Quotients of the curve complex

We consider three kinds of quotients of the curve complex which are obtained by coning off uniformly quasi-convex subspaces: symmetric curve sets, non-maximal train track sets, and compression body disc sets. We show that the actions of the mapping class group on those quotients are strongly WPD, which implies that the actions are non-elementary and those quotients are of infinite diameter.


Introduction
If X is a Gromov hyperbolic space, and Y = {Y i } is a collection uniformly quasiconvex subsets of X, then the electrification X Y is Gromov hyperbolic.Furthermore, if a group G acts on X by isometries, and Y is G-equivariant, then G also acts on the electrification X Y by isometries.
However, if the action of G on X has some useful property (e.g.nonelementary, acylindrical, WPD), then it is not known whether the corresponding action on X Y has this property.In particular, in general, X Y need not be infinite diameter.
We shall consider the action of the mapping class group of a finite type orientable surface on the curve complex of the surface C(S).Our results apply to the following families of uniformly quasiconvex subsets.
The symmetric curve sets: If q : S → S ′ is an orbifold quotient of S, then we shall denote by C q (S) the collection of all q-equivariant curves in S. The collection {C q (S)} as q runs over all orbifold quotients of S is a mapping class group invariant, uniformly quasiconvex family of subsets of C(S) [RS09].We shall denote the electrification by C q (S).
Non-maximal train track sets: A train track τ on S is maximal if every complementary region is a triangle or a monogon containing a single puncture.Given a train track τ , we shall write C(τ ) for all simple closed curves carried by τ .The set {C(τ )} as τ runs over all non-maximal train tracks in S is a mapping class group invariant, uniformly quasiconvex family of subsets of C(S) [MM04].We shall denote the electrification by C τ (S).
Compression body disc sets: Let V be a compression body with boundary S, and let D(V ) be the collection of isotopy classes of essential simple closed curves in S which bound discs in V .The set {D(V )}, as V runs over all compression bodies, forms a mapping class group invariant, uniformly quasiconvex family of subsets of C(S) [MM04].We shall write C D (S) for the electrification, which is quasi-isometric to the compression body graph.
We shall show: Theorem 1.1.Both C q (S), C τ (S) and C D (S) are infinite diameter Gromov hyperbolic spaces, and furthermore, the action of the mapping class group on these spaces is strongly WPD.
See Sections 3, 4 and 2.4 for more detailed statements.In the case of C τ (S) this is a special case of a much more general result of Hamenstädt, which has not yet appeared in print.In the case of C D (S), the compression body graph was previously shown to have infinite diameter in [MS20].
For simplicity in this introduction we only consider random walks on the mapping class group MCG(S) whose transition probability has the support generating the whole MCG(S).In [MT18], Maher-Tiozzo proved that in our situation, any random walk gives rise to a loxodoromic element with asymptotic probability one.Hence in particular, the following two previously known results are now immediate corollaries of this results.Furthermore, this gives largely independent proofs of these results.
Theorem 1.2.[Mas17a] Let S be a surface of finite type of gnerus g with n punctures, and with 2g + n > 4. Then a random walk on the mapping class group of S gives rise to an asymmetric pseudo-Anosov element with asymptotic probability one.
We remark that here we say a pseudo-Anosov map on S is asymmetric if it is not a lift of a pseudo-Anosov map on a surface S ′ which admits an orbifold cover p : S → S ′ .This cover need not be regular, so this is a more general condition than requiring some power of the pseudo-Anosov element commute with a finite order element.The condition 2g + n > 4 includes all surfaces which support a pseudo-Anosov map, but excludes the four exceptional surfaces S 0,4 , S 1,1 , S 1,2 and S 2,0 .The result does not hold for these surfaces, as all pseudo-Anosov maps on these surfaces are symmetric with respect to the hyperelliptic involutions on the surfaces.
We say a pseudo-Anosov map has generic singularities if its invariant foliations have 1-prong singularities at each puncture, and all other singularities are trivalent.
Theorem 1.3.[GM18] Let S be a surface of finite type of genus g with n punctures, and with 2g + n > 3. Then a random walk on the mapping class group of S gives rise to a pseudo-Anosov element whose invariant foliations have generic singularities with asymptotic probability one.
The condition 2g + n > 3 includes all finite type surfaces which support a pseudo-Anosov map with the exception of S 1,1 for which the result does not hold, see Remark 4.1.
Furthermore, our result also has an application to the Poisson boundary.In [KM96], it is proved that the space of projective measured foliations PMF has a unique harmonic measure ν, and the measure space (PMF, ν) is a Poisson boundary of the random walk on MCG(S).Maher-Tiozzo [MT18] proved that if a group acts acylindrically on a Gromov hyperbolic space, then the Gromov boundary equipped with the hitting (harmonic) measure is a Poisson boundary.By the work of Osin [Osi16], any WPD action has a quotient with acylindrical action.Combining these with a characterization of the boundary of quotient (Proposition 2.6 below) Theorem 1.1 gives the following.
Corollary 1.4.Let S be a surface of finite type of genus g with n punctures, and with 2g + n > 4. Let ν be the harmonic measure arising from a random walk on MCG(S).Then the harmonic measure ν on PMF(S) is concentrated on asymmetric and maximal foliations.

Quotients of hyperbolic graphs
Given a metric space (X, d), and a collection of subsets Y = {Y i } i∈I , the electrification of X with respect to Y is the metric space X Y obtained by adding a new vertex y i for each set Y i , and coning off Y i by attaching edges of length 1 2 from each y ∈ Y i to y i .The metric on X Y is the induced path metric, and so the image of each set Y i in X Y has diameter at most one.
We say the family of sets Y is G-equivariant if for all g ∈ G and each Y i ∈ Y then gY i is also in Y .
Theorem 2.1.Let X be a Gromov hyperbolic space and let Y = {Y i } i∈I be a collection of uniformly quasi-convex subsets of X.Then X Y , the electrification of X with respect to Y , is Gromov hyperbolic.
Furthermore, if a group G acts on X by isometries, and Y is G-equivariant, then G also acts on X Y by isometries.

