A proof, again without this bound on rank, is given in [5] (Theorem 7.3). However, one may verify that under the assumptions of Theorem 2.5, properties (P1)–(P4) of Section 10 of [7] are satisfied, with , and hence, by Proposition 10.2 of that paper, is coarse median of rank at most . Another result on rank for coarse median spaces is the following, Theorem 2.1 of [7] (see also Corollary 4.3 of [30]).

Theorem 2.6.

Let be a coarse median space of rank 1. Then is Gromov hyperbolic.

2.4. Subsurfaces in

Recall that we denote by the set of those subsurfaces of which every separating curve intersects non\hyptrivially. We will show that has a hierarchically hyperbolic structure with respect to , where the associated hyperbolic spaces are the curve graphs of the subsurfaces in . We briefly describe here what the subsurfaces in look like.

Let . Then every boundary component of is non\hypseparating in and no component of contains a separating curve of . Hence, each component of (or, more correctly, ) is a planar subsurface containing at most one boundary component of , and, conversely, any subsurface with this property is in . See Figure 1 for examples and Figure 2 for non\hypexamples.

\thesubsubfigure
\thesubsubfigure
Figure 1. Examples of subsurfaces that every separating curve must intersect.
\thesubsubfigure
\thesubsubfigure
Figure 2. Examples of subsurfaces where there is a disjoint separating curve.

The relation of orthogonality for elements of will correspond to disjointness, so to obtain Corollary 1.4, we need to consider when a collection of subsurfaces in can be pairwise disjoint. Because of the restrictions on the complement of each of these subsurfaces, if has at least three boundary components then there is no pair of disjoint subsurfaces in . If has at most two boundary components then a set of pairwise disjoint elements of can have cardinality 2, but not 3. If and are disjoint elements of , where , then each of them is a copy of , and they meet along their boundary components. Similarly, when either they are both copies of or one is and one is , and when we will have two copies of (Figure 3). Notice that in most cases, if is a subsurface in such that there exists disjoint from , then must be equal to and hence is completely determined by . The exception is when and is a copy of . Then we may choose a curve in such that one component of is a copy of containing and the other component is in .

\thesubsubfigure
\thesubsubfigure
\thesubsubfigure
Figure 3. Examples of pairs of disjoint subsurfaces in , for , and .

3. A graph of multicurves

In this section, we introduce a graph associated to a surface whose vertices are certain multicurves, and prove that it is hierarchically hyperbolic. We shall show in Section 4 that this graph is quasi\hypisometric to .

3.1. Definition of

Let be a surface as in Theorem 1.1. We define a graph whose vertices are multicurves which cut into subsurfaces which are not in the set . In particular, every separating curve is a vertex of . Also note that, since for any , any subsurface containing is also in , the addition of a disjoint curve to any vertex of gives another vertex of . When we remove a multicurve from , we will really want to remove a regular open neighbourhood in order to obtain compact subsurfaces. However, we shall abuse notation and simply write .

Definition 3.1.

The graph has:

  • a vertex for each multicurve in such that for every component of , there is a separating curve of disjoint from this component,

  • an edge between vertices and if one of the following holds:

    1. is obtained either by adding a single curve to or by removing a single curve from ,

    2. is obtained by replacing a curve in with a curve , where the component of containing is an subsurface in and and intersect exactly twice.

The second type of edge can arise only when is , , or , since these are the cases where there are subsurfaces in which are copies of . Since we assume that satisfies the hypotheses of Theorem 1.1, there is no subsurface in which is a copy of or which has complexity less than 1.

Figure 4. An example of a path in .
Figure 5. Another example of a path in .

Note that connectedness of is implied by connectedness of the pants graph as follows. Every pants decomposition of is a vertex of and a pants move corresponds to either one or two moves in . Moreover, each vertex of is connected to a pants decomposition by adding curves one by one. For closed surfaces, connectedness of the pants graph was first proved by Hatcher and Thurston [19]. Connectedness in the general case follows, for example, from the distance formula indicated in [26]. As usual, from now on we shall treat as a discrete set of vertices equipped with the combinatorial metric induced from the graph.

Proposition 3.2.

Let be the set of subsurfaces which every vertex of must intersect. Then .

Proof.

