EXTREMAL LENGTH ESTIMATES AND PRODUCT REGIONS IN TEICHMÜLLER SPACE

YAIR N. MINSKY

June 6, 1995

Abstract. We study the Teichmüller metric on the Teichmüller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the Teichmüller metric is approximated up to bounded additive distortion by the sup metric on a product of lower dimensional spaces. The main technical tool in the proof is the use of estimates of extremal lengths of curves in a surface based on the geometry of their hyperbolic geodesic representatives.

1. Introduction

There is a longstanding but imperfect analogy between the geometry of the Teichmüller space T (S) of a surface and that of a complete, negatively curved space. For example, T (S) admits a boundary at infinity similar to that of hyperbolic space (See e.g. [8, 23]), and the Teichmüller geodesic flow on its quotient, the moduli space, is ergodic (see [22]). This paper studies one of the strong ways in which this analogy fails, namely the existence of large regions in the space which are closely approximated by products of lower-dimensional metric spaces.

There are several natural metrics on T (S); we will consider throughout only the Teichmüller metric (§2.5). The first indication of positively curved behavior of the Teichmüller metric came from Masur [20], in which examples were given of geodesic rays with a common basepoint, which remain a bounded distance apart for all time. This contradicts non-positive curvature on small scales at every point (though it also involves large-scale behavior of the geodesics in a subtle way). More recently, Masur and Wolf [24] have given examples of geodesic triangles which fail, arbitrarily badly, the “thin triangle” condition for hyperbolicity in the sense of Gromov and Cannon [5, 6, 12]; this implies that the large-scale behavior of the metric is not negatively curved. The main result of this paper describes regions of T (S) where the large scale behavior exhibits some characteristics of positive curvature.

The main theorem can be summarized as follows (a complete statement appears in §6). Let T (S) be the Teichmüller space of a surface S of finite type, endowed with the Teichmüller metric. Let γ = {γ_{1}, …, γ_{k}} be a system of disjoint, homotopically distinct simple closed curves on S, and let T hin_{ϵ}(S, γ) denote the set of σ ∈ T (S) for which ℓ_{σ}([γ_{i}]) ≤ ϵ for all i, where ℓ_{σ}([γ_{i}]) denotes hyperbolic length of the σ-geodesic representative of γ_{i}. Let X_{γ} denote the product space T (S\γ)×H_{1}×⋯×H_{k}, where S \ γ is considered as a punctured surface, and each H_{i} is a copy of the hyperbolic plane. Endow X_{γ} with the sup metric, d_{X} = max{d_{T (S\γ)}, d_{H_{1}}, …, d_{H_{k}}}, of the metrics on the factors.

This work was partially supported by an NSF postdoctoral fellowship.

Theorem 6.1 The Fenchel-Nielsen coordinates on T (S) give rise to a natural homeomorphism Π : T (S) → X_{γ}, and for ϵ sufficiently small this homeomorphism restricted to T hin_{ϵ}(S, γ) distorts distances by a bounded additive amount. That is,

|d_{T (S)}(σ, τ) − d_{X}(Π(σ), Π(τ))| ≤ c

For σ, τ ∈ T hin_{ϵ}(S, γ), and c = c(χ(S), ϵ).

Note in particular that the distortion of Π is additive and not multiplicative, as is more commonly the case with quasi-isometries. For this reason it is significant that we endow X_{γ} with the sup metric, and not the L^{2} metric; these are equivalent up to bounded multiplicative, but not additive, distortion. A product of spaces with the sup metric exhibits not just non-negative, but actually positive curvature characteristics in the large. See §7 for a fuller discussion of this point.

The idea of decomposing Teichmüller space as a product using length and twist parameters of simple closed curves dates back to Fenchel-Nielsen [10] (see Abikoff [1, Chap. II] and Fricke-Klein [11]). Such decompositions have particular significance when the curves in question are short, and this has been used by several authors to study the degeneration of Riemann surfaces towards “noded” surfaces, in which a short curve has been replaced by two punctures. An incomplete list of treatments of this phenomenon from complex-analytic, algebraic-geometric and differentialgeometric points of view includes Bers [4], Earle-Marden [7], Fay [9], Hejhal [13], Kra [16], Maskit [17, 18], Masur [21], Wolf-Wolpert [27] and Wolpert [29, 30, 28]. In all these cases, the emphasis is on careful local analysis, concentrating on the complex structure of the moduli space in a neighborhood of a noded surface, or on differential geometric structure such as the Weil-Petersson metric in the case of [21, 28], or a precise expression for the hyperbolic metric on the surface in [30]. By contrast, our approach ignores most local analytical details and concentrates on large-scale estimates in the Teichmüller space. Extremal length estimates. The estimates of Teichmüller distance necessary for theorem 6.1 are obtained using Kerckhoff’s theorem (see §2.5), which relates Teichmüller distance to ratios of extremal lengths. Therefore our main tool is a theorem that allows us to estimate extremal lengths of arbitrary curves on a surface, in terms of the behavior of their hyperbolic geodesic representatives.

The relationship between hyperbolic length ℓ_{σ} and extremal length λ_{σ} (see §2.3) of curves on a Riemann surface (S, σ) is not completely straightforward. For very short curves, ℓ_{σ} and λ_{σ} are nearly proportional. For long curves, the relation depends on the geometry of the surface. In particular an upper bound for λ_{σ} is given by λ_{σ} ≤ (1/2)ℓ_{σ}e^{ℓ_{σ}/2} (see Maskit [19]), and this is close to sharp in general. But on the other hand λ_{σ} ≥ ℓ_{σ}^{2}/2π|χ(S)| is the general lower bound that comes from the definition of extremal length, and for any fixed conformal structure σ one can show (see e.g. [26] and lemma 4.3) that it is the lower bound which estimates λ_{σ} to within a bounded factor.

Theorem 5.1 gives an estimate for λ_{σ} which is correct, up to a bounded factor that depends only on the topological type of the surface, for all cases. The main observation is that, after decomposing a Riemann surface (S, σ) into thick and thin parts, or what we call here an (ϵ_{0}, ϵ_{1}) collar decomposition (see §2.4), each component of the decomposition contributes individually to the extremal length of a curve β in S. The contribution λ_{P,σ}(β) of a component P of the thick part (see definition (4.2) and lemma 4.3) is approximately ℓ_{σ}^{2}(β^{σ} ∩ P), where β^{σ} is the σ-geodesic representing β. A component A of the thin part, which is an annulus, contributes a quantity λ_{A,σ}(β) which depends on the modulus of A and the amount of “twisting” that β does in A (see §3 and definition (4.3)). The theorem then states:

Theorem 5.1 Let S be a surface of finite type with a standard hyperbolic metric σ in which boundary lengths are at most ℓ_{0}, and let Q be the set of components of the (ϵ_{0}, ϵ_{1}) collar decomposition. Then, for any curve system β in S, the extremal length λ_{S,σ}(β) is approximated by

max_{Q∈Q} λ_{Q,σ}(β)

up to multiplicative factor depending only on ϵ_{0}, ϵ_{1}, ℓ_{0} and the topological type of S.