WPD action on quotients
We recall the definition of weakly properly discontinuous action defined by Bestvina-Fujiwara [BF02].
Definition 2.2.Let X be a connected metric graph and G a group of isometries of X.An element g ∈ G acts loxodromically, or is loxodromic if the map Z → X given by n → g n x is a quasi-isometric embedding for any x ∈ X.A loxodromic element g ∈ G is said to act weakly properly discontinuously(WPD) or is WPD if for any r > 0 and x ∈ X, there exists N ∈ N such that We say that the action of G on X is WPD if • there is at least one loxodromic element in G which is WPD.
We say that G is strongly WPD if every loxodromic element is WPD.Note that if there is a loxodromic element acting on a graph, then the graph must have infinite diameter.Moreover, Bestvina-Fujiwara observed the following.
Proposition 2.3 ([BF02, Proposition 6]).Suppose the action of G on a Gromov hyperbolic space X is WPD, then the action of G on X is non-elementary.
We now show that under certain conditions, if g acts WPD on a Gromov hyperbolic metric graph, then g is also WPD on the quotient graph we get by coning off.
Theorem 2.4.Let the group G act by isometries on the hyperbolic metric space X, and let Y be a family of uniformly quasiconvex G-equivariant subsets of X.
Let g ∈ G act loxodromically on X and α be a quasi-geodesic axis of g.We denote by π α the nearest projection map to α on X. Suppose (1) the action of g on X is WPD, and (2) there is a constant C > 0 such that for every element Then the action of g on X Y is also WPD.
Proof.For notational simplicity let Z := X Y .We first observe that α has infinite diameter in Z. Suppose, not then there is a D such that any pair of points x and y in α are distance at most D apart in Z. Then there is a path consisting of D segments in the Y i connecting x to y.By assumption (2), the nearest point projection of each path to α in X has diameter at most C, the distance between x and y in X is at most CD, a contradiction.
Let z be a point in X, and hence also a point in Z, and let B Z (z, r) denote the ball of radius r centered at z in Z.We denote by B Z (z, r) the restriction of B Z (z, r) to X. Then by assumption (2), we see that diam X (π α ( B(z, r)) < Cr.Recall that in a Gromov hyperbolic space, if the nearest projections to a quasigeodesic α of given two points x, y are reasonably far apart, then the geodesic connecting x and y must have a subarc fellow traveling with α.Therefore for large enough N , there is a subarc β ⊂ α and D 1 > 0 such that any geodesic in X connecting a point in B Z (z, r) and a point in B Z (g N z, r) must contain β in its D 1 -neighbourhood.Note that we may suppose β as long as we need by taking large enough N .Let R := d X (z, α).Suppose there is h ∈ G such that d Z (z, hz) < r and d Z (g N z, hg N z) < r.Then any geodesic γ in X connecting hz and hg N z contains β in its D 1 -neighbourhood.This implies that there exists a constant D 2 > Cr such that for any point b ∈ β at least R + D 2 apart from the endpoints, we have d X (b, hb) ≤ R + 2D 1 + D 2 .Note that the constant R + 2D 1 + D 2 is independent of N .Therefore by taking N large enough, we may apply the fact that g acts WPD on X to conclude that g acts WPD on Z.

The Gromov boundary of quotients
In this subsection, we discuss the boundary of the quotient graph we obtain by coning off.We first recall the work of Dowdall-Taylor.
Let X be a Gromov hyperbolic graph and Z a quotient graph obtained by coning off a family Y = {Y i } i∈I of quasi-convex subsets of X. Recall that the Gromov boundary of a Gromov hyperbolic space X is defined as the space of equivalence classes of quasi-geodesic rays.Given a quasi-geodesic ray γ in X, let γ(∞) ∈ ∂X denote the corresponding point in the Gromov boundary ∂X.
Let proj : X → Z denote the map given by the inclusion X ⊂ Z.To give a description of the Gromov boundary of Z, Dowdall-Taylor defines a space ∂ Z X by {γ(∞) ∈ ∂X | γ : R + → X is a quasi-geodesic ray with diam Z (proj(Imγ)) = ∞}.
Then the work of Dowdall-Taylor which we need here can be stated as follows.
Proposition 2.5 (see the proof of [DT17, Theorem 4.8]).Let X be a Gromov hyperbolic space, and let Y = {Y i } i∈I be a collection of uniformly quasi-convex subsets of X.Let Z be the space formed by coning off X with respect to Y .Then the Gromov boundary of Z is homeomorphic to ∂ Z (X).
We shall define ∂Y to be the union of the limit sets of each Y i , i.e. ∂Y = i∈I ∂Y i .This need not be equal to the limit set of the union of the Y i , which will often be all of ∂X if, for example, G acts coarsely transitively on X.
Proposition 2.6.Let X be a Gromov hyperbolic space, and let Y = {Y i } i∈I be a collection of uniformly quasi-convex subsets of X.Let Z be the space formed by coning off X with respect to Y .Let ∂Y := i ∂Y i .Then the space ∂ Z (X) is contained in ∂X \ ∂Y .
Proof.Let γ be a quasi-geodesic ray such that γ(∞) ∈ ∂Y i for some Y i ∈ Y .Since Y i is quasi-convex, we see that γ eventually fellow travels with Y i , and thus the diameter of the image of γ in Z is finite.
Definition 2.7.Let X be a path-connected hyperbolic space on which G acts isometrically.Let H < G be a finitely generated subgroup.Then embedding H → X for any p ∈ X.
• the limit set in X of H is the set of accumulation points in ∂X of any orbit of H in X.
Remark 2.8.If H qi-embeds into a geodesic Gromov hyperbolic space, then H is also Gromov hyperbolic.
We now characterize qi-embedded subgroups in the quotients.
Theorem 2.9.Let X, Y , Z := X Y , and G as in Proposition 2.1.Let H be a finitely generated subgroup of G. Then (a) H qi-embeds in X with limit set contained in ∂X \ ∂Y in X if and only if H qi-embeds into Z (b) If H qi-embeds in X with limit set contained in ∂X \ ∂Y , then no element of infinite order in H preserves any Y i ∈ Y setwise.
Proof of Theorem 3.10.Suppose H qi-embeds into Z, then H must qi-embed into X.Furthermore, if the limit set contains an element λ in ∂Y , then there must be a sequence (g i ) i≥0 in H such that {g i x} i is quasi-isometrically embedded in X and g i x → λ ∈ ∂X for some x ∈ X.As λ ∈ ∂Y , the orbit g i x must fellow travel with some Y ′ ∈ Y .But this means that g i x is in a bounded region in Z which contradicts the assumption that H qi-embeds into Z.
Conversely, suppose H qi-embeds in X and the limit set is contained in ∂X \ ∂Y in X.Now we will prove that for every D ≥ 0, there exists N ≥ 0 such that d Z (x, gx) ≥ D whenever |g| ≥ N, g ∈ H. Suppose not, then there must be D ≥ 0 and (g i ) i≥0 such that d Z (x, g i x) < D and |g i | → ∞.Then after taking subsequences, g i converges to a point in the boundary ∂H.But then corresponding limit point in ∂X must be in ∂Y , which contradicts our assumption.Now (a) follows from [DT17, Lemma 3.4].
To prove (b), we note that if H qi-embeds into Z, every element of infinite order in H acts loxodromically on Z. Then if H contains an element of infinite order which preserves Y ′ ∈ Y , then the limit set in X must contain a point in ∂Y ′ .Thus (b) follows.

