{'fraction_non_alphanumeric': 0.07133099402673237, 'fraction_numerical': 0.012562612166180055, 'mean_word_length': 3.5176669893514037, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 18, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Hyperbolic spaces in Teichm\"uller spaces

We prove, for any n, that there is a closed connected orientable surface S so that the hyperbolic space H^n almost-isometrically embeds into the Teichm\"uller space of S, with quasi-convex image lying in the thick part. As a consequence, H^n quasi-isometrically embeds in the curve complex of S.

Our main theorem states that if we drop the assumption of equivariance, then quasiisometric embeddings with all the remaining properties exist. On the other hand, as was shown in [6], one can find cocompact lattices Γ < Isom(H 2 ) and Γ-equivariant quasiisometries into T (S) with image in T (S)-for these examples the image is not quasi-convex.
The main theorem for n = 2 also contrasts with the situation of isometrically embedding hyperbolic planes in T (S). More precisely, every geodesic in T (S) is contained in an isometrically embedded hyperbolic plane (with the Poincaré metric) called a Teichmüller disk. However, it is well-known that no Teichmüller disk lies in any thick part-this follows from [21] which guarantees that along a dense set of geodesic rays in the Teichmüller disk the hyperbolic length of some curve on S tends to zero.
The curve complex of S is a metric simplicial complex C(S) whose vertices are isotopy classes of essential simple closed curves, and for which k + 1 distinct isotopy classes of curves span a k-simplex if they can be realized disjointly. In [23], Masur and Minsky proved that C(S) is δ-hyperbolic. One of the key ingredients in their proof is the construction of a coarsely Lipschitz map T (S) → C(S). The restriction of this map to any quasi-convex subset of T (S) is a quasi-isometry (see for example [27,Lemma 4.4] or [15,Theorem 7.6]). Composing the almost-isometry of Theorem 1.1 with the map T (S) → C(S) we have the following corollary. Corollary 1.3. For every n ≥ 2, there exists a surface of finite type S and a quasi-isometric embedding H n → C(S). The case of n = 2 here can be compared to the result of Bonk and Kleiner [5] in which it is shown that every δ-hyperbolic group which is not virtually free contains a quasi-isometrically embedding hyperbolic plane. The assumption that the group is not virtually free implies the existence of an arc in the boundary. According to [9] (see also [19,18]) with the exception of a few small surfaces, there are indeed arcs in the boundary of C(S). In [5] however, essential use is made of the fact that there is an action of the group, and so even in the case n = 2, Corollary 1.3 does not follow from [5].
We now explain the idea for the construction in the case n = 2. Given a closed Riemann surface Z and a point z ∈ Z, the Teichmüller space T (Z, z) is naturally a H 2 -bundle over T (Z); see Section 4.3. Given a biinfinite geodesic τ in T (Z), the preimage of τ in T (Z, z) is a 3-manifold. The parameterization t → τ (t) lifts to a flow on the preimage of τ for which the flow lines are geodesics in T (Z, z). The fiber over τ (0) admits a pair of transverse 1-dimensional singular foliations-these are naturally associated to the vertical and horizontal foliations of the quadratic differential defining τ . Any two flow lines meeting the same nonsingular leaf of the vertical foliation are forward asymptotic. Therefore, we have a 1-parameter family of forward asymptotic geodesics in T (Z, z). We use this to define a map from H 2 to T (Z, z): we pick a horocycle C ⊂ H 2 and send the pencil of geodesics perpendicular to C to our set of forward asymptotic geodesics in T (Z, z).
At the beginning of Section 5.2 we give a brief explanation of how this can be modified to give the construction for n = 3. The idea for n ≥ 4 is then a straightforward inductive construction.
Definition 2.1. A map F : X → Y is a K-almost-isometric embedding if for all x, x ∈ X we have |dX (x, x ) − dY (F (x), F (x ))| ≤ K.
For two points p, q ∈ H n we use d H (p, q) to denote the distance between them. The exponential model of hyperbolic space is related to the upper-half space model U = R n−1 × (0, ∞) by the map H n → U given by (x, t) → (x, e t ). In the exponential model, for every x ∈ R n−1 the path ηx(t) = (x, t) is a vertical geodesic and is parameterized by arc-length.
Lemma 2.2. Suppose (X, dX ) is a geodesic metric space and δ, , R > 0 are constants. Suppose F : H n → X is a function with the following properties.
1. F • ηx is a geodesic for all x ∈ R n−1 .
2. For distinct x, x ∈ R n−1 the geodesics F • ηx and F • η x are two sides of an ideal δ-slim triangle in (X, dX ).