In specific cases this theorem recovers the above exponential and quadratic bounds from [19] and [26]. Proofs. The proof of theorem 5.1 is essentially a long exercise in elementary hyperbolic geometry. Given a homotopy class of curves β in S we must arrange it in a sufficiently evenly-spaced way so as to obtain an embedded annulus which gives us an upper bound for extremal length (lower bounds are easier). The part of the argument which is not straightforward has to do with finding an appropriate metric, conformally equivalent to the hyperbolic metric, in which to make the spacing construction. The scaling factors, which vary among components of the thick-thin decomposition, must depend on the way in which β is distributed on S. The construction of this metric is given in lemma 5.2 and the discussion that follows it. (Note that the metric we obtain is in some way mimicking the actual extremal metric, which is given by a quadratic differential. However our construction is explicitly related to the hyperbolic characteristics of β, which makes it more suited to our purposes.)

The proof of theorem 6.1 is an almost direct application of theorem 5.1 and Kerckhoff’s theorem 2.1, except for one point. In the passage from a component of S \ γ with (short) boundary to a punctured surface, we must at some point compare extremal length ratios λ_{σ}/λ_{τ} for properly embedded arcs with endpoints on the boundary to similar ratios for simple closed curves in the interior. That the suprema of these ratios are within a bounded factor of each other is shown in lemma 6.2. The main trick in this lemma is to start with a (possibly non-closed) curve β that maximizes the reciprocal ratio λ_{τ}/λ_{σ}, and then find a closed curve which is sufficiently close to being “orthogonal” to β.

A different approach to the proof of theorem 6.1 than the one we have taken might involve estimating Teichmüller distances directly by constructing appropriate quasiconformal maps. One might hope that, since a short curve corresponds to a high-modulus annulus in the surface, a quasi-conformal map on the complement of this annulus might be “spliced” across the annulus in a way that takes the appropriate modulus and twist parameters into account, and thus the quasi-conformal dilatation of the spliced map would be estimated by the dilatations of the component maps (and the modulus and twisting information in the annulus). It is therefore of interest that, in our experience, this approach actually runs into quite serious difficulties, since “splicing” quasi-conformal maps is, in general, not very easily done without great loss of control over dilatation. In particular, estimates of dilatation of such spliced maps tend to be, at best, polynomially related to the dilatations of the component maps, and not linearly as we would need for theorem 6.1. It is for this reason that we have adopted the extremal length approach, which circumvents these problems. Summary of contents. In section 2 we set our notation and recall some wellknown facts about conformal structures on surfaces, curves and their lengths, and Teichmüller distance. The only proof we give is an elementary one for lemma 2.2.

In section 3 we define twisting numbers, which are a convenient way to measure how many times one geodesic twists around another in a hyperbolic surface, and in particular inside a Margulis annulus. We prove a few elementary facts about them, and relate them to Fenchel-Nielsen twist parameters in lemma 3.5.

Section 4 gives some well-known estimates of extremal lengths, in particular an estimate in terms of hyperbolic length in a component of the thick part (lemma 4.3). Lemma 4.4 describes briefly how to space a curve evenly in a component of the thick part, and this will be used in §5. At the end of the section we define the annulus contribution λ_{A,σ} to extremal length, in terms of twisting numbers.

Sections 5 and 6 give the proof of theorems 5.1, and 6.1, respectively. Section 7 gives a brief discussion of the geometric consequences of theorem 6.1. Acknowledgements. The author is grateful to the Long Island Rail Road for its generosity in providing space and time for contemplation.

2. Definitions and background

2.1. Metrics and conformal structures on surfaces. Let S be a smooth orientable surface of finite topological type, by which we mean that S is homeomorphic to a connected, compact surface (possibly with boundary) minus a finite set, which we call punctures or cusps.

By conformal structure on S we shall mean a conformal structure on the interior, for which each puncture has a neighborhood conformally equivalent to a punctured disk, and each boundary component has a neighborhood conformally equivalent to an annulus of finite modulus. Let T (S) denote the Teichmüller space of conformal structures on S considered up to conformal homeomorphisms homotopic to the identity (with no prescribed behavior on the boundary; see [1]). These are called marked conformal structures. If S has no boundary (but possibly punctures) then T (S) is the space of analytically finite marked conformal structures.

A conformal structure also determines a class of metrics on the interior, all those that can be described in a local conformal coordinate as a positive function times the Euclidean metric. In this paper we will use metrics which are smooth or mildly singular (non-differentiable on some curves), and which have a completion to the boundary of S. The conformal class of a metric σ will be denoted [σ].

Call S hyperbolic if its Euler characteristic is negative, which implies that it admits a complete finite-volume hyperbolic metric in which the boundary is geodesic. We will call such a hyperbolic metric on S standard. We say a Euclidean metric on an annulus is standard if the annulus can be obtained isometrically from a Euclidean rectangle by identifying a pair of opposite edges. Note that every conformal annulus of finite modulus admits a standard Euclidean metric in the prescribed conformal class, unique up to scaling.

2.2. Curve systems. A curve system on S is a disjoint union of homotopically non-trivial properly embedded simple arcs and closed curves (for an arc, non-trivial means not deformable into ∂S with endpoints fixed, and for a closed curve it means not deformable to a point or a puncture). For a hyperbolic S, let C(S, ∂S) denote the set of homotopy classes of curve systems under homotopies that keep the endpoints (if any) on the boundary. (We admit non-simple curves to the homotopy classes for convenience.) Let C(S) ⊂ C(S, ∂S) denote the subset of curve systems represented by unions of simple closed curves. Note that C(S) includes curves that are homotopic to components of ∂S. These are called peripheral, and to exclude them we define C_{0}(S) to be the subset of C(S) consisting of curve systems with no peripheral components. Also define (for brief use in §3) C′(S, ∂S) to be the set of homotopy classes of curve systems which are allowed to terminate in punctures as well as boundary components.

For an annulus A, we make a different definition: Let C(A, ∂A) denote the set of curve systems on A with equivalence under homotopies that fix the endpoints.

We call a subsurface M ⊂ S hyperbolic if no component of ∂M is homotopically trivial or homotopic to another component of ∂M. Equivalently, if σ denotes any smooth non-positively curved metric on S in which the boundary is geodesic (or convex), then M is isotopic to a subsurface M^{σ} whose boundary is geodesic in σ.