The compression body graph
Finally, we observe that these results apply in the case of C D (S), the electrification of the curve complex along the discs sets of compression bodies.These sets are unifomly quasiconvex [MM04], the electrification has infinite diameter, and there are pseudo-Ansosov elements which act loxodromically on C D (S) [MS20].It remains to verify condition (2) from Theorem 2.4, which is a corollary of the following stability result.
Theorem 2.10.[MS20, Theorem 4.2] Given a closed orientable surface S and a constant k, there is a constant k ′ > k with the following property.Suppose that γ is a geodesic ray in C(S), and V i is a sequence of compression bodies such that, for all i, the segment γ|[0, i] lies in a k-neighborhood of D(V i ).Then there is a constant k ′ and a non-trivial compression body W , contained in infinitely many of the V i , such that that γ is contained in a k ′ -neighborhood of D(W ).
We now verify condition (2) from Theorem 2.4 in this setting.
Corollary 2.11.Let S be a closed orientable surface, and let g be a pseudo-Ansosov map which acts loxodormically on C D (S), and let α be an axis for g in C(S).Then there is a constant C such that for any disc set D(V ), the nearest point projection of D(V ) ot α has diameter at most C.
Proof.Suppose there is a sequence of compression bodies V i such that diam(π α (D(V i ))) tends to infinity.As the D(V i ) are quasiconvex, and g acts coarsely transitively on α, there is a constant k such that we may translate each D(V i ) by a power n i of g so that α|[0, t i ] is contained in a k-neighbourhood of g ni D(V i ), and t i tends to infinity as i tends to infinity.Theorem 2.10 then implies that there is a k ′ such that α is contained in a k ′ -neighbourhood of a single disc set D(W ).This implies that the image of α in C D (S) is finite, contradicting the fact that g acts loxodromically on C D (S).n > 0, then the mapping class group of S contains pseudo-Anosov elements.There is a finite list of exceptional cases in which the mapping class group contains a hyperelliptic involution which acts trivially on the curve graph and the Teichmüller space of the surface, and for which every pseudo-Anosov map is symmetric with respect to this involution.We shall call these special cases the exceptional surfaces, and they consist precisely of the surfaces S 0,4 , S 1,1 , S 1,2 and S 2,0 , see for example [FM12, Section 12.1].We may exclude precisely these surfaces from the collection of surfaces satisfying 3g − 3 + n > 0, by replacing the condition 3g − 3 + n > 0 with the condition 2g + n > 4.
By the word curve on S, we mean an isotopy class of essential simple closed curve on S. The curve graph, denoted C(S), of S is the graph each of whose vertices corresponds to a curve, and two vertices are connected by an edge of length 1 if corresponding curves can be realized disjointly.We also need to consider orbifolds covered by S, so from now on any finite cover may be an orbifold or branched cover of S. For such an orbifold, we define the curve graph as the one for the surface we obtain by puncturing all the orbifold points.Given a finite covering of surfaces, there is a map between the corresponding curve graphs.Definition 3.1.Let p : S → S ′ be a finite covering.We denote by Π : C(S ′ ) → C(S) the associated (one-to-finite) map on the curve graphs which is defined so that a ∈ Π(b) if and only if p(a) = b.We call Π(C(S ′ )) the subspace of symmetric curves with respect to p : S → S ′ .
By the work of Rafi-Schleimer, the subspaces of symmetric curves are quasiisometrically embedded.

Theorem 3.2 ([RS09]
).The map Π : C(S ′ ) → C(S) is a quasi-isometric embedding with quasi-isometry constants depending only on S and the degree of the finite covering p : S → S ′ .Remark 3.3.In a Gromov hyperbolic space, every quasi-isometrically embedded subspace is quasi-convex and the quasi-convexity constant depends only on the quasi-isometry constants and the hyperbolicity constant.
The main object we consider in this section is the following.
Definition 3.4.The co-symmetric curve graph C q (S) is the graph obtained by coning off the family Π := {Π(C(S ′ ))} of subspaces of symmetric curves, where we consider all finite (possibly orbifold) coverings of type p : S → S ′ .
Proposition 3.5.The co-symmetric curve graph C q (S) is Gromov hyperbolic and MCG(S) acts isometrically on C q (S).
Proof.Given a finite covering p : S → S ′ , the map p•g −1 is also a finite covering for any g ∈ MCG(S).Therefore the family Π is setwise MCG(S)-invariant.Furthermore, as there is a lower bound of Euler characteristics of 2-dimensional orbifolds, the degree of the coverings from S is bounded.Hence by Theorem 3.2 and Remark 3.3, the quasi-convexity constants of the elements in the family Π is bounded.Hence, we get the conclusion by Proposition 2.1.Note that we have not yet eliminated the possibility that C q (S) has bounded diameter.To discuss further properties of C q (S), the following terminology is useful.
Definition 3.6.Let A be a mapping class, a foliation, or a lamination.Then A is said to be symmetric if it is a lift with respect to some finite covering p : S → S ′ .If A is not symmetric, it is said to be asymmetric.
In subsection 3.1, we discuss the MCG(S)-action on C q (S).By using Theorem 2.4, we will prove the following theorem which is a detailed version of Theorem 1.1 for C q (S).
Theorem 3.7.Let S be an orientable surface of genus g with n punctures.Suppose 2g + n > 4. For the action of MCG(S) on C q (S), we have (a) g ∈ MCG(S) is loxodromic if and only if g is asymmetric and pseudo-Anosov, (b) the action of MCG(S) on C q (S) is WPD.
In particular, the action of MCG(S) on C q (S) is non-elementary.
We now give a characterization of ∂C q (S).Recall that by the work of Klarreich [Kla99], the boundary of C(S) is identified with the space of ending laminations, denoted EL(S).We call an ending lamination symmetric if it is a lift of an ending lamination with respect to a finite covering p : S → S ′ .An ending lamination is called asymmetric if it is not symmetric.Let AEL(S) denote the subspace of EL(S) consisting of all asymmetric ending laminations.Then by Proposition 2.5 and Proposition 2.6 we have the following.
In [FM02], Farb-Mosher defines the notion of convex cocompact subgroups of MCG(S) as an analogue of convex cocompact Kleinian groups.Instead of giving the definition, we regard the following theorem due to Kent-Leininger as a definition of convex cocompactness.