Firstly is contained in since each separating curve is a vertex of . Suppose is in and is a vertex of . If does not cut then is contained in a single component of . But then has a separating curve in its complement, which contradicts that it is in . ∎

In Sections 3.2 and 3.3, we shall prove the following theorem.

Theorem 3.3.

Let be as in Theorem 1.1. The graph is a hierarchically hyperbolic space with respect to the set .

3.2. Verification of Axioms 1–8

As above, let be the set of subsurfaces which every vertex of (or equivalently of ) must intersect. For each , the -hyperbolic space is the curve graph of . The constant need not depend on the surface , since curve graphs are uniformly hyperbolic [1] [8] [13] [20]. Most of the axioms follow easily from known results on subsurface projections. The only significant new work needed is the verification of Axiom 9. We reserve this for a separate section.

1. Projections Let be the usual subsurface projection. The image of a vertex is never empty since every vertex of intersects each in . Let and be at distance 1 in . Unless they are connected by a move in an subsurface, is a multicurve so its projection to any for has diameter at most 2. Suppose and are connected by a move in a subsurface . If , then the projection of to is two adjacent curves and has diameter 1. Suppose . Since no subsurface of can be in , some curve of intersects . This curve is disjoint from every curve of so the diameter of the projection is at most 4. Hence, the projection is 4\hypLipschitz.

2. Nesting. The partial order on is inclusion of subsurfaces. The unique \hypmaximal element is . If , then we can take , that is, all boundary curves of which are non\hypperipheral in . This has diameter at most 1 in as the curves are pairwise disjoint. The projection is the subsurface projection from to .

3. Orthogonality. The orthogonality relation on is disjointness of subsurfaces. If is disjoint from then it is disjoint from any subsurface of . Suppose and . Then either no other subsurface of disjoint from is in , or the complement is in and any which is disjoint from is nested in . Finally, if and are disjoint then neither is nested in the other.

4. Transversality and consistency. Two subsurfaces and in are transverse, , if they are neither disjoint nor nested. If , let be the subsurface projection of to , and similarly for . These each have diameter at most 2 by Proposition 2.1. By Behrstock’s lemma [3], for each there exists such that for any and any multicurve projecting to both (and hence any vertex of ),

For a more elementary proof due to Leininger, with a uniform value of , see [24]. Given , and in consider

The second term compares projecting directly to from and projecting first to and then to . This gives the same result, so this quantity is . Also, if , then the union of their boundary components is a multicurve in , so for any such that or and , .

5. Finite complexity. The length of a chain of nested subsurfaces in is bounded above by .

6. Large links. Let and , with . Assume for now that . Let be a geodesic in , where and . For each , let be the component of containing the adjacent curves of the geodesic. This is not necessarily in . Suppose satisfies and , where is the constant of Theorem 3.1 of [26] (Bounded Geodesic Image). This cannot be the case if every cuts . Hence some does not cut and so is contained in a single component of . Suppose that this component is not . Then the adjacent curves to in the geodesic also do not cut . Since is contained in or , so too is . Hence, is contained in some . Now suppose that this is not in . But this contradicts that is in , so we need only those which are subsurfaces in . In particular, if , there are no subsurfaces of properly nested in so trivially for every with . Moreover, for each , .

7. Bounded geodesic image. This is implied by Theorem 3.1 of [26]. For a proof that the constant does not depend on the surface , see [34].

8. Partial realisation. Any set of pairwise disjoint subsurfaces in contains at most two elements (at most one if has at least three boundary components). First suppose the set contains only one element . Let be a curve in . Consider the multicurve . We may complete this to a vertex of by, for example, adding curves to obtain a pants decomposition of . Firstly, the projection of to is a multicurve containing , so . Let be a subsurface of containing . Then , by Proposition 2.1, since contains . Let be transverse to . Then similarly . Now suppose and are distinct and disjoint subsurfaces in . Let be a curve in for each . Again, there exists in containing , , and . Moreover, as before, for each , , for every containing , and for every transverse to .

We remark that all of the above constants, apart from the complexity, may be taken to be independent of the surface . Our proof below that Axiom 9 holds gives constants which do depend on the surface and are probably far from optimal. It would be interesting to consider how far they can be improved. The quasi\hypisometry constants in Section 4 also a priori depend on the surface.

3.3. Verification of Axiom 9

The most significant part of the proof of Theorem 3.3 is the verification of the final axiom.

Proposition 3.4.