Given a hyperbolic subsurface M, there is a natural map

R = R_{M} : C(S, ∂S) → C(M, ∂M),

defined as follows: Let σ be a standard hyperbolic metric on S. If γ is a curve system on S let γ^{σ} denote its representative of shortest σ-length (see §2.3). Let R([γ]) be the class represented by γ^{σ} ∩ M^{σ}. It is not hard to see that R does not depend on the choice of σ.

2.3. Lengths and geodesic representatives. For any metric σ on S and any curve α we denote by ℓ_{σ}(α) the length of α as measured in σ. If [γ] ∈ C(S, ∂S) then γ^{σ} denotes a representative of minimal length – if σ is a standard hyperbolic metric then this is unique, and orthogonal to ∂S. For an annulus A where the homotopies must fix endpoints, there is also a minimal representative γ^{σ}, but it may not be orthogonal to ∂S. In both cases we may denote ℓ_{σ}(γ^{σ}) by ℓ_{σ}([γ]). Note also that any γ ∈ C′(S, ∂S) has a unique geodesic representative γ^{σ}, though its length may be infinite. Extremal length. For a metric σ and any class [γ] of curves in S, we can define the extremal length λ_{σ}([γ]) as

λ_{σ}([γ]) = λ_{S,σ}([γ]) = sup_{σ′∈[σ]} \frac{inf_{γ′∈[γ]} ℓ_{σ′}^{2}(γ′)}{Area(σ′)}.  (2.1)

Where the supremum is taken over all metrics σ′ conformally equivalent to σ. (See Ahlfors [2].) In particular, λ_{σ} only depends on the conformal class of σ.

If A is an annulus with conformal structure [σ] then its conformal modulus m(A, σ) (or m(A) if σ is implicit) is the extremal length of the class of arcs joining its boundaries. The extremal length of the class of simple closed curves homotopic to either boundary is 1/m(A). A standard Euclidean metric on A is unique in its conformal class up to constant multiple, and realizes the supremum in the definition of the modulus. Thus m(A) = H/L where H is the distance between the boundaries in this metric, and L is the length of either boundary.

2.4. The Thick-Thin decomposition. Fixing a standard hyperbolic metric σ on S, the ϵ-thin part of (S, σ) is the subset of S where the injectivity radius is at most ϵ. The closure of its complement is called the ϵ-thick part. If ϵ is less than the Margulis constant for H^{2} (see e.g. [14]), then the ϵ-thin part consists of compact annular neighborhoods of short geodesics, and non-compact cuspidal components, or neighborhoods of punctures. Let us call a compact annulus A of the thin part peripheral if its core is homotopic to ∂S (in which case one of its boundary components lies on ∂S), and internal otherwise. In either case call a boundary component of A internal if it lies in the interior of S.

For our later convenience we shall describe the following slight variation on the standard decomposition, which we shall call the (ϵ_{0}, ϵ_{1}) collar decomposition, for two numbers 0 < ϵ_{1} < ϵ_{0} less than the Margulis constant. Let A_{1}, ..., A_{k} denote the annular neighborhoods of the geodesics of length δ_{i} ≤ ϵ_{1}, such that the internal boundary components of A_{i} have length ϵ_{0}. It is not hard to see that the A_{i} are contained in separate components of the ϵ_{0}-thin part, and that S − ∪A_{i}, minus neighborhoods of the cusps, are in the ϵ′_{1}-thick part, for some ϵ′_{1} possibly slightly smaller than ϵ_{1}.

Let A denote the set of A_{i}, and let P denote the set of closures P of components of S − ∪A_{i}. We call P the set of hyperbolic components of the decomposition. Let Q = P ∪ A denote the entire decomposition.

It is well-known that ϵ_{0} may be chosen sufficiently small (independently of the surface) that, for any simple geodesic γ in S, the components of γ ∩ A_{j} are either segments that traverse A_{j} from one boundary to the other, or are equal to the geodesic core of A_{j}. (Roughly, this is because if a geodesic penetrates sufficiently deeply into a Margulis tube and exits on the same side that it entered, it must have gone around the tube more than once, and hence intersects itself).

One can see from an explicit computation (in the upper half-plane, for example) that the modulus of each A_{i} is

m(A_{i}) = \frac{π}{δ_{i}} − \frac{2}{ϵ_{0}} ,  (2.2)

and we can guarantee by choosing ϵ_{1} sufficiently small that m(A_{i}) is larger than any given constant m_{0}.

2.5. The Teichmüller metric, and Kerckhoff’s theorem. The Teichmüller distance on T (S) is defined for two conformal structures [σ], [τ] as

d_{T (S)}(σ, τ) = \frac{1}{2} log K(σ, τ)  (2.3)

where K ≥ 1 is the least number such that there exists a homeomorphism homotopic to the identity on S which is a K-quasiconformal map between the σ and the τ conformal structures. We shall be using a different characterization of K(σ, τ), due to Kerckhoff, namely:

Theorem 2.1. For a surface of finite topological type S, the following holds:

K(σ, τ) = sup_{α∈C(S,∂S)} \frac{λ_{S,σ}(α)}{λ_{S,τ}(α)}  (2.4)

Remarks. Kerckhoff proved this in [15] for a closed surface S. However the same techniques apply to surfaces of finite analytic type, and the case with boundary can also be deduced easily using a doubling argument (see e.g. Abikoff [1, §II.1.5] for an example of such arguments). An interesting feature of the equivalence of the two definitions of K is that the first one is fairly easily seen to be symmetric in σ and τ (since the inverse of a K-quasiconformal map is also K-quasiconformal), but the symmetry of the second is less evident. We shall use this to advantage in section 6.

The Teichmüller space of the torus is isometric to the hyperbolic plane H^{2}, and Kerckhoff’s theorem in this case can be written as a simple formula, after proper interpretation. We will use this formula not for the torus but in conjunction with twisting numbers and the annulus contributions to extremal length, λ_{A,σ} (definition (4.3)) in section 6. For the sake of exposition we shall give an elementary proof:

Lemma 2.2. Let H denote the upper half-plane {z : Im(z) > 0}, and define for z_{j} = x_{j} + iy_{j}

K(z_{1}, z_{2}) = sup_{t∈R} \frac{y_{2} + (t + x_{2})^{2}/y_{2}}{y_{1} + (t + x_{1})^{2}/y_{1}} .

Then

d_{H}(z_{1}, z_{2}) = \frac{1}{2} log K(z_{1}, z_{2})  (2.5)

where d_{H}(z_{1}, z_{2}) is the hyperbolic distance function.

(Note: actually the common definition of d_{H} does not include the factor of a half; however we will retain it in order to be consistent with (2.3).)