Theorem 3.9 ([KL08, Section 7]). A finitely generated subgroup H < MCG(S) is convex cocompact if and only if H qi-embeds into C(S).
We call a subgroup H < MCG(S) purely asymmetric if every element in H of infinite order is asymmetric.This is equivalent to saying every element in H of infinite order does not preserve any symmetric subspaces.Then by Theorem 2.9, we have: Theorem 3.10.Let S be as in Theorem 3.7.Let H be a finitely generated subgroup of MCG(S).Then (a) H is convex cocompact with every element in the limit set asymmetric in C(S) if and only if H qi-embeds into C q (S).
(b) If H satisfies the equivalent condition in (a), then H is purely asymmetric.
For any group with non-elementary action by isometries on a Gromov hyperbolic space X, Taylor-Tiozzo [TT16] proved that random subgroups given by independent random walks are qi-embedded free groups.Putting Theorem 3.7, Theorem 3.10 and [TT16] together we get the following.

The MCG(S) action on C q (S) is WPD
In this subsection, we prove Theorem 3.7.First, we prepare the following.Lemma 3.12.Let g ∈ MCG(S) be an asymmetric pseudo-Anosov element with quasi-geodesic axis α.We denote by π α the nearest projection map to α on C(S).Then there exists C > 0 such that for any finite covering p : S → S ′ and the corresponding symmetric subspace Π(C(S ′ )), we have diam C (π α (Π(C(S ′ )))) < C, where diam C is the diameter in C(S).
A key ingredient of a proof of Lemma 3.12 is the uniform finiteness of the number of parallel symmetric subspaces, which is proved in [Mas17a].We need the following corollary to Lemma 3.13 to prove Lemma 3.12.This corollary may be of independent interest.Corollary 3.14.Let g ∈ MCG(S) be pseudo-Anosov.Then g is symmetric if and only if one of its stable or unstable foliation is symmetric.
Proof of Lemma 3.12.Suppose the contrary that there is a sequence Π n of symmetric subspaces such that diam C (π α (Π n )) → ∞.Since in any finitely generated group, the number of subgroups of a given index is finite, the number of coverings from S up to conjugacy is finite.Hence after taking a subsequence, we may suppose that Π n = g n Π(C(S ′ )) for some g n ∈ MCG(S) and some fixed symmetric subspace Π(C(S ′ )).We fix a point p ∈ α.By applying suitable power of g to Π n we may suppose that all π α (Π n ) are coarsely centered at p.By Gromov hyperbolicity of C(S), if diam C (π α (Π n )) is greater than a constant which depends only on the hyperbolicity constant, then Π n must fellow travel with α.By Corollary 3.14, the stable and unstable foliations of g are asymmetric, which implies that {Π n } must contain infinitely many distinct elements.Thus we get a contradiction to Lemma 3.13.Finally, we verify the existence of asymmetric pseudo-Anosov maps.We remark that this result does not hold for the exceptional surfaces, as every map is a lift with respect to the covering given by a hyperelliptic involution, and we now enumerate them for the convenience of the reader.Given an orbifold cover p : S → S ′ , we shall write S ′ as S ′ g ′ (d 1 , . . .d n ′ ), where g ′ is the genus of S ′ , n ′ is the number of orbifold points, and the d i 2 are the orders of the orbifold points, so each orbifold points has metric angle 2π/d i .We allow d i = ∞, corresponding to a puncture.With this notation the exceptional cases are: Recall that the dimension of the Teichmüller space of an orbifold S ′ g ′ (d 1 , . . .d n ′ ) is 6g ′ − 6 + n ′ , and so the dimension only depends on the number of orbifold points, not on their orders.This is because any two orbifold points have small neighbourhoods which are conformally equivalent.Therefore the set of conformal structures on S ′ g ′ (d 1 , . . ., d n ′ ) is equal to the set of conformal structures on S g ′ ,n ′ , and due to work of Picard [Pic91] in the closed case, and Heins [Hei62] for the punctured case, given a choice of order d i at each orbifold point, there is a unique hyperbolic metric with cone points of that order in a given conformal class.
The following result is well-known, but we provide the details for convenience.
Proposition 3.15.Let S be a surface of genus g with n punctures, with 2g+n > 4, and let p : S → S ′ be an orbifold cover.Then dim(T (S ′ )) < dim(T (S)).
Proof.The hyperbolic metric on S ′ lifts to a hyperbolic metric on S, and this gives an isometric embedding T (S ′ ) → T (S), see for example [RS09, Section 7].In particular, if the dimension of T (S ′ ) is equal to the dimension of T (S), then the image of T (S ′ ) is equal to T (S).
First consider the case in which the cover is regular, with finite deck transformation group F < MCG(S).The image of T (S ′ ) in T (S) is the set of points fixed by all elements of F , and so the fixed point set of F is all of T (S).However, the mapping class group MCG(S) acts faithfully on T (S), except for the finite list of exceptional cases S 0,4 , S 1,1 , S 1,2 and S 2,0 in which there are hyperelliptic involutions which act trivially on T (S), see for example [FM12, Section 12.1].However, these are precisely the surfaces we have excluded with the condition 2g + n > 4. Now suppose that the cover p : S → S ′ is not regular.Let S ′ have genus g ′ , and n ′ orbifold points.The generalization of Royden's Theorem [Roy71] due to Earle and Kra [EK74] states that if T (S) is isometric to T (S ′ ) then S is homeomorphic to S ′ , unless 2g + n 4, but this is precisely the exceptional cases we have excluded.This implies that g = g ′ and n = n ′ .We now show that in fact there can be no such orbifold cover with these properties.
Recall that the Euler characteristics of the two surfaces are given by The order of an orbifold point satisfies d i 2, and so as g = g ′ and n = n ′ : Euler characteristic is multiplicative under covers, so χ(S ′ ) = 1 d χ(S ′ ), where d is the degree of the covering map: This gives: Which we may rewrite as: As we are assuming that the cover is irregular, d 3. The only possible nonnegative integer solutions have g = 0 or g = 1.If g = 1, then n = 0, but the torus does not have a hyperbolic metric.If g = 0, then For d 7 this implies that n < 5, so the only possibility is the exceptional surface S 0,4 .The remaining finite list of possibilities consists of: S 0,8 , d = 3, S 0,7 , d = 3, S 0,6 , d = 3, 4, S 0,5 , d = 3, 4, 5, 6 We now consider these in turn.d = 3 First consider the case in which the cover is degree d = 3, and the surface S is S 0,n , with 5 n 8.The Euler characteristic of S is χ(S) = 2 − n, so χ(S ′ ) = 2−n 3 .Let S ′ have a orbifold points of order 3 and n − a orbifold points of order ∞.Then which implies that a = 2n − 4.However, a n, which implies that n 4, a contradiction.d = 4 Now consider the case in which the cover is degree d = 4, and the surface S is S 0,n , with 5 n 6.The Euler characteristic of S is χ(S) = 2 − n, so χ(S ′ ) = 2−n 4 .Let S ′ have a orbifold points of order 2, b orbifold points of order 4 and n − a − b orbifold points of order ∞.Then which we may rewrite as 2a + b = 3n − 6.
When n = 5 we obtain 2a + b = 9, but a + b 5, so the only possible solution with non-negative integers is a = 4 and b = 1, but this means that S ′ has no punctures, a contradiction.
When n = 6 we obtain 2a + b = 12, but a + b 6, so the only possible solution with non-negative integers is a = 6 and b = 0, but again, this means that S ′ has no punctures, a contradiction.
which we may rewrite as: However, each of a, b and c is at most 5, so the only possible solution with non-negative integers is a = 5, b = 0 and c = 0.This implies that the number of punctures for S ′ is also zero, a contradiction.
This completes the proof of Proposition 3.15.
Proposition 3.16.Let S be a surface of genus g and n.Suppose 2g + n > 4.
Then there exists an asymmetric pseudo-Anosov element in MCG(S).
Proof.An orbifold cover p : S → S ′ gives an isometric embedding of T (S ′ ) in T (S), and as we are not in one of the exceptional cases, by Proposition 3.15, the dimension of T (S ′ ) is strictly less than the dimension of T (S).
There is a finite index subgroup H < MCG(S ′ ) consisting of mapping classes which lift to maps of S. By abuse of notation, we shall also refer to the corresponding subgroup in MCG(S) as H, and this subgroup preserves the image of T (S ′ ) in T (S).
Let B(x, r) be a ball in T (S), then there is an ǫ > 0 such that B(x, r) lies in the ǫ-thick part of T (S).Pick a basepoint on T (S ′ ).The subgroup H acts coarsely transitively on the ǫ-thick part of T (S ′ ), and so if some translate gT (S ′ ), for g in MCG(S), intersects B(x, r), then there is a translate of the basepoint a bounded distance from x.As MCG(S) acts properly on T (S) there are only finitely many distinct translates of T (S ′ ) intersecting B(x, r).
As there are only finitely many quotients p : S → S ′ up to covering space isomorphism, there are only finitely many isometrically embedded symmetric subsets T (S ′ ) intersecting B(x, r).As we are in the non-exceptional case, they all have strictly smaller dimension than T (S).In particular, there is a ball B(x ′ , r ′ ) ⊆ B(x, r) in T (S) disjoint from all of the symmetric subsets (i.e. the union of the gT (S ′ ) over all g and all covers).
Periodic geodesics are dense in T (S), see for example [EM11, Section 6].Therefore, there is a pseudo-Anosov mapping class with geodesic axis intersecting B(x ′ , r ′ ), and this pseudo-Anosov mapping class is asymmetric, as required.
Proof of Theorem 3.7.By Lemma 3.12, we see that the assumption (2) of Theorem 2.4 is satisfied for X = C(S), Y = {Π(C(S ′ ))} and an asymmetric pseudo-Anosov g.Then Z = C q (S).As every pseudo-Anosov element is WPD on C(S) [BF02, Proposition 11], every asymmetric pseudo-Anosov is also WPD on C q (S), especially it is loxodromic.On the other hand, note that if an element in MCG(S) acts loxodromically on C q (S), then it must act loxodromically on C(S).Since every symmetric pseudo-Anosov element must fix corresponding symmetric subspaces, it can not be loxodromic.Thus (a) follows.
We now prove (b).If 2g + n > 4, then MCG(S) contains a free group of rank 2, and in particular it is not virtually cyclic.Then by Proposition 3.16, at least one loxodromic element exists.We have already seen that every loxodromic element acts WPD, hence the action of MCG(S) on C q (S) is strongly WPD.That the action is non-elementary is the consequence of Proposition 2.3.