Let satisfy the hypotheses of Theorem 1.1. For every , there exists , depending only on and , such that if and are two vertices of , and if for every subsurface in , then .

In order to prove this, we make use of a combinatorial construction based on that described in Section 10 of [10]. This will give us a way of representing a sequence of multicurves in . We shall construct this sequence inductively so that eventually it will be a path in . We shall consider the product , for a non\hyptrivial closed interval . We consider to be the horizontal direction and to be the vertical direction. We have a horizontal projection and a vertical projection . When we denote a subset of by , will be a subset of the horizontal factor, , and of the vertical factor, . To ensure that curves in are pairwise in minimal position, we will fix a hyperbolic structure on with totally geodesic boundary and take the geodesic representative of each isotopy class of curves.

Definition 3.5.

A vertical annulus in is a product , where is a curve in and is a non\hyptrivial closed subinterval of . The curve is the base curve of the annulus.

Definition 3.6.

An annulus system in is a finite collection of disjoint vertical annuli. An annulus system is generic if whenever and are two distinct annuli in , .

We denote by and by . Each is a (possibly empty) multicurve, and there is a discrete set of points in where the multicurve changes. Hence the annulus system is a way of recording a sequence of multicurves in .

Definition 3.7.

Let . A tight geodesic in between curves and is a sequence , where:

  • each is a multicurve in ,

  • for any , and any curves , , ,

  • for each , is the boundary multicurve of the subsurface spanned by and .

If , then a tight geodesic is an ordinary geodesic in .

This definition comes from [26], although the tight geodesics of [26] are equipped with some additional data which will not be relevant here. A tight geodesic can be realised as an annulus system as follows.

Definition 3.8.

A tight ladder in is a generic annulus system so that:

  • there exists a tight geodesic in so that the curves appearing in the tight geodesic correspond exactly to the base curves of the annuli in ,

  • for two annuli and in , the intervals and intersect if and only if and are disjoint,

  • there exist in such that for each the multicurve .

In the case where , this corresponds to moving from to by adding in the curves of one at a time then removing the curves of one at a time (Figure 6). In the case where , this corresponds to moving from to by removing the curve then adding in the curve after a vertical interval with no annuli (Figure 6).

\thesubsubfigure Complexity .
\thesubsubfigure Complexity .
Figure 6. Illustrations of tight ladders in .

From now on, we will assume that satisfies the hypotheses of Theorem 1.1.

Definition 3.9.

Let , and let be a component of . Let be the maximal interval containing such that is a component of for every . The product is a brick of . The surface is the base surface of the brick.

We remark that this differs slightly from the definition of “brick” in [10]. Note that the interiors of any two distinct bricks are disjoint, and that we may decompose as a union of regular neighbourhoods of all bricks of (recall that when we remove a multicurve from , we also remove a regular open neighbourhood of ). In order to obtain a path in , we want to decompose into bricks whose base surfaces are not in .

Definition 3.10.

A brick is small if one of the following holds.

  1. The base surface is not in .

  2. The base surface is a copy of and is in . Moreover, and each intersect in an essential non\hypperipheral curve, and the two curves are adjacent in .

Notice that a generic annulus system where every brick is small realises a path in , as follows. First assume there are no copies of in . Consider the multicurves for . These change precisely at the points in the interior of which are the endpoints of vertical projections of annuli in . Let denote this set of points. Let be the components of in the order in which they appear in , and for each pick any from . Let be the multicurve . The sequence is a path in .

In the case where there are copies of in , we place an additional restriction on a generic annulus system, requiring that whenever we have a Type 2 small brick, the endpoints of its vertical projection to are consecutive points of . This can be achieved by appropriate isotopies. Again, let be a generic annulus system where every brick is small. Construct the sequence of curves as above and suppose that, for some , has a component which is an subsurface in (and hence is not a vertex of ). Then by the restriction on the endpoints of Type 2 small bricks, is not a component of or , nor is any other subsurface in . Then and are adjacent vertices of . Hence we obtain a path in as for the previous case except that we remove any multicurves in the sequence which are not vertices of .

Definition 3.11.

The \hypcomplexity of an annulus system is , where, for each , is the total number of non\hypsmall bricks of whose base surface is a subsurface in of complexity . We give this the lexicographical ordering.

Since there are no subsurfaces in of complexity less than 1, the \hypcomplexity is precisely when every brick is small.