Proof. Consider first the case that z_{1} and z_{2} are purely imaginary. It is an easy exercise that K(iy_{1}, iy_{2}) = max\{y_{2}/y_{1}, y_{1}/y_{2}\}. Since \frac{1}{2}| log(y_{2}/y_{1})| is exactly the hyperbolic distance function, this completes the proof in this case.

Since for general z_{1}, z_{2} ∈ H there is always some hyperbolic isometry A ∈ SL_{2}(R) such that A(z_{1}), A(z_{2}) are purely imaginary, it would suffice to show that K is invariant under the action of SL_{2}(R).

One can see this conveniently by reducing it to a linear algebra problem. Let Ĥ denote {(α, β) ∈ C^{2} : Im(β/α) > 0}. For (α, β) ∈ Ĥ and u, v ∈ R, define

λ_{α,β}(u, v) = \frac{|uα + vβ|^{2}}{Im(β\overline{α})} .

Then define K̂ on Ĥ × Ĥ by

K̂(α_{1}, β_{1}, α_{2}, β_{2}) = sup_{u,v∈R} \frac{λ_{α_{2},β_{2}}(u, v)}{λ_{α_{1},β_{1}}(u, v)}.

One can check that the left action of SL_{2}(R) on C^{2} (viewed as column vectors) preserves Im(βᾱ) = |α|^{2}Im(β/α), and therefore in particular preserves Ĥ. Furthermore note that the denominator of λ_{α,β}(u, v) is the squared norm of an inner product (u, v)·(α, β), so it is easy to see that λ_{A(α,β)}(u, v) = λ_{α,β}((u, v)A). It follows that A ∈ SL_{2}(R) leaves K̂ invariant, since the supremal ratio over all (u, v) is the same as the supremum over all (u, v)A.

It is an easy computation to show that λ_{1,z}(t, 1) = y + (t + x)^{2}/y if z = x + iy, and that λ_{1,β/α}(u, v) = λ_{α,β}(u, v). Moreover, λ_{α,β}(cu, cv) = c^{2}λ_{α,β}(u, v). It follows that

K̂(α_{1}, β_{1}, α_{2}, β_{2}) = K(β_{1}/α_{1}, β_{2}/α_{2}).

The map (α, β) ↦ β/α is a fibration of Ĥ over H, which takes the linear action of SL_{2}(R) upstairs to its action by hyperbolic isometries downstairs. We conclude that K is also invariant under SL_{2}(R).

Remark. One can give a geometric interpretation to this proof by associating to (α, β) ∈ Ĥ a torus, obtained by identifying opposite sides on a parallelogram spanned by α and β in C. The quantity Im(β\overline{α}) is the area of this parallelogram. For u, v ∈ Z, the quantity λ_{α,β}(u, v) is the extremal length of the homotopy class in the torus represented by (u, v) in the generators α and β (for general u, v ∈ R, one can give an interpretation in terms of measured foliations on the torus). Now K̂ can be seen as a supremum of extremal length ratios, just as in Kerckhoff’s theorem. The fibration from Ĥ to H is just identification under conformal equivalence.

3. Twisting numbers and Fenchel-Nielsen coordinates

In this section we will describe a measure, called t_{γ,σ}(α), for the amount that a simple curve α in a hyperbolic surface (S, σ) “twists” around another curve γ. In particular if γ is short then this number will approximate the amount of twisting that α undergoes in the thin annulus corresponding to γ, and this will determine the contribution of this annulus to the extremal length of α. At the end of the section we will relate this construction to the familiar Fenchel-Nielsen twist coordinates.

Let σ be a standard hyperbolic metric on S. Let γ denote a (homotopically non-trivial) oriented simple closed curve, and let α denote any element of C′(S, ∂S), which intersects γ. Let x be a point of the intersection γ^{σ} ∩ α^{σ}. In the universal cover S̃ of S, which is a convex subset of H^{2}, let L_{γ} be a geodesic in the lift of γ^{σ}, and L_{α} be a geodesic in the lift of α^{σ} (extended to an infinite geodesic if it meets ∂S̃), such that L_{γ} and L_{α} meet in a point that projects to x. Identify L_{γ} isometrically with R, consistent with the orientation of γ, and let let a_{R}, a_{L} be the endpoints of L_{α} to the right and left, respectively, of L_{γ}. Let p_{γ} : H^{2} ∪ S^{1} → L_{γ} be the orthogonal projection, and define the (signed) twisting number

t′_{γ,σ}(α, x) = \frac{p_{γ}(a_{R}) − p_{γ}(a_{L})}{ℓ_{σ}(γ)}.  (3.1)

(see figure 1). It is clear that the definition does not depend on the particular choice of lifts.

Figure 1

Lemma 3.1. Let γ be a homotopically non-trivial oriented simple closed curve, α ∈ C(S, ∂S), and σ a standard hyperbolic metric on S. If x and x′ are two points of γ^{σ} ∩ α^{σ}, then

|t′_{γ,σ}(α, x) − t′_{γ,σ}(α, x′)| ≤ 1.

Proof. Let g be the indivisible element of π_{1}(S) whose axis is L_{γ}. Choosing appropriate lifts L_{α} and L′_{α} corresponding to x and x′, we may assume that L′_{α} is trapped between L_{α} and g(L_{α}). The proof is completed by figure 2, and the fact that if a < s < a + 1 and b < t < b + 1 then (b − a) − 1 < (t − s) < (b − a) + 1.

Figure 2

Thus it is useful to define the quantity

t_{γ,σ}(α) = min_{x} t′_{γ,σ}(α, x)  (3.2)

where the minimum is taken over all x in γ^{σ} ∩ α^{σ}.

We can also measure twisting inside an annulus. If A is an annulus with conformal structure [σ] then let σ be a standard Euclidean metric on A in this conformal class. The univeral cover of A is isometric to a strip [0, L] × R in the plane, with fundamental domain of height H. If α is a curve connecting the boundaries of A, it lifts to a curve with endpoints (0, y_{0}) and (L, y_{1}), and we may define

t_{A,σ}(α) = (y_{1} − y_{0})/H

(the sign of t is again determined after a choice of orientation of A).

Now consider again a standard hyperbolic metric σ on S, and an annulus A of the (ϵ_{0}, ϵ_{1}) collar decomposition of (S, σ). If α ∈ C(S) intersects A essentially then its geodesic representative α^{σ} intersects A in several disjoint components, whose twists therefore differ by no more than one. Thus we may take t_{A,σ}(α) to be the minimum of these. Note in particular that if A is peripheral then t_{A,σ}(α) = 0 since α^{σ} is perpendicular to ∂A.

Lemma 3.2. There exist ϵ_{0}, ϵ_{1} less than the Margulis constant such that, for any standard hyperbolic metric σ on a surface S, if A is an (ϵ_{0}, ϵ_{1}) collar with (nonperipheral) core γ and α is any element of C(S) that crosses γ,

|t_{A,σ}(α) − t_{γ,σ}(α)| ≤ 1.