Non-maximal train tracks
We say a pair of curves a and b on a surface is filling if when put in minimal position, there is no essential closed curve disjoint from a ∪ b.In particular, this implies that every complementary region is a disc containing at most one puncture.We say a pair of curves is maximally filling if when put in minimal position every complementary region is a square, hexagon or a bigon containing a single puncture.
We say a pair of curves a and b is K-maximally filling, if for any pair of curves a ′ and b ′ with d C(S) (a, a ′ ) K and d C(S) (b, b ′ ) K, then a ′ and b ′ are maximally filling.
In this section we will consider surfaces S g,n which support a pair of curves which are maximally filling, and which also support a pair of curves which fill the surface, but are not maximally filling.This consists of all surfaces which support a pseudo-Anosov map, with the exceptions of S 0,4 and S 1,1 .The fourpunctured sphere S 0,4 has the property that all filling curves are maximally filling.The punctured torus S 1,1 has the property that there are no maximally filling pairs of curves, as any pair of curves in minimal position may be realised as a pair of geodesics in a flat metric on the torus with a marked point, and the marked point is then contained in a square, not a bigon.We shall therefore consider surfaces S g,n with 3g + n > 4, as this condition includes all surfaces which support a pseudo-Anosov map, excluding S 0,4 and S 1,1 .
Remark 4.1.In the case that the surface S is either S 0,4 or S 1,1 , the Teichmüller space T (S) has complex dimension 1, so there is a single stratum of quadratic differentials, and all the pseudo-Anosov maps have invariant foliations with the same collection of singularities.In the case of S 0,4 , the invariant foliation is the quotient of an invariant foliation for an Anosov map on a torus by the hyperelliptic involution, with the order two cone points replaced by punctures.This foliation then has exactly four 1-prong singularities, and so Theorem 1.3 holds in this case.However, in the case of S 1,1 , Theorem 1.3 does not hold, as the invariant foliations are preserved by the hyperelliptic involution, which fixes the puncture, and so the foliations cannot have 1-prong singularities at the puncture.
Lemma 4.2.Let S = S g,n be a surface of finite type of genus g with n punctures, with 3g + n > 4. Let g be a pseudo-Anosov element on S with maximal invariant laminations and let α be an axis for g.Then for any K there is an R such that any two curves on α distance at least R apart are K-maximally filling.
We shall prove this result in this section.We start by reviewing some useful properties of train tracks.