For any
Then there exists a constant K so that F is a K-almost isometric embedding.
A useful consequence of Property 3 is that for any x, x , t ∈ R we have The remainder of this section gives the proof of Lemma 2.2. We begin by controlling how F moves the centers of ideal triangles. To be precise: Suppose that T = P ∪ Q ∪ R ⊂ H n is an ideal triangle where P and Q are distinct vertical geodesics. Let r ∈ R denote the point of R with maximal t-coordinate. We call r the midpoint of R. Thus r serves as a center for Observe, say from the upper-half space model, that for all t ≥ t(r) we have Thus, by Inequality (1) we have dX (F (x, t), F (x , t)) ≤ 2R/ +R. Define ∆ = max{3δ, 2R/ + R} and define the displaced height of T to be It follows that h(T ) ≤ t(r). Note that for any vertical triangle T , Property 2 implies that h(T ) > −∞.
As mentioned above, for every vertical triangle T we have h(T ) > −∞ and hence t(r) − h(T ) < ∞. We now obtain a uniform bound on this quantity.
Proof. Suppose not. Then we are given a sequence of vertical triangles Note that by Equation (2) Thus e −h k |x k − x k | tends to infinity with k. From Property 4 we deduce that the quantity dX (F (x k , h k ), F (x k , h k )) also tends to infinity with k. This last, however, contradicts Claim 2.3.
We give the proof of Lemma 2.2. Fix any p, q ∈ H n . If x(p) = x(q) then we are done by Property 1. Suppose instead that x(p) = x(q). Let P ∪ Q ∪ R denote the vertical triangle having vertical sides P and Q so that x(P ) = x(p), x(Q) = x(q); let r ∈ R be the midpoint of the non-vertical side. Define C1 = 2C0 + 5∆ + 1. There are now two cases to consider.
Let p ∈ P and q ∈ Q be the points with t(p ) = t(q ) = max{t(p), t(r)}. Then by the triangle inequality and Equation (2) we have It follows that d H (p, q) is estimated by d H (q , q) = |t(q ) − t(q)| up to an additive error at most C0 + C1 + 2. Appealing to Property 1, Inequality (1), and the triangle inequality we similarly have Thus dX (F (p), F (q)) is estimated by dX (F (q ), F (q)) = d H (q , q) with an additive error at most C0 + C1 + 2R/ + R. This completes the proof in this case.
In this case, since the triangle P ∪ Q ∪ R is slim in H n , we find that that d H (p, q) is estimated by t(r) − t(p) + t(r) − t(q) up to an additive error of at most 2. We now show that dX (F (p), F (q)) is also estimated by the latter quantity, with a uniformly bounded error. Using Property 1 and Inequality (1) deduce We now give a lower bound for dX (F (p), F (q)). Recall that F (P ) and F (Q) are two sides of a δ-slim triangle in X. Let RX be the third side of this triangle. Since dX (F (p), F (Q)), dX (F (P ), F (q)) > ∆ ≥ δ it follows that there are points pX , qX ∈ RX so that dX (F (p), pX ), dX (qX , F (q)) ≤ δ. Thus the distance dX (pX , qX ) is an estimate for dX (F (p), F (q)) with an additive error at most 2δ.
Define a = (x, hT ), b = (x , hT ). Again, as in the previous paragraph, there are points aX , bX ∈ RX within distance δ of F (a), Note that the geodesic segments [pX , aX ], [bX , qX ] ⊂ RX have length at least h(T )−t(p)−2δ and h(T ) − t(q) − 2δ respectively. Each of these is greater than C1 − 2δ.

Foliations and projections
Let Z be a closed surface of genus at least 2 and z a set of marked points. A measured singular foliation F on (Z, z) is a singular topological foliation so that • F has only prong-type singularties, • all one-prong singularties of F appear at points of z, and • F is equipped with a transverse measure of full support.
We refer the reader to [8,20] for a detailed discussion of measured foliations. Two measured (respectively, topological) foliations are measure equivalent (respectively, topologically equivalent) if they differ by isotopy and Whitehead moves. We will only be concerned with those foliations which appear as the vertical foliation for some meromorphic quadratic differential on Z (see Section 4.1). Every measured singular foliation is measure equivalent to such a foliation for a fixed complex structure on Z; see [13].
The space of measure classes of measured foliation on (Z, z) is denoted by MF(Z, z) and its projectivization by PMF(Z, z). A measured foliation F ∈ MF(Z, z) is arational if it has no closed leaf cycles. We say that F is uniquely ergodic if whenever F ∈ MF (Z, z) is topologically equivalent to F, then F and F project to the same point in PMF(Z, z). Both of these notions depend only on the topological classes of the foliations, and not the transverse measures.
If F is a measured foliation representing an element of MF(Z), and z ⊂ Z is a set of marked points, then F also determines an element of MF(Z, z). We note that it is important in this case that F be a foliation, and not an equivalence class of foliations. If F is arational as an element of MF(Z), and if z = {z} is a single point, then F is also arational as an element of MF(Z, z); see [19].
By a strict subsurface Y ⊂ Z − z we mean a properly embedded surface with nonempty boundary and a set of punctures, possibly empty, such that every component of ∂Y is an essential curve in Z − z; that is, homotopically nontrivial and nonperipheral. We also assume that Y is not a sphere with k punctures and j boundary components where k +j = 3. We will only refer to subsurfaces in one context, and that is as follows. Given a pair of arational measured foliation F, G ∈ MF(Z, z) and a proper subsurface between F and G in Y . This is the distance in the arc-and-curve complex of Y between the the subsurface projections of F and G to Y . For a detailed discussion, see [23,24]. All we use is that dY satisfies a triangle inequality for all arational measured foliations F1, F2, G ∈ MF(Z, z). This relates to Teichmüller geometry by Theorem 4.2 below.