Proof. In case A is peripheral both twists are 0, so let us assume A is internal. Consider figure 3, where in the upper half-plane model of H^{2} we place a lift L_{γ} of γ^{σ}, and the corresponding lift Ã of A. Pick x ∈ γ^{σ} ∩ α^{σ} and choose a component L_{α} of the lift of α^{σ} which intersects L_{γ} in a point projecting to x. Let α_{x} denote the component of α^{σ} ∩ A which contains x. We may arrange the figure so that L_{α} exits Ã as shown, terminating at infinity in the interval J.

Figure 3

Let ϵ ≤ ϵ_{1} be the translation length of γ along its axis. The condition that A is an (ϵ_{0}, ϵ_{1}) collar implies that the angle θ between L_{γ} and ∂Ã satisfies cos θ = ϵ/ϵ_{0}. The left endpoint of J is determined by the limiting case where L_{α} is vertical, so its projection to L_{γ} has height r = sin θ. It follows that the distance between the projection to L_{γ} of the endpoint of α_{x} and the endpoint at infinity of L_{α} is at most | log sin θ|. The quantity |t′_{γ,σ}(α, x) − t_{A,σ}(α_{x})| is equal to exactly double this distance (counting the other endpoint), scaled by 1/ϵ. We obtain:

|t′_{γ,σ}(α, x) − t_{A,σ}(α_{x})| < \frac{1}{ϵ} |log(1 − ϵ^{2}/ϵ_{0}^{2})| .

Thus given ϵ_{0} we may choose ϵ_{1} and hence ϵ sufficiently small that this is at most 1. Since both t′_{γ,σ}(α, x) and t_{A,σ}(α_{x}) are minimized for the same x (both depend monotonically on the angle at which γ^{σ} and α_{x} cross), we have the desired inequality.

The following lemma observes that for a thin annulus A any twisting number can be achieved, up to an additive constant, by some curve that spends “most” of its time in A. This fact will be used in the proof of theorem 6.1.

Lemma 3.3. Let (Q, τ) be a surface of finite topological type with standard hyperbolic metric τ. Let ϵ_{0} > ϵ_{1} > 0 be as in lemma 3.2, and fix ℓ_{1} > 0. Suppose that ℓ_{τ}(γ) ≤ ℓ_{1} for all boundary components γ of Q, and let A ⊂ Q be an internal annulus of the (ϵ_{0}, ϵ_{1}) collar decomposition. Given t ∈ R there exists α ∈ C(Q) such that

|t_{A,τ}(α) − t| ≤ c_{1},

and α consists of one or two arcs traversing A, together with one or two segments of hyperbolic length at most c_{2}. The constants c_{1}, c_{2} depend only on ϵ_{0}, ϵ_{1}, ℓ_{1} and χ(Q).

Proof. We claim that, for some r_{1} depending on ϵ_{0}, ϵ_{1}, ℓ_{1} and χ(Q), there is, for each boundary component of A, a geodesic arc outside A of length at most r_{1}, which either joins that component to the other, or to itself. To see this, let A_{r} denote the r-neighborhood of A in Q; A_{r} is the union of A with two pieces, each adjacent to a boundary component. If at least one of them is a collar, i.e. an embedded annulus of radius r in the interior of Q, we have Area(A_{r}) > ϵ_{1}e^{r}, and since Area(Q) ≤ 2π|χ(Q)| we have an upper bound r < r_{0} = log 2π|χ(Q)|/ϵ_{1}. Thus, if r ≥ r_{0} then both added pieces fail to be collars. If for some r ≤ r_{0} a collar meets itself or the other collar we have the desired geodesic arc of length r_{1} = 2r_{0}. The other possibility is for a collar to hit the boundary at some r ≤ r_{0}. In this case there is an arc of length at most r_{1} = 2r_{0} +ℓ_{1} which runs once around the boundary and back to A.

Now if there is an arc joining both components, we can take α_{0} to be the union of this short arc with an arc that runs through A. If there are two arcs, joining each boundary to itself, we can join these arcs to a union of two parallel arcs through A. Applying n Dehn twists around the core of A to α_{0} will clearly increase (or decrease) its twisting number by about n (see also lemma 3.5), so that by an approriate twist we may obtain α with the desired twisting number.

In the proof of corollary 5.4, we will need to measure twisting using a hyperbolic metric that is not quite standard. In particular, suppose that σ is a standard hyperbolic metric on S and that P is a subsurface consisting of a union of components of the (ϵ_{0}, ϵ_{1}) collar decomposition, such that any annular components in P are nonperipheral in P. Let σ̂ be the standard hyperbolic metric on P which is conformally equivalent to σ. Now if γ ⊂ P is a non-peripheral geodesic of length ℓ_{σ}(γ) < ϵ_{1}, we can measure twisting around γ with respect to both metrics. The following lemma compares the two:

Lemma 3.4. Let P, S, σ and σ̂ be as above. There exist ϵ_{0}, ϵ_{1} sufficiently small and c > 0 such that, if ℓ_{σ}(γ^{σ}) < ϵ_{1} for a non-peripheral simple closed curve γ ⊂ P, then

|t_{γ,σ}(β) − t_{γ,σ̂}(R_{P} (β))| ≤ c

for any β ∈ C(S, ∂S).

Proof. We shall give the proof in the case that S has no boundary. The general case can be reduced to this by means of a doubling argument. Let β^{σ} be the σgeodesic representative of β. Let P_{0} be the complement in P of ϵ_{M}-thin annulus neighborhoods of ∂P (measured in σ̂), where ϵ_{M} > ϵ_{0} is the Margulis constant. We claim that, given ν > 0 there is a choice of ϵ_{0} so that in P_{0} the ratio σ/σ̂ is ν-close to 1, and its derivatives (in normal σ̂ coordinates) are bounded by ν. This will imply that β^{σ} ∩ P_{0} is a curve system with very low (O(ν)) curvature in the metric σ̂ . Thus the σ̂-geodesic β̂_{0} homotopic to β^{σ} ∩ P_{0} rel endpoints is O(ν)-close to it. Now, we are interested in R_{P} (β)σ̂, the σ̂-geodesic representative of β^{σ} ∩ P, rel ∂P. But by lemma 4.5 applied in (P, σ̂), the intersection R_{P} (β)σ̂ ∩ P_{0} is obtained by a bounded homotopy from β̂_{0}. It follows that the twisting number around γ, which is certainly in P_{0}, has not changed by more than a bounded amount.