Train tracks
We briefly recall some of the results we use about train tracks on surfaces.For more details see for example Penner and Harer [PH92].
Recall that a train track on a surface S is a smoothly embedded graph, such that the edges at each vertex are all mutually tangent, and there is at least one edge in each of the two possible directed tangent directions.Furthermore, there are no complementary regions which are nullgons, monogons, bigons or annuli.The vertices are commonly referred to as switches and the edges as branches.We will always assume that all switches have valence at least three.A trivalent switch is illustrated below in Figure 1.An assignment of non-negative numbers to the branches of τ , known as weights, satisfies the switch equality if the sum of weights in each of the two possible directed tangent directions is equal: that is a = b + c in Figure 1 above.
An assignment of non-negative numbers to the branches of τ , known as weights, satisfies the switch equality if the sum of weights in each of the two possible directed tangent directions is equal: that is a = b + c in Figure 1 above.A train track with integral weights defines a simple closed multicurve on the surface, and we say that this curve is carried by τ .We shall write C(τ ) for the subset of the curve complex consisting of the simple closed curves carried by τ .
More generally, for weights with non-integral values, a weighted train track defines a measured foliation on the surface.We say that the corresponding foliation is carried by the train track.We shall write MF (S) for the space of measured laminations on the surface S, and P (τ ) for the set of measured foliations carried by τ .The set P (τ ) is projectively invariant, and we shall write P (τ ) for its projectivization in PMF(S), the space of projective measured foliations on the surface.The set P (τ ) is a polytope in PMF(S).Let V (τ ) be the set of vertices of P (τ ).Every v ∈ V (τ ) gives a vertex cycle: a simple closed curve, carried by τ , that puts weight at most two on each branch of τ .By abuse of notation, we shall also write V (τ ) for the corresponding set of simple closed curves in C(S).
We say a train track τ is maximal if every complementary region is a triangle or a monogon containing a single puncture.Every surface which contains a maximally filling pair of curves also contains a maximal train track, as given a choice of orientations on the curves, one may smooth the intersections compatibly to produce a maximal train track.

Non-classical interval exchange transformations
Train tracks are closely related to non-classical interval exchanges, which we now describe, see for example [BL09].
Definition 4.3.A non-classical interval exchange consists of the following: 1. Given positive numbers δ and ǫ, let B 0 be a Euclidean rectangle [0, δ]×[0, ǫ] which we shall call the base rectangle.We will refer to the sides of length ǫ as the vertical sides and the sides of length δ as the horizontal sides.We shall call {0} × [0, ǫ] the initial vertical side and {δ} × [0, e] the terminal vertical side.We shall call the horizontal side [0, δ] × {ǫ} the upper side and label it I + , and the other side [0, δ] × {0} the lower side and label it I − .
2. Let B 1 , . . .B n be a finite collection of metric Euclidean rectangles, which we shall call bands.Each band B i has one pair of opposite sides called the horizontal sides, and one pair called the vertical sides.The length of the horizontal sides of the band B i is called the width w i of the band B i , and the total width of the bands w i is equal to L.
3. For each horizontal side of a band there is a Euclidean isometry from the horizontal side to the disjoint union I + ⊔ I − , with the following properties: (a) The images of the interiors of any two horizontal sides are disjoint.
(b) The quotient space obtained from gluing a band to the base rectangle along its two horizontal sides is orientable, i.e. homeomorphic to an annulus and not a Möbius band.
If a band has one horizontal side mapped to I + , and the other mapped to I − , then we call it an orientation preserving band.Otherwise we call it an orientation reversing band.
Every generalized interval exchange transformation gives rise to a measured train track with a single vertex, by collapsing the base rectangle to a vertex, and each band to a edge of weight w i .
In the description above we attached the bands to the horizontal sides of the rectangle, but we could instead have attached them all to the vertical sides.

Quadratic differentials and transversals
A pseudo-Anosov map g preserves a unique geodesic axis γ in Teichmüller space.Choose a unit speed parameterization γ t such that g(γ 0 ) = γ t for some positive t.Let q be the quadratic differential at γ 0 determined by the geodesic γ.The quadratic differential q determines a flat structure on S which we shall denote S q .Let F + be the vertical measured foliation determined by S q , whose projectivization F + is the stable invariant projective measured foliation for g.
Similarly, let F − be the horizontal measured foliation for S q , whose projectivization F − is the unstable invariant foliation for g.
A transversal t for a foliation F in S is an embedded arc in S which is disjoint from the singularities of F , and is transverse to the leaves of F .We say a transversal for F + is horizontal if it is a horizontal geodesic in the flat metric S q corresponding to q. Similarly we say a transversal for F − is vertical if it is a vertical geodesic.
The transversal t determines a non-classical interval exchange.This may be thought of as arising by cutting the surface along the singular flow lines, and then completing each maximal open interval of non-singular flow lines by adding a pair of non-singular edges at each end.We now give a detailed description of this construction for horizontal transversals for the vertical foliation F + .
Let t be a horizontal transversal for the vertical foliation F + .As t is a horizontal geodesic disjoint from the singular set of F , there is an ǫ > 0 such that the Euclidean rectangle t × [−ǫ, ǫ] is also disjoint from the singular set.We shall choose this to be the base rectangle B 0 in the non-classical interval exchange map.Recall that one horizontal side of the base rectangle is called I + and the other horizontal side is called I − .
A flow line ℓ is the closure of a connected component of a leaf of the foliation F in S \ B 0 .A flow line is non-singular if it is a properly embedded arc with distinct endpoint in the horizontal boundary of B 0 .A flow line is singular if it contains a singular point of the foliation F .A tripod is the topological space homeomorphic to the connected and simply connected graph consisting of three edges meeting at a common vertex.The foliation F + contains finitely many singular points, all of which are trivalent, so there are finitely many singular flow lines, which are tripods, properly embedded with distinct endpoints in the horizontal boundary of B 0 .All other flow lines are non-singular, and this determines a first return map on I + ⊔ I − , defined on all but finitely many points, which is an isometry on connected intervals for which it is defined.Furthermore, as S is orientable, any subsurface consisting of B 0 and any collection of non-singular flow lines is orientable, so if we choose a band for each maximal subinterval of I + ⊔ I − on which the first return map is defined, this is a non-classical interval exchange transformation.
The same construction applies to a vertical transversal for the horizontal foliation F − , but with the vertical and horizontal directions swapped.