Teichmüller spaces
Here we set notation and recall some basic properties of Teichmüller space. For background on Teichmüller space, we refer the reader to any of [2, 10, 1, 14].

Teichmüller space, quadratic differentials and geodesics
Given a closed Riemann surface Z with a finite (possibly empty) set of marked points z ⊂ Z, let T (Z, z) denote the Teichmüller space of equivalence classes of marked Riemann surfaces f is an orientation preserving homeomorphism to the Riemann surface X .
The equivalence relation is defined by where K h is the dilatation of h and where h : (Y, y) → (X, x) ranges over all quasi-conformal maps isotopic (rel marked points) to f • g −1 .
Given > 0, the -thick part of Teichmüller space T (Z, z) ⊂ T (Z, z) is the set of points [f : (Z, z) → (X, x)] ∈ T (Z, z) where the unique complete hyperbolic surface in the conformal class of X − x has its shortest geodesic of length at least . When is understood from context we will simply refer to T (Z, z) as the the thick part of Teichmüller space.
Let T (Z, z) → M(Z, z) denote the projection to moduli space obtained by forgetting the marking or, equivalently, by taking the quotient by the mapping class group. Mumford's compactness criterion [3] now implies: For any > 0, the thick part T (Z, z) projects to a compact subset of M(Z, z). Conversely, the preimage of any compact subset of is a closed Riemann surface with marked points and q ∈ Q(X, x) is a unit norm, meromorphic quadratic differential with all poles simple and contained in x. We also use q to denote the associated Euclidean cone metric on X. We note that Q(X) ⊂ Q(X, x), for any set of marked point x ⊂ X. Given q ∈ Q(X) we view it as an element of Q(X, x) whenever it is convenient.
Given q ∈ Q(X, x) and t ∈ R, let gt : (X, x) → (Xt, gt(x)) denote the e 2t -quasiconformal Teichmüller mapping defined by (q, t). Let qt ∈ Q(Xt, gt(x)) denote the terminal quadratic differential. For any point p ∈ X which is not a zero or pole of q we have a preferred coordinate z0 for (X, q) near p and preferred coordinate zt for (Xt, qt) near gt(p). In these coordinates q = dz 2 0 and qt = dz 2 t , and gt is given by being a Teichmüller geodesic through [f : (Z, z) → (X, x)]; note that every Teichmüller geodesic can be described in this way. The Teichmüller geodesic τ is -thick if the image of τ lies in T (Z, z). We also simply say a geodesic τ is thick if it is -thick for some > 0. A collection of geodesics {τα} is uniformly thick if there is an > 0 so that each τα is -thick.
Given q ∈ Q(X, x) we will let F(q), G(q) denote the vertical and horizontal foliations respectively; that is, the preimage in preferred coordinates of the foliations of C by vertical and horizontal lines. For q ∈ Q(X, x) and t ∈ R consider the associated Teichmüller mapping gt : (X, x) → (Xt, gt(x)) as above. If c : R → X is a nonsingular leaf of F(q) parameterized by arc-length with respect to the q-metric, then composing with gt we obtain a nonsingular leaf of the vertical foliation for the terminal quadratic differential F(qt), From the description of gt in local coordinates we see that this is parameterized proportional to arc-length and, in fact, the qt-length is given by

Properties of Teichmüller geodesics
and q ∈ Q(X, x). The forward asymptotic behavior of τ is reflected in the structure of the vertical foliation F(q). For us, the most important instance of this is a result of Masur [22].
• τq(t k ) ∈ T (Z, z) for all k then F(q) is arational and uniquely ergodic.
In particular, if τq is thick then both F(q) and G(q) are uniquely ergodic. We say a pair of arational foliations F and G are K-cobounded if for all strict subsurfaces Y ⊂ X − x we have dY (F, G) ≤ K. A result of Rafi [26,Theorem 1.5] relates the thickness of a geodsic τq ⊂ T to the coboundedness of the associated vertical and horizontal foliations.
Conversely, for all K > 0 there exists > 0 so that if q ∈ Q(X, x) has F(q) and G(q) being K-cobounded then τq is -thick.