It remains to prove our claim about σ/σ̂, but this is an elementary argument in complex analysis. If we consider the universal coverings of P in (subsets of) the unit disk which are obtained from the two metrics σ and σ̂, we see that the identity on P lifts to a conformal map Φ between these covers. For x ∈ P_{0}, the hyperbolic distance to ∂P, in either metric, is bounded below by roughly log ϵ_{M}/ϵ_{0}, which we can make as large as we please. If x lifts to 0 in both covers we see (using the formulas for hyperbolic distance) that Φ must map a disk of radius roughly (ϵ_{M} − ϵ_{0})/(ϵ_{M} + ϵ_{0}) univalently over a disk of similar radius. The Schwarz lemma applied to Φ and Φ^{−1} implies that |Φ′(z)| is close to 1 in a neighborhood of 0, and an application of the Cauchy integral formula serves to bound Φ′′(0). Our claim about σ/σ̂ follows.

Recall now the construction of Fenchel-Nielsen coordinates on a Teichmüller space T (S) (see e.g. [1]) Let γ_{1}, …, γ_{n} denote a system of oriented simple closed curves which cuts S into a union of pairs of pants. Suppose, to start, that S has no punctures and p boundary components, so that n = −3χ(S)/2 and each pair of pants is compact. For any standard hyperbolic structure σ on S we immediately obtain n positive numbers {ℓ_{σ}(γ_{i}^{σ})}, the length parameters. Another n − p real numbers describe “twisting” parameters for the gluings of the pairs of pants. Let us make this concrete as follows.

A pair of pants P with boundary curves γ_{1}, γ_{2}, γ_{3} contains three unique homotopy classes of simple properly embedded arcs α_{12}, α_{23}, α_{13} (the “seams”), such that α_{ij} joins γ_{i} to γ_{j}. Fix a set of representatives of the seams which match on opposite sides of each non-peripheral γ_{i}. This determines a homotopy class of curve systems, μ in C(S, ∂S). We call the (oriented) γ_{i} together with μ a marking, and note that it is purely topological information.

Now for any standard hyperbolic structure σ on S, the seams have geodesic representatives α_{ij}^{σ} which are orthonal to γ_{i}^{σ} and γ_{j}^{σ}. The seams cut each pair of pants into two congruent right-angled hexagons, and in particular they bisect the boundary components. Now for each γ_{j} and each pair of corresponding seam endpoints there is a unique geodesic path along γ_{j} that must be spliced between the endpoints, so that the resulting curve is in the homotopy class μ. The length m_{j}(σ) of this path (which can be given a sign, since γ_{j} is oriented) is the same for both pairs of seam endpoints on γ_{j}. We define our twist parameter s_{j}(σ) to be m_{j}(σ)/ℓ_{σ}(γ_{j}).

This construction gives a homeomorphism F : T (S) → R_{+}^{n} × R^{n−p}, taking σ to (ℓ_{σ}(γ_{1}), …, ℓ_{σ}(γ_{n}), s_{1}(σ), …, s_{n−p}(σ)). We note also that a positive Dehn twist on γ_{j} has the effect of incrementing s_{j}(σ) by one and leaving the other coordinates invariant.

In the case with punctures the construction is similar, except that one or two ends of a pair of pants may be a puncture, and we allow the seams to be non-compact arcs terminating in boundaries or punctures, so that μ is in C′(S, ∂S).

To conclude this section, we give a comparison between our twisting numbers and the Fenchel-Nielsen twist coordinates.

Lemma 3.5. Let γ_{1}, …, γ_{n} and μ be a marking for S, and s_{j} : T (S) → R be the associated twist coordinates. Then for any standard hyperbolic metric σ on S,

|s_{j}(σ) − t_{γ_{j},σ}(μ)| ≤ 1.

Furthermore, given any α ∈ C(S, ∂S) and two standard hyperbolic metrics σ and σ′,

|(t_{γ_{j},σ}(α) − t_{γ_{j},σ′}(α)) − (s_{j}(σ) − s_{j}(σ′))| ≤ 4

Proof. To obtain the first inequality, lift γ_{j}^{σ} to a geodesic L_{j} in H^{2}, and consider a component μ̃ of the preimage of μ^{σ} in H^{2} that intersects L_{j}. This geodesic is homotopic to a curve that follows two (lifts of) seam curves α_{jk}^{σ}, α_{jm}^{σ} whose endpoints on L_{j} are separated by s_{j}(σ)ℓ_{σ}(γ_{j}) (see figure 4). The other endpoints of these seams meet lifts L_{k} and L_{m} of γ_{k}^{σ}, γ_{m}^{σ}. Sinc L_{k} (resp. L_{m}) is disjoint from its translates, its projection to L_{j} is a segment of length at most ℓ_{σ}(γ_{j}) centered on an endpoint of α_{jk}^{σ} (resp. α_{jm}^{σ}). The endpoints of μ̃ at infinity must be separated from L_{j} by L_{k} and L_{m}, respectively. It follows that the length of the projection of μ to L_{j}, divided by ℓ_{σ}(γ_{j}), differs from s_{j}(σ) by at most 1.

Figure 4

For the rest of the lemma we note the fact that, for any two curve systems α, β ∈ C′(S, ∂S), the difference t_{γ_{j},σ}(α)−t_{γ_{j},σ}(β) is independent of σ up to a bounded error (of 1). This is because we can detect this difference topologically: lifting to the universal cover, let L_{α} be a lift of a component of α which crosses a lift L_{j} of γ_{j}. The translates of L_{α} along L_{j} divide the plane into a sequence of consecutive strips, and a component of a lift of β that crosses L_{j} meets some number δ of these strips. (The number obtained from a different lift of β may differ by at most 1). It is easy to see that δ, appropriately signed, gives this twist difference, up to an error of 1.

The second inequality now follows from this fact applied to α and μ, together with the first inequality.

4. Basic extremal length estimates

In this section we shall discuss and derive some fairly well-known estimates for extremal lengths, which will be the building blocks of the proof in section 5.

Here and in the rest of the paper, we shall use “a ≍ b” to mean that the ratio a/b is bounded above and below by positive constants, which usually depend on topological data, and on previously chosen constants. Similarly, “a ≺ b” and “b ≻ a” mean that a/b is bounded above. Where necessary we will indicate explicitly the dependence of the constants.

4.1. Spacing and upper bounds. Lower bounds on extremal lengths are relatively easy to obtain, by exhibiting some metric and computing the length in that metric. The following lemma is helpful for obtaining upper bounds.

For a representative curve β̂ of a class β in C(S, ∂S) and a metric σ on S, define the spacing ν_{S,σ}(β̂) as the supremum of numbers ν such that the ν-neighborhood N_{ν}(β̂) is a standard product β̂ × [0, 1]. We have:

Lemma 4.1. Given a metric σ on S and a representative β̂ of β ∈ C(S, ∂S), the extremal length of β is bounded by

λ_{S,σ}(β) ≤ \frac{Area(S, σ)}{4ν_{S,σ}^{2}(β̂)}.