Rauzy induction
We say a transversal t = [t 0 , t 1 ] for a foliation F is admissible if it determines a non-classical interval exchange such that the terminal point t 1 is the endpoint of a singular flow line.In this case, let t ′ 1 ∈ [t 0 , t 1 ) be the closest endpoint of a singular flow line to t 1 .The transversal [t 0 , t ′ 1 ] is a subset of [t 0 , t 1 ] and determines a new non-classical interval exchange, again with the property that the terminal endpoint meets a singular flow line.An example of this is illustrated in Figure 2.
This process is called Rauzy induction and is a special case of splitting a train track.For general non-classical interval exchanges, Rauzy induction need not always be defined, but the following result shows that it is always defined for the invariant foliations of a pseudo-Anosov map.
Proposition 4.4.Let g be a pseudo-Anosov map on a surface S, let q be a choice of quadratic differential corresponding to the invariant Teichmüller geodesic, and let S q be the corresponding flat surface.Let t = t 1 be an admissible horizontal transversal.Then there is an infinite sequence of horizontal transversals (t n ) n∈N , in which t n+1 is obtained from t n by Rauzy induction.Furthermore, the length of the horizontal transversals t n tends to zero as n tends to infinity.
Replacing g by g −1 in Proposition 4.4 switches the horizontal and vertical directions, so the result also holds for a vertical transversal for the horizontal foliation F − .Proposition 4.4 is an immediate consequence of the next two results, Lemma 4.5 and Proposition 4.6.
A saddle connection for a flat surface is a geodesic connecting two singular points, whose interior is disjoint from the singular set.we say a saddle connection is vertical if it is contained in the vertical foliation.The following fact is well known, see for example [FM12,Lemma 14.11].
Lemma 4.5.[FM12, Lemma 14.11]The invariant foliations for a pseudo-Anosov map do not contain any saddle connections.
In this case, Rauzy induction is always defined, and gives an infinite sequence of transversals t n determining non-classical interval exchanges: Proposition 4.6.[BL09, Proposition 4.2] If the vertical foliation F + has no saddle connections, then Rauzy induction starting at a horizontal transversal t 1 gives an infinite sequence of transversals (t n ) n∈N determining non-classical interval exchanges, such that the horizontal length of the transversals t n tends to zero as n tends to infinity.

Maximally filling curves
Given a non-singular point x in the flat surface S q there are positive numbers δ and ǫ such that x is the bottom left corner of an embedded Euclidean rectangle B 0 = s×t, disjoint from the singular set, where |s| = δ and |t| = ǫ.In particular, s is a horizontal transversal for the vertical foliation F + and t is a vertical transversal for the horizontal foliation F − .We may therefore use B 0 as the base rectangle for a non-classical interval exchange in both the horizontal and vertical directions.Furthermore, possibly after passing to a subrectangle [0, δ ′ ]×[0, ǫ ′ ] ⊆ [0, δ] × [0, ǫ], we may assume that both non-classical interval exchanges are admissible.By abuse of notation we shall relabel the new rectangle as B 0 , and relabel the sides as s and t of lengths δ and ǫ.
For the vertical non-classical interval exchange, let d + be the union of the singular flow lines for the vertical foliation F + , together with the flow lines incident to the initial vertical edge {0} × [0, ǫ].For the horizontal non-classical interval exchange, let d − be the union of the singular flow lines for the horizontal foliation F − , together with the flow lines incident to the initial horizontal edge [0, δ] × {0}.
This determines a cell decomposition of the surface as follows.
1.The vertices V are the singular points of S q , together with all intersection points d + ∩ d − , and all endpoints of d + and d − with the base rectangle B 0 .
2. The edges E are the connected components of (∂B The collection of faces consists of the base rectangle B 0 , together with a collection of discs, each of which is a Euclidean rectangle in S q with two vertical edges, each of which is contained either in d + or the vertical boundary of B 0 , and two horizontal edges, each of which is contained either in d − or the horizontal boundary of B 0 .
The vertices are one of the following types: 1. Trivalent vertices in the interiors of the sides of B 0 .
2. 4-valent vertices corresponding to the corners of B 0 , and all non-singular intersection points of d + ∩ d − .
3. 1-valent and 6-valent vertices corresponding respectively to the 1-prong and trivalent singular points of S q .
Proposition 4.7.Let v be a simple closed curve carried by the vertical train track, which passes over every vertical band at least twice.Let w be a horizontal curve carried by the horizontal train track, which passes over every horizontal band at least twice.Then v and w are maximally filling.
Proof.We may isotope v to consist of a union of vertical flow lines, together with geodesic arcs in B 0 , each of which has endpoints in the opposite horizontal sides of B 0 .Similarly, we may isotope w to consist of a union of horizontal flow lines, together with geodesic arcs in B 0 , each of which has endpoints in the opposite vertical sides of B 0 .
The intersections of v and w with each face of the cell structure for S q is: B 0 If a complementary region of v ∪ w is contained in the interior of a single face, then it is a square.
If it intersects an edge of the cell structure, but is disjoint from the vertices, then it is a square.Finally, if it contains a singular point of S q , then this singular point is either a 1-prong singularity, or a trivalent singular point.In the first case the component is a bigon containing a puncture, and in the second case the component is a hexagon.This is illustrated in Figure 6.