We now begin the proof of Proposition 3.4. Let . We shall construct a generic annulus system in , with \hypcomplexity , which realises a path in from to , and show that the length of this path is bounded in terms of and .

We construct the annulus system inductively, while ensuring that the \hypcomplexity is strictly decreasing. We start by choosing distinct points for each curve of and for each curve of and defining an annulus system .

Now suppose, for induction, we have constructed a generic annulus system . We will describe how to construct the next stage . See Figure 7 for an illustration. Consider the bricks of . If every brick is small, then the \hypcomplexity of is and we are done. Suppose this is not the case, and choose a brick , where is in and has maximal complexity among such bricks. (Note that a priori the same subsurface might appear as the base surface of more than one brick.) Decreasing past and increasing past , the components of change to not include . Since has maximal complexity among base surfaces of in , it is not a proper subsurface of any component of for any . Hence, the intersection of and of with must be non empty, and, since is generic, it is in each case a single curve, which we call and respectively. Slightly extend on each side to so that the subset now contains vertical annuli corresponding to each of these curves but still intersects no other annuli. We may consider annulus systems in as for . Add a tight ladder in , corresponding to a tight geodesic in from to , arranging that the resulting annulus system is generic by slightly moving the endpoints of intervals if necessary. The annulus system is the union of and the tight ladder in . Notice that the \hypcomplexity of is strictly less than that of .

Figure 7. Constructing from by adding a tight ladder in a brick .

At each stage, we add a tight ladder in some brick, , increasing the length of the sequence of multicurves determined by the annulus system, where these multicurves are not yet necessarily vertices of . Let us consider the maximal increase in the length of this sequence. Let be the set of points in the interior of corresponding to the endpoints of the vertical projections of bricks to . First suppose . The transition from to gives a point of for every curve in and every curve in , so . Now suppose . Then the number of points of is . Hence between and , when we add a tight ladder of length in a brick , we add at most to the length of the corresponding sequences of curves.

The length of the tight ladder we add between and is equal to . We now show that this quantity is bounded in terms of .

Claim 1.

Let be the set of the base curves of all annuli in and as in the statement of Proposition 3.4. Then for each .

We prove this by an induction on . The base case is when and holds since, by hypothesis, for every . Suppose at stage the projection has diameter at most . At stage , we add a tight geodesic in for some , where and are curves which already appear as base curves in . By the induction hypothesis, . There are several cases depending on how and intersect.

Case 1: is disjoint from . Then none of the curves added in contributes to the projection to so the diameter is unchanged.

Case 2: intersects and is not nested in . Then there is a curve in which intersects non\hyptrivially. Such a curve is also a base curve in . Every curve added in is disjoint from . Hence every curve added either does not intersect so does not change the projection to , or projects to a curve at distance at most 2 from . Hence, the diameter of the projection increases by at most 4.

Case 3: . Suppose that some multicurves and in the tight geodesic do not cut , for . Then there is a curve in which intersects neither, so . Moreover, if then also does not intersect since it is the boundary of the subsurface spanned by and . Hence, any multicurves in the geodesic which do not cut are consecutive terms. Let and be respectively the first and last terms which do not intersect . Suppose and . Then the increase in diameter between and is at most the sum of the maximal possible distances from to and from to . By Proposition 2.1, . Similarly, if or then we have only one of these two terms and again . If every term in the tight geodesic cuts then the increase in diameter from is bounded above by the maximal distance from or to the middle term. In any case, .

This proves Claim 1.

In order to find an upper bound on the length of the final path in , we will find upper bounds on the length of the sequence of curves at certain stages of the induction. For each , let be minimal such that for all . In particular , and is the stage where the \hypcomplexity of the annulus system reaches . For , define by and for , and define by and for .

Claim 2.

For each , and the length of the sequence of curves corresponding to is at most .

We shall prove this by a reverse induction on . We start with the annulus system defined above. Between and , we add a tight ladder in the maximal complexity brick, the length of which is at most . There is now no brick of complexity , so . The length of the sequence of multicurves given by is at most .

Now assume for induction that and that the length of the sequence of multicurves given by is at most . If there are no bricks of complexity , then and we are done, so suppose there is at least one. For each multicurve, there are at most two complementary components which are in , since a set of pairwise disjoint subsurfaces in has cardinality at most 2 (see Section 2.4). Hence, . The maximal complexity is now