Proof. Let β_{1}, …, β_{n} denote the components of β̂. The neighborhoods R_{i} = N_{ν}(β_{i}) for any ν < ν_{S,σ}(β̂) are, by assumption, either annuli or rectangles. In either case, the extremal length of the family of transverse arcs α_{i} connecting the boundaries β_{i} × {0, 1} is, applying the definition,

λ_{R_{i},σ}(α_{i}) ≥ \frac{4ν^{2}}{Area(R_{i}, σ)}.  (4.1)

Let [β_{i}] denote the family of curves in R_{i} freely homotopic to β_{i} when β_{i} is a closed curve, and homotopic with endpoints on the edges ∂β_{i} × [0, 1] when β_{i} is a segment. Then it is a standard fact that

λ_{R_{i},σ}([β_{i}]) = 1/λ_{R_{i},σ}(α_{i}).

This together with (4.1) bounds λ_{R_{i},σ}([β_{i}]) from above. Now let β_{R} denote the class of all curve systems that are a union of representatives from each [β_{i}]. Since the R_{i} are all disjoint it is easy to see that

λ_{∪R_{i},σ}(β_{R}) = ∑_{i=1}^{n} λ_{R_{i},σ}([β_{i}])

by an application of the Schwarz inequality (actually only the “≤” direction is needed, but the other direction is even easier).

The proof is completed by a standard monotonicity argument: Since the class β in C(S, ∂S) is bigger than the class β_{R} (it has representatives not limited to the R_{i}), its length in any metric is smaller. Furthermore the area of S in any metric is at least that of ∪R_{i}. Thus λ_{S,σ}(β) ≤ λ_{∪R_{i},σ}(β_{R}), and the lemma follows.

4.2. Extremal length in hyperbolic subsurfaces. If a hyperbolic subsurface M of S is “insulated” from S by sufficiently thick annuli on its boundary, then extremal lengths in C(S) and C_{0}(M) (curves that are non-peripheral in M) are approximately the same:

Lemma 4.2. [25, lemma 8.4] Let (S, σ) be any Riemann surface. There exist m_{0} and C, depending only on the topological type of S, such that if M ⊂ S is a hyperbolic subsurface for which each component γ of ∂M bounds an annulus A_{γ} ⊂ M with modulus m(A_{γ}) ≥ m_{0}, then for any α ∈ C_{0}(M),

1 ≤ \frac{λ_{M,σ}(α)}{λ_{S,σ}(α)} ≤ C.

More generally, a curve in S may cut through M, and if M is a nicely chosen subsurface then we can say something about the extremal length of the intersection. For a standard hyperbolic metric σ, recall the (ϵ_{0}, ϵ_{1}) collar decomposition Q = P ∪ A. For a hyperbolic component P ∈ P, define

λ_{P,σ}(β) = λ_{P,σ}(R_{P} (β)).  (4.2)

Note that, provided ϵ_{0} is chosen sufficiently small, R_{P} (β) is represented by β^{σ} ∩ P. This hyperbolic geodesic also gives a good estimate for extremal length on P, via the following:

Lemma 4.3. Suppose that σ is a standard hyperbolic metric for S, and suppose that ℓ_{σ}(γ) ≤ ℓ_{0} for any component γ of ∂S. For any β ∈ C(S, ∂S) and a hyperbolic component P of the (ϵ_{0}, ϵ_{1}) collar decomposition,

λ_{P,σ}(β) ≍ ℓ_{σ}^{2}(β^{σ} ∩ P)

Where the multiplicative constants for ≍ depend only on the topological type of S and on ϵ_{0}, ϵ_{1} and ℓ_{0}.

Proof. Because of the absence of thin parts in P we can rearrange β^{σ} ∩ P to have sufficiently large spacing in the hyperbolic metric σ. Indeed,

Lemma 4.4. There is a curve system β′ in P homotopic to β^{σ} ∩ P by a homotopy that moves no point of β^{σ} ∩ ∂P more than halfway around the boundary, and such that

ν_{P,σ}(β′) ≥ \frac{c}{ℓ_{σ}(β^{σ} ∩ P)}

where the constant c depends only on the topological type of S, and on ϵ_{0}, ϵ_{1} and ℓ_{0} Proof. Subdivide P into pairs of pants Q_{1}, …, Q_{p}, using the shortest collection of pairwise disjoint simple geodesics in P. The number p of pieces is given by p = |χ(P)|, and we note that some pairs of pants may have one or two punctures rather than boundaries. The lengths of all boundary components of Q_{i} are bounded below by ϵ_{1}, and above by a constant ℓ_{1} depending on ℓ_{0} and the topological type of S. (This is a standard fact, whose proof is similar to the proof that the injectivity radius of (S, σ) is bounded above in terms of the hyperbolic area of S).

Begin by performing a homotopy on β^{σ} ∩ Q_{j}, for each j, which moves the points of β^{σ} ∩ ∂Q_{j} around ∂Q_{j} until they are evenly spaced, and keeps the rest of the curve geodesic. Call the new curve β_{1}. Clearly this can be done so that no endpoint moves more than halfway around its boundary component. Furthermore, provided ϵ_{0} has been chosen sufficiently small, β_{1} will still be contained in P.

The bound ℓ_{1} on boundary lengths means that the length of no arc has changed by more than an additive constant by this homotopy. It also means that there is a lower bound ℓ_{2} on the length of any path joining two boundaries of any Q_{j}. Furthermore no closed component of β^{σ} ∩ P has length less than ϵ_{1}. We conclude that

ℓ_{σ}(β_{1}) ≤ c_{1}ℓ_{σ}(β^{σ} ∩ P)

where c_{1} depends on the previous constants.

Now we deform β_{1} in each Q_{k} separately to obtain a curve β_{2} with maximal spacing. Consider first the case where β_{1} ∩ Q_{k} has no closed curve components, and where Q_{k} has three boundaries and no cusps.

One way to make the construction concrete is to form the following Voronoi diagram: For i = 1, 2, 3 let C_{i} be the collar defined by

C_{i} = {x ∈ Q_{k} : d(x, γ_{i}) ≤ d(x, γ_{j}) for j ≠ i}.

Then each C_{i} is an annulus whose whose width at any point is bounded between two constants c_{2}, c_{3}, and whose boundary in the interior of Q_{k} consists of two geodesic arcs α_{ij} = C_{i} ∩ C_{j} (for j ≠ i) of length s_{ij} ∈ [c_{4}, c_{5}] (The estimates here are uniform since the lengths of γ_{i} are bounded above and below, so we have a compact set of possible pairs of pants. Alternatively one can obtain explicit estimates using hyperbolic trigonometry.)