Widely separated curves are maximally filling
It is well known that if the vertical foliation is uniquely ergodic, then the intersection of the projective measured foliations carried by the train tracks arising from Rauzy induction is equal to the vertical foliation.We provide a proof for the convenience of the reader.Proposition 4.8.Let S q be a flat surface with uniquely ergodic vertical foliation F + .Let t n be a sequence of nested horizontal transversals in S q such that the length of t n tends to zero as n tends to infinity.Let τ n be the train track determined by t n .Then P (τ n ) = {F + } in PMF (S).
Proof.Let τ n be the train track corresponding to t n .As the transversals are nested, t n+1 ⊂ t n , each train track is carried by the previous one, τ n+1 ≺ τ n , and so P (τ n+1 ) ⊆ P (τ n ).Every train track τ n in the splitting sequence carries the vertical foliation F + , so F + ∈ P (τ n ).We now show P (τ n ) = {F + }.
Let F + (a) be the measure of a with respect to the vertical measured foliation F + , and let F − (a) be the measure of a with respect to the horizontal foliation F − .Consider the ratio between these two measure, ρ q (a) = F − (a)/F + (a), which is projectively invariant, i.e. ρ q (λa) = F − (λa)/F + (λa) = λF − (a)/λF + (a) = ρ q (a), and so defines a function on PMF(S).
If a is carried by τ n , then it has a piecewise geodesic representative in S q consisting of flow lines in bands, which are vertical geodesics segments, and geodesic segments in the base rectangle B 0 , each of which has one endpoint in the lower horizontal side, and one endpoint in the upper horizontal side.Flow lines have zero vertical measure in F + , and positive horizontal measure in F − .Each segment in B 0 contributes at least 2ǫ to the horizontal measure of a, and at most F + (t n ) to the vertical measure of a. Therefore ρ q (a) 2ǫ/F + (t n ), where F + (t n ) is equal to the length of the horizontal geodesic t n in the flat metric S q .As the length of t n tends to zero, any fixed curve a can only be carried by finitely many τ n .In particular, for any simple closed curve a, there is an n such that a ∈ P (τ n ).
This argument works for elements F ∈ PMF (S).Suppose that (a n ) n∈N is a sequence of simple closed curves converging to F .Then for any representative F in MF (S) for F , there is a sequence of numbers λ n such that λ n a n → F .Then F + (λ n a n ) → F + (F ), where F + (F ) is equal to the intersection number i(F + , F ) of the two measured foliations.In particular, if i(F + , F ) > 0, then there is an n such that F ∈ P (τ n ).Therefore all foliations in P (τ n ) have zero intersection number with F + , but as F + is uniquely ergodic, the set of all projective measured foliations with zero intersection number with F + consists just of {F + }.
We will also use the following results of Masur and Minsky and Klarreich.The first says that the vertex sequences arising from splitting sequences of train tracks are unparameterized quasigeodesics in the curve complex.
Theorem 4.9.[MM04, Theorem 1.3] Given a surface S, there are constants Q and c such that the vertices of a nested train-track splitting sequence form an unparameterized (Q, c)-quasigeodesic in C(S).
The next says that a 1-neighbourhood of a simple closed curve which runs over every branch of a train track is in fact carried by the train track.Finally, the boundary of the curve complex is the space of minimal foliations.
Theorem 4.11.[Kla99, Theorem 1.3]The boundary at infinity of the curve complex C(S) is the space of minimal foliations on S.
We may now prove Lemma 4.2.
Proof.Choose a non-singular point in S q and construct a base rectangle B 0 = t 1 × s 1 disjoint from the singular set, such that both vertical and horizontal non-classical interval exchanges are admissible.Let (t n ) n∈N and (s n ) n∈N be the train tracks arising from applying Rauzy induction to t 1 and s 1 respectively.
After applying Rauzy induction to t 1 , there is a transversal t k1 such that each band runs at least twice over every band of the original non-classical interval exchange.Similarly, after applying Rauzy induction to s 1 , there is a transversal s k2 and such that each band runs at least twice over every band of the original non-classical interval exchange.To simplify notation, choose k = max{k 1 , k 2 }, and consider the train tracks τ k determined by t k , and σ k determined by s k .
By Proposition 4.8, Theorem 4.9 and Theorem 4.11, the vertex sequence (V (τ n )) n∈N is a (Q, c)-quasigeodesic, whose limit point in ∂C(S) is equal to F + .The positive limit point of the (1, K 1 )-quasigeodesic α g is also equal to F + , so there is a constant L such that all of the V (τ n ) are contained in an L-neighbourhood of any (1, K 1 )-quasiaxis for g.
Recall that the Rauzy induction moves may be realised in S q by extending the flow lines giving the boundaries of the bands.As F + is minimal, every leaf is dense, and so for every τ k there is a k ′ > k such that every band of τ k ′ runs at least once over every band of τ k .The nesting lemma, Lemma 4.10, then implies that N 1 (C(τ k ′ )) ⊆ C(τ k ).By repeating this process, for any K there is a k ′ such that N K (C(τ k ′ )) ⊆ C(τ k ).In particular, for any K + L there is a k ′ such that N K+L (C(τ k ′ )) ⊂ C(τ k ) in C(S).Let α + = α g ([r + , ∞)) be the terminal ray of α such that α + ⊆ N L ((V (τ n )) n k ′ ).Then N K (α + ) ⊆ N K+L (C(τ k ′ )) ⊆ C(τ k ).
As g acts by translations on α g , given any two points v and w on α g distance at least R = d C (α − , α + ) + d C (a, ga) + O(δ) apart, we may translate them by powers of g such that, up to relabelling, v lies in α − and w lies in α + .Let v ′ and w ′ be curves distance at most K from v and w respectively.Then v ′ ∈ C(σ k ) and w ′ ∈ C(τ k ), so v ′ and w ′ are maximally filling, by Proposition 4.7.Therefore, v and w are K-maximally filling, as required.
Lemma 4.12.Let g be a pseudo-Anosov element with maximal invariant laminations and axis α.Then there is a D such that for any non-maximal train track τ , the nearest point projection of C(τ ) to α has diameter at most D.
Proof.The set C(τ ) is uniformly quasiconvex, so if there is a projection of size R, then there are curves a 1 , a 2 in C(τ ) with d C(S) (α, a i ) K and d C(S) (a 1 , a 2 ) R − 2K − O(δ), where K is a uniform constant independent of C(τ ).
As a 1 and a 2 are both carried by τ , they have a complementary region which is not either a triangle or a monogon with exactly one puncture, so this contradicts the lemma above.Now by Lemma 4.12 combined with Theorem 2.4, we have the following theorem which corresponds to the statement for C τ (S) in Theorem 1.1.Theorem 4.13.Let S be an orientable surface of genus g with n punctures.Suppose 2g + n > 3.For the action of MCG(S) on C τ (S), we have In particular, the action of MCG(S) on C τ (S) is non-elementary.
Lemma 4.12 holds for all surfaces with 3g + n > 4.However, part (a) of this result also holds for S 0,4 by Remark 4.1.In fact, (b) also holds, as the non-maximal train tracks in S 0,4 have finite diameter in C(S 0,4 ), and so the result holds for all surfaces with 2g + n > 3.Among surfaces suppporting pseudo-Anosov elements this excludes only S 1,1 , for which the result does not hold.
Lemma 3.13 ([Mas17a, Lemma 4.5.]).Let p : S → S ′ be a finite covering and denote the corresponding symmetric subspace by Π(C(S ′ )).Then for any D 0 > 0, there exist D 1 , D 2 > 0 which depend only on S and D 0 such that for any a, b ∈ C(S) with d C (a, b) > D 1 , the number of distinct elements in {gΠ(C(S ′ )) | d C (a, gΠ(C(S ′ ))) < D 0 and d C (b, gΠ(C(S ′ ))) < D 0 } is bounded from above by D 2 .Here we count the number of distinct images i.e. if g 1 Π(C(S ′ )) = g 2 Π(C(S ′ )) as subsets, we just count one time.

Figure 1 :
Figure 1: A trivalent switch for a train track.

3.
The 2-cells or faces F are the connected components of S \ (∂B ∪ d + ∪ d − ).

Figure 3 :
Figure 3: The curves v and w.

Figure 4 :
Figure 4: A complementary region intersecting an edge.

Figure 6 :
Figure 6: Complementary regions containing singularities of S q .
Lemma 4.10.[MM04, Lemma 3.4] If τ is a maximal birecurrent train track and a is a simple closed curve which runs over every branch of τ , then N 1 (a) ⊆ P (τ ).
(a) g ∈ MCG(S) is loxodromic if and only if g has maximal invariant laminations, and (b) the action of MCG(S) on C τ (S) is strongly WPD.