Forgetting the marked point: the Bers fibration
Suppose now that Z is a closed surface and z ∈ Z is a single marked point; we use (Z, z) to denote (Z, {z}). Let p : Z → Z denote the universal covering. Given [f : (Z, z) → (X, f (z))] we can forget the marked point to obtain an element [f : Z → X] ∈ T (Z). This defines a holomorphic map Π : called the Bers fibration [4]. The fiber of this map over [f : Z → X] is holomorphically identified with X, the universal covering of X. Moreover, this identification is canonical, up to the action of the covering group on X.
The projection of Teichmüller spaces Π : T (Z, z) → T (Z) descends to a projection of moduli spacesΠ : M(Z, z) → M(Z). The fiber ofΠ over X ∈ M(Z) is just X/ Aut(X) and this is compact.
Recall that puncturing a closed surface once increases the hyperbolic systole. (Lift to universal covers and apply the Schwarz-Pick lemma.) It follows that the preimage of T (Z) by Π −1 is contained in T (Z, z).
By a theorem of Royden [28] the Teichmüller metric agrees with the Kobayashi metric on Teichmüller space. Recall that the inclusion of the universal covering X → T (Z, z) is a holomorphic embedding [4]. Thus, if we give X the Poincaré metric ρ0 -one-half the hyperbolic metric -then ( X, ρ0) → (T (Z, z), dT ) is a contraction [16]. Kra [17] further proved the following.
The function h can be described concretely in terms of the solution to a certain extremal mapping problem for the hyperbolic plane which was solved by Teichmüller [29] and Gehring [11]. We will extend h to a nondecreasing function, h :

Branched covers
Here we use branched covers to induce maps on Teichmüller space.
Suppose P : Σ → Z is a branched cover, branched over some finite set of points z ⊂ Z. Then any complex structure on Z pulls back to a complex structure on Σ, and thus induces a map P * : T (Z, z) → T (Σ). Regarding Teichmüller space as the space of marked Riemann surfaces, T (Z, z) = {[f : (Z, z) → (X, x)]}, the embedding is described as follows. The branched covering P : Σ → (Z, z) induces a branched covering U : Ω → (X, x), for some Riemann surface Ω, namely the branched cover induced by the subgroup (f • P ) * (π1(Σ − P −1 (z))) < π1(X − x). By construction, there is a lift of the marking homeomorphism φ : Σ → Ω. This is described by the following commutative diagram.
Proof. When P is a covering then P * is an isometric embedding; see [27,Section 7]. The proof is identical in the presence of nontrivial branching, as a one-prong singularity at a point of z lifts to a regular point or to a three-prong or higher singularity.
Let M(Z, z) be the quotient of T (Z, z) by the group of mapping classes of (Z, z) that lift to Σ. Note that M(Z, z) → M(Z, z) is a finite sheeted (orbifold) covering. The embedding P * : T (Z, z) → T (Σ) descends to a map M(Z, z) → M(Σ), giving a commutative square.
By Mumford's compactness criteria [3], the image of T (Z, z) in M(Z, z) is compact, and hence so is the image in M(Σ). Appealing to Mumford's criteria again (for M(Σ)), it follows that for some > 0 we have P * (T (Z, z)) ⊂ T (Σ).
In general, for any branched cover P : Σ → Z, branched over z ⊂ Z, consider σ = P −1 (z) as a set of marked points on Σ. Then again there is an isometric embedding P * : T (Z, z) → T (Σ, σ).
When ω is empty we may omit the subscript. In this notation, the composition Π • P * gives the map of Proposition 4.4. So, if P is non-trivially branched at all points of σ then Π • P * is an isometric embedding. If P is not branched at all points of σ then Π • P * fails to be an isometric embedding; however it remains 1-Lipschitz.
Proof. The Bers fibration is a holomorphic map [4] and, by forgetting the points of σ − ω one at a time, we see that Πω : T (Σ, σ) → T (Σ, ω) is a composition of holomorphic maps, hence holomorphic. In particular, because the Teichmüller metric agrees with the Kobayashi metric [28], it follows that Πω is 1-Lipschitz [16]. Since P * is an isometric embedding, the composition is 1-Lipschitz.

An inductive construction
The proof of Theorem 1.1 is constructive, but also appeals to an inductive procedure. We begin by constructing the required embedding of H 2 into some Teichmüller space as the base case of the induction, then produce an embedding of H 3 into some other Teichmüller space, then an embedding of H 4 , and so on. All the main ideas and technical difficulties are present in the construction of the embedding of H 2 and then the embedding of H 3 from that of H 2 . The only further complications which arise to describe the embedding of H n from H n−1 for n ≥ 4 are in the notation, which becomes increasingly messy as n increases. This is due to the fact that the proof for n really depends on the proof for all 2 ≤ k < n (rather than just n − 1). For this reason, we carefully describe the cases n = 2 and n = 3, and sketch the general inductive step indicating only those things that require modification.