In fact, one can use (for example) the foliation of each C_{i} by geodesics perpendicular to γ_{i} to obtain a Euclidean metric σ̂ on C_{i} which is bilipschitz equivalent to σ, with a uniform constant.

Now for each arc τ of β_{1} ∩ Q_{k}, associate a point x(τ) on α_{ij} if the endpoints of τ lie on γ_{i} and γ_{j}. Arrange the points {x(τ)} so that they are uniformly spaced on each α_{ij}. There is a unique arc τ′, homotopic to τ, which spirals in C_{i} from γ_{i} to x(τ), and then in C_{j} from x(τ) to γ_{j}, and is a σ̂-geodesic in each C_{i}. The homotopy from τ to τ′ will move no point more than a bounded distance (this is easiest to see in the universal cover – see figure 5). It is also easy to see that the resulting arcs will be disjoint Figure 5. (a) The decomposition of a pair of pants into three annuli. (b) In the lifts of two adjacent annuli to H^{2}, an example of an arc τ and the homotopic arc τ′.

We can treat the case with cusps similarly, except that we first excise a horospherical neighborhood of each puncture with boundary length ϵ_{0}, and work in the complement.

If β_{1} has a closed component in Q_{k}, it must be homotopic to a boundary component γ_{j}, and no part of β_{1} can cross that component. Thus we can take a collar neighborhood of γ_{j} of definite width and space the corresponding components evenly (and maximally) there.

Now for maximally spaced curves in a Euclidean annulus it is clear that the spacing is approximately the area divided by the total length. Where two collars meet along a common boundary segment, the local picture is, up to uniform bilipschitz distortion, as in figure 6, and it is an exercise in Euclidean geometry to see that the spacing is equal to the minimum of the spacings on either side.

Figure 6 Thus we have, since the area of each Q_{k} is bounded,

ν_{Q_{k},σ}(β_{2} ∩ Q_{k}) ≥ \frac{C}{ℓ_{σ}(β^{σ} ∩ Q_{k})}

for each Q_{k}, and for all of P we obtain

ν_{P,σ}(β_{2}) ≥ C′ min_{k} ν_{Q_{k},σ}(β_{2} ∩ Q_{k})

since again the spacing is not reduced by more than a constant factor at the interfaces between subsurfaces.

Setting β′ = β_{2} and observing that ℓ_{σ}(β^{σ} ∩ P) ≥ max_{k} ℓ_{σ}(β^{σ} ∩ Q_{k}), the statement of lemma 4.4 follows.

Now we can apply the spacing lemma 4.1 and the fact that the area of P is bounded in terms of its Euler characteristic to conclude that

λ_{P,σ}(β) ≤ \frac{C}{ν_{P,σ}^{2}(β′)} ≤ Cℓ_{σ}^{2}(β^{σ} ∩ P).

To obtain the inequality in the other direction we observe that the minimal length in the homotopy class R_{P} (β) is nearly attained by β^{σ} ∩ P. In fact, the shortest representative β^{P} can be obtained by sliding the endpoints of β^{σ} ∩ P around ∂P until an orthogonal arc is obtained. The sliding distance is uniformly bounded by lemma 4.5 which appears below. Thus, ℓ_{σ}(β^{P}) ≥ cℓ_{σ}(β^{σ} ∩ P) for uniform c, and we conclude by the definition of extremal length that

λ_{P,σ}(β) ≥ c^{2} \frac{ℓ_{σ}^{2}(β^{σ})}{Area(S, σ)}.

Since Area(S, σ) is determined by the Euler characteristic of S, we are done with lemma 4.3.

Lemma 4.5. Let A_{1} and A_{2} be annuli (possibly the same) of the (ϵ_{0}, ϵ_{1}) collar decomposition of (S, σ), and let α,β be geodesic arcs in S −(A_{1} ∪A_{2}) with endpoints on ∂A_{1} ∪∂A_{2}, such that α is homotopic to β keeping the endpoints on the boundary. Then the homotopy can be taken to move the endpoints of α by no more than c = c(ϵ_{0}, ϵ_{1}).

Proof. Lifting to Ã_{1}, Ã_{2}, α̃ and β̃ in H^{2}, we see that the endpoints of α̃ and β̃ must lie in the arcs a_{1} and a_{2} in figure 7. It is fairly easy to see that the extremal case (longest a_{i}) occurs when Ã_{1} and Ã_{2} are actually tangent, and each is the rneighborhood of a geodesic, where r = cosh^{−1}(ϵ_{0}/ϵ_{1}). The length of a_{1} and a_{2} in this case gives the constant c.

Figure 7

4.3. Extremal length on annuli. For annuli A in S we will define a quantity λ_{A,σ}(β) which is not exactly extremal length, but will quantify the extremal length contribution of A to the whole curve β ∈ C(S, ∂S).

In particular, if A is an annulus of the (ϵ_{0}, ϵ_{1}) collar decomposition of (S, σ) with core curve γ, then for β ∈ C(S, ∂S) let i(β, A) or i(β, γ) denote the (minimal) intersection number of β with γ (note, this includes the case that A is peripheral, γ ⊂ ∂S and β has i(β, γ) endpoints on γ). The geodesic β^{σ} can meet A either in a collection of i(β, A) properly embedded arcs, or if i(β, A) = 0, in a union of closed curves homotopic to γ. In either case let n(β, A) denote the number of components of β^{σ} ∩ A. We define

λ_{A,σ}(β) = \begin{cases}i(β, A)^{2} (m(A) + \frac{t_{A,σ}(β)^{2}}{m(A)} ) & if i(β, A) > 0, \\ n(β, A)^{2}/m(A) & otherwise. \end{cases}  (4.3)

It is worth nothing that, if t_{A,σ}(β) is an integer and a gluing together (without twisting) of the boundaries of A takes β to a closed curve on the torus, then (4.3) gives the extremal length of the homotopy class of this curve. Compare also to lemma 2.2.

5. The main length estimate

For the remainder of the paper we shall fix a choice of ϵ_{0} > ϵ_{1} > 0 so that the following hold for any standard hyperbolic metric σ on S.

(1) ϵ_{0} is less than ϵ_{M}, the Margulis constant of H^{2}.

(2) Each annulus of the (ϵ_{0}, ϵ_{1}) collar decomposition has modulus at least 1. It is furthermore contained in a slightly larger annulus, so that each internal boundary component bounds an annulus in the thick part of modulus m_{0}, where m_{0} is the modulus bound in lemma 4.2.

(3) A simple geodesic in (S, σ), if it intersects an annulus of the (ϵ_{0}, ϵ_{1}) collar decomposition, either does so in arcs that connect the boundaries or it is the geodesic core of the annulus (see §2.4).

(4) Lemma 3.2 holds.

Let σ denote a fixed standard hyperbolic metric on S, and let Q = P ∪ A denote the (ϵ