The hyperbolic plane case
Let Z be a closed hyperbolic surface. Let q ∈ Q(Z) be a nonzero holomorphic quadratic differential on Z so that the associated Teichmüller geodesic [gt : Z → Zt] is thick. Write F = F(q) and G = G(q) for the vertical and horizontal foliations of q, respectively. Next, let c : R → Z be a nonsingular leaf of F parameterized by arc-length with respect to q and let z = c(0) be a marked point on Z; see Section 4.
Our goal is to construct an almost-isometric embedding We consider an isotopy We further assume that f x preserves F for all x ∈ R. To construct such an isotopy we can, for example, take the flow of a complete, time-dependent vector field ξx on Z which is tangent to F for all x ∈ R and with ξx(c(x)) = c (x). We think of the isotopy as "pushing z along c". This determines the horocyclic coordinate Applying the Teichmüller mapping gt : Z → Zt determined by q and t ∈ R gives the height coordinate. These coordinates together define Z : Here we are using the coordinates (x, t) on H 2 described in Section 2.
Since the marking homeomorphisms are determined by x and t, we simplify notation and denote the values in Teichmüller space by We also write Z(x, 0) = c(x) = (Z, c(x)).
As the notation suggests, ct : R → Zt ⊂ T (Z, z) is a lift of gt • c : R → Zt to the universal cover Zt, thought of as the fiber over [gt : Z → Zt].
Theorem 5.1. The map Z : H 2 → T (Z, z) is an almost-isometric embedding. Moreover, the image lies in the thick part and is quasi-convex.
Proof. We verify the hypothesis of Lemma 2.2 to prove that Z is an almost-isometric embedding and, along the way, prove that the image is quasi-convex and lies in the thick part. First, fix any x ∈ R so that ηx(t) = (x, t) is a vertical geodesic in H 2 . Then t → Z • ηx(t) = Z(x, t) = (Zt, gt(c(x))) is a Teichmüller geodesic, and hence Property 1 of Lemma 2.2 holds. Furthermore, since t → Zt is a thick geodesic, we see that {Z • ηx(t)} x∈R are uniformly thick geodesics. That is, that the union of these geodesics, over all x ∈ R, project into a compact subset of M(Z, z); namely, the preimage of the compact subset of M(Z) containing the image of t → Zt (see Section 4.3). In particular, the image of Z lies in the thick part of T (Z, z).
For each x ∈ R, the geodesic Z • ηx is defined by the quadratic differential q ∈ Q(Z) viewed as a quadratic differential in Q(Z, c(x)). We denote the vertical and horizontal foliations of q ∈ Q(Z, c(x)) by F x and G x , respectively, and consider them as measured foliations in MF(Z, z) by pulling them back via f x . Since f x preserves F, it follows that Now, since t → Zt is a thick geodesic, by Theorem 4.1 the foliations F and G are arational. Puncturing an arational foliation once gives an arational foliation in the punctured surface.
Hence F x and G x are also arational for all x. Since F x = F 0 for all x ∈ R and since the geodesics {Z • ηx} x∈R are uniformly thick, Theorem 4.2 implies that there exists K > 0 so that the pairs (F 0 , G x ) = (F x , G x ) are K-cobounded for all x. By the triangle inequality (applied to each subsurface Y ) we see that for all x, x ∈ R the pair (G Appealing to the other direction in Theorem 4.2 the geodesic Γ x,x , determined by G x and G x for distinct x, x ∈ R, is uniformly thick, independent of x and x . From this and [15,Theorem 4.4] it follows that there is a δ > 0 so that Z • ηx, Z • η x and Γ x,x are the sides of a δ-slim triangle for every pair of distinct points x, x ∈ R, and hence Property 2 of Lemma 2.2 holds. From this, it follows that Z(H 2 ) has Hausdorff distance at most δ from the union of the geodesics This is precisely the weak hull of {G x } x∈R ∪ {F 0 } ⊂ PMF(Z, z), and so according to [15,Theorem 4.5], this set, hence also Z(H 2 ), is quasi-convex (the assumption in [15] that the subset of PMF(Z) be closed was not used in the proof).
Finally, we must prove that Properties 3 and 4 of Lemma 2.2 hold. For this we can appeal directly to Theorem 4.3. More precisely, observe that because {Zt} t∈R lies in the thick part, the pull-back of the flat metric on Zt (which we also denote qt) is uniformly quasi-isometric to the Poincaré metric ρ0 on Zt. That is, there exist constants A, B ≥ 0 so that for all t ∈ R and z, z ∈ Zt (see for example [7, Lemma 2.2]).
Applying (4), the upper bound of Theorem 4.3, (5) and (3), in that order, we find From this, and because h is a homeomorphism on [0, ∞) and hence proper, Property 4 of Lemma 2.2 also holds. This completes the proof of Theorem 5.1.

Hyperbolic 3-space
Before diving into the construction, we explain the basic idea. Our embedding of the hyperbolic plane in Section 5.1 sends (x, t) to Z(x, t) ∈ T (Z, z) by pushing the marked point z distance x along a leaf of the vertical foliation of a quadratic differential then travelling distance t along the Teichmüller flow. There is a simple extension of this construction which produces a map of hyperbolic 3-space into Teichmüller space T (Z, {z, w}). Take z and w to lie on distinct leaves and send (x, y, t) to the point of T (Z, {z, w}) obtained by pushing z a distance x along its leaf, pushing w a distance y along its leaf, and applying the Teichmüller flow for time t.
The problem is that whenever z and w move close to each other on Z, the corresponding point in T (Z, {z, w}) is in the thin part of Teichmüller space; if z and w are very close to each other then there is a simple closed curve surrounding z and w having an annular neighborhood of large modulus. This also shows that this map (x, y, t) → T (Z, {z, w}) is not a quasi-isometric embedding. In fact the map is not even coarsely Lipschitz.
A more subtle construction is required. We first choose a branched cover P : Σ → Z, nontrivially branched at each point of P −1 (z). According to Proposition 4.4, this induces an isometric embedding of T (Z, z) into T (Σ). Fix a suitably generic point w ∈ (Z, z) and pick a point σ ∈ P −1 (w). Roughly, we map our three parameters (x, y, t) into T (Σ, σ) as follows. The coordinates (x, t) determine Z(x, t) ∈ T (Z, z) as in Section 5.1. The map P * applied to Z(x, t) gives a point in T (Σ) as in Section 4.4. Finally use y to determine a point Σ(x, y, t) ∈ T (Σ, σ), lying in the Bers fiber above P * • Z(x, t). On its face, this new construction avoids the problem we had before. In (Z, z) we have only one marked point; after taking the branched covering over z we forget all of the branch points over z. The single image of σ can now move freely enough so that we stay in the thick part of T (Σ, σ). We now explain this construction in more detail and prove that the resulting map has all the required properties.

The construction
The notation from Section 5.1 carries over to this section without change. Let P : Σ → Z be a branched cover, branched over the marked point z ∈ Z so that P is nontrivially branched at every point of P −1 (z). This determines an isometric embedding of Teichmüller spaces by Proposition 4.4. We write so that φ x t is a lift of the marking gt • f x , and P x t is the induced branched cover making the following commute: The quadratic differentials qt pull back to quadratic differentials µ x t on Σ x t , and gt lifts to Teichmüller mappings of the covers We have another commutative diagram which may be helpful in organizing all the maps: Denote the vertical foliation for µ x t by Φ x t . Each nonsingular leaf of Φ x t maps isometrically to a nonsingular leaf of the vertical foliation Ft for qt via the branched covering Σ x t → Zt since µ x t is the pull back of qt. Choose any nonsingular leaf γ 0 0 : R → Σ 0 0 = Σ, parameterized by arc-length. Observe that γ 0 0 maps isometrically by P to a leaf γ : R → Z for F. Note that c and γ are distinct leaves; the preimage of c in Σ consists entirely of singular leaves, namely the leaves that meet the branch points of P .
As we vary x, we can continuously choose lifts of γ to leaves γ x 0 : R → Σ x 0 which agrees with our initial leaf γ 0 0 when x = 0. Specifically, we define the lift to be Composing with the lifts ψ x t , we obtain leaves γ x t = ψ x t • γ x 0 . Observe that via the branched covering P x t : Σ x t → Zt, γ x t projects to the leaf gt • γ, independent of x. Furthermore, this shows that the µ x t -length of the arc γ x t ([y, y ]) is the qt-length of gt • γ which is e −t |y − y |. We pick a basepoint σ = γ 0 0 (0) ∈ Σ, and consider the surface (Σ, σ), marked by the identity Id = φ 0 0 as a point in T (Σ, σ). Just as we constructed f x by pushing along c to c(x), we push σ along γ x t to γ x t (y) to obtain maps ξ x,y t : (Σ, σ) → (Σ x t , γ x t (y)).
Specifically, we take ξ x,y 0 to be the composition of φ x 0 and a map isotopic to the identity on Σ x 0 which preserves the foliation Φ x 0 and pushes φ x 0 (σ) along γ x 0 to γ x 0 (y). Then ξ x,y t = ψ x t • ξ x,y 0 maps the foliation Φ 0 0 to Φ x t . We denote the associated point in Teichmüller space [ξ x,y t : (Σ, σ) → (Σ x t , γ x t (y))] ∈ T (Σ, σ) simply by (Σ x t , γ x t (y)) as this point is uniquely determined in this construction by (x, y, t).

Fibration over H 2 case
We also require a slightly different description of the map Σ to take advantage of the construction in the H 2 case. Observe that P * • Z : H 2 → T (Z, z) → T (Σ) is an almostisometric embedding, and is given by with γ x t (y) = (Σ x t , γ x t (y)) when it is convenient to do so.
Finally we record the equation which holds for all (x, y, t) ∈ H 3 . The fact that Π is 1-Lipschitz and P * • Z is an almostisometric embedding provides us with useful metric information about Σ.
Theorem 5.2. The map Σ : H 3 → T (Σ, σ) is an almost-isometric embedding. Moreover, the image lies in the thick part and is quasi-convex.
Proof. As before, we will verify the hypothesis of Lemma 2.2 to prove that Σ is an almostisometry and, along the way, prove that the image is quasi-convex and lies in the thick part. For all (x, y) ∈ R 2 , the geodesic η (x,y) (t) in H 3 is sent to This is a geodesic in T (Σ, σ) because ψ x t : Σ x 0 → Σ x t is a Teichmüller mapping; thus Property 1 follows. Furthermore, note that Σ • η (x,y) (t) lies over P * • Z • ηx(t) for all (x, y, t). Since P is nontrivially branched over every point, the uniform thickness of the set of geodesics {Z • ηx(t)} x∈R implies the same for {P * • Z • ηx(t)} x∈R by Proposition 4.4, and hence also for {Σ • η (x,y) (t) | (x, y) ∈ R 2 } by (6) as discussed in Section 4.3. That is, Σ(H 3 ) lies in the thick part.
By our choice of maps ξ x,y 0 , if we pull back the vertical foliation Φ x 0 of µ x 0 to a foliation Φ x,y 0 ∈ MF (Σ, σ) the result is independent of x and y. Furthermore, Theorem 4.1 implies that these foliations, as well as the pull backs of the horizontal foliations, are arational. Thus all strict subsurface projection distances are defined. Theorem 4.2 and the results of [15] can be applied as in the H 2 case to prove that Property 2 of Lemma 2.2 is satisfied for some δ > 0. Furthermore, Σ(H 3 ) is quasi-convex.
We now come to the subtle point of the proof, which is verifying Properties 3 and 4 of Lemma 2.2. We start with Property 3. This is because we are taking a branched covering, Σ x t → Zt, and then forgetting all but one of the marked points in Σ x t . Since Zt lies in some fixed thick part of T (Z) for all t ∈ R, there exists > 0 so that the 2 -ball about Cx in the qt metric, Bq t (Cx, 2 ) is a disk for all t, x ∈ R (that is, we have a lower bound on the qt-injectivity radius of Zt, independent of t). Now suppose Γy lies outside this ball Γy ∈ Bq t (Cx, 2 ).
Using again the fact that Zt lies in some thick part of T (Z) for all t ∈ R, it follows that there is some R > 0 with the property that for any point z ∈ Bq t (Cx, ) we have Here the marking homeomorphism for (Zt, {z , Γy}) is assumed to differ from that of (Zt, {Cx, Γy}) by composition with a homeomorphism of Zt that is the identity outside Bq t (Cx, 2 ). In particular, if e −t |x − x | < and, crucially, Γy ∈ Bq t (Cx, 2 ) then deduce that C x ∈ Bq t (Cx, ) and, from Equation (9), that On the other hand, because the leaves of F are geodesics for qt and because Bq t (Cx, 2 ) is a disk, if Γy ∈ Bq t (Cx, 2 ) then there exists y ∈ R so that e −t |y − y| ≤ 2 and Γ y ∈ Bq t (Cx, 2 ).
All that remains to show is Property 4 of Lemma 2.2. Suppose we have a sequence of pairs {(xn, yn, tn), (x n , y n , tn)} ∞ n=1 with e tn |(xn, yn) − (x n , y n )| → ∞ as n → ∞. Then, up to subsequence, we must be in one of two cases.
If we also have e tn |xn − x n | → ∞, then we can appeal to the previous case and we are done. So we assume, as we may, that e tn |xn − x n | < M , for some constant M > 0. Since we have already shown that there are , R > 0 so that part 3 from Lemma 2.2 holds, it follows from (1)that dT Σ(xn, y n , tn), Σ(x n , y n , tn) ≤ R e −tn |xn − x n | + R ≤ R M + R.

Now, by the triangle inequality we have
dT Σ(xn, yn, tn), Σ(x n , y n , tn) ≥ dT Σ(xn, yn, tn), Σ(xn, y n , tn) − dT Σ(xn, y n , tn), Σ(x n , y n , tn) We can now appeal to Theorem 4.3 as in our proof for Z : H 2 → T (Z, z) to find A, B so that dT ( γ xn tn (yn), γ xn tn (y n )) ≥ h 1 A e −tn |y n − yn| − B The right-hand side tends to infinity by the properness of h, so we can combine this with (12) to obtain lim n→∞ dT (Σ(xn, yn, tn), Σ(x n , y n , tn)) = ∞ as required. Therefore, Property 4 from Lemma 2.2 holds, and the proof of Theorem 5.2 is complete.

The general case
The previous arguments set up an inductive scheme for producing almost-isometric embeddings of H n into Teichmüller spaces. The idea is as follows.
For n − 1 ≥ 3, induction gives us an almost-isometric embedding W : H n−1 → T (W, w) satisfying all the hypotheses of Lemma 2.2 for some closed surface W with a marked point w. We again take a branched cover P : Ω → W nontrivially branched over each point in P −1 (w) ⊂ Ω. This determines a map P * • W : H n−1 → T (Ω).
Using the coordinates (x, t) = (x1, x2, . . . , xn−2, t) ∈ H n−1 we write Inductively, we assume that the foliation of H n−1 by asymptotic geodesics {ηx(t)} x∈R n−2 are all mapped by W to uniformly thick geodesics in T (W, w), so the same is true for P * • W . These geodesics are obtained by applying the Teichmüller mapping ψ x t : for all x ∈ R n−2 and t ∈ R. Furthermore, the defining quadratic differentials all have the same vertical foliation.
We pick a leaf of this foliation γ : R → Ω, and arguing as before, this determines a leaf in each surface γ x t : R → Ω x t with γ 0 0 = γ : R → Ω 0 0 = Ω. Now, pick ω = γ 0 0 (0) to be our marked point, add a factor of R to H n−1 with coordinate y = xn−1 to obtain H n with coordinates (x, y, t) = (x1, . . . , xn−2, xn−1, t), and define by Ω(x, y, t) = (Ω x t , γ x t (y)). So we are again pushing a point along a leaf of the vertical foliation. Sketch of proof. Again, we must verify the hypotheses of Lemma 2.2 and prove that the image of Ω is quasi-convex in the thick part, assuming that this is true in all previous steps of the construction.
We can argue exactly as in the case of H 3 to prove Properties 1 and 2 of Lemma 2.2 as well as the fact that the image of Ω is quasi-convex in the thick part. Property 3 requires more care. However, once established, Property 4 follows formally, just as in the case of H 3 .
We elaborate on the proof that Property 3 holds for some and R. For this, we must give a more precise description of the construction. Write Ωn−1 = Ω, Ωn−2 = W and Pn−2 = P : Ωn−1 → Ωn−2 for the branched cover used in the construction. Inductively, we have a tower of branched covers Ωn−1 In this tower, Pj is nontrivially branched at every point P −1 j (ωj) where ωj ∈ Ωj is the marked point. To clarify, we note that Ω1 = Z, ω1 = z, Ω2 = Σ and ω2 = σ from the preceding discussion.

3
→ Ω x 1 2 → Ω1 and we push this along an appropriate lift ζ x 1 ,x 2 3 of ζ3 to a point ζ x 1 ,x 2 3 (x3) in the preimage of ζ3(x3). We continue in this way to produce a tower of branched covers induced by P1, P2, . . . , Pn−3, Pn−2: To find Ω(x, y, t) for any t, we apply the appropriate Teichmüller deformation to Ω(x, y, 0). This is the Teichmüller deformation determined by t and the pull back of ν1 (via the composition of branched covers). We can pull back ν1 by any of the branched covers, and since the resulting quadratic differential depends only on the surface in this construction, we will simply write Φt for the associated Teichmüller deformation on any of the surfaces Ω x 1 ,...,x j−1 j . In particular, we have Ω(x, y, t) = Φt(Ω(x, y, 0)). Set x = (x 1 , x 2 , . . . , x n−2 ). We now must find an and R so that if dT Ω(x, y, t), Ω(x , y , t) ≤ R.
As in the case of H 3 , appealing to the triangle inequality it suffices to find an and R so that if (x1, . . . , xn−2, y) and (x 1 , . . . , x n−2 , y ) agree in all but one coordinate, and in that coordinate differ by at most , then dT Ω(x, y, t), Ω(x , y , t) ≤ R .
Note that this required three point pushes in two different auxiliary surfaces. We varied the (n − 1) st coordinate twice, in the highest surface, and varied the (n − 2) nd coordinate once. Now suppose that x differs from x in the (n − 3) rd coordinate only. We view Φt(Ω (xn−3)). We can move this last point a small amount, changing the Teichmüller distance a bounded amount, provided the other two points, higher in the tower, are not too close to it. If they are too close, we first move them out of the way (as in the first two pushes above), move the third point, then move the two higher points back. The triangle inequality together with the 1-Lipschitz property of the branched cover map applied as before, implies a uniform bound on the change in Teichmüller distance dT Ω(x1, . . . , xn−3, xn−2, xn−1, t), Ω(x1, . . . , x n−3 , xn−2, xn−1, t) .
It follows that varying xn−3 requires at most five point pushes in the three highest auxiliary surfaces.
In general, varying x n−k in this way requires 2k − 1 point pushes in the k highest auxiliary surfaces. Thus we can change any coordinate by a small amount and change the Teichmüller distance by a bounded amount R , as required. This completes the sketch of the proof of Theorem 5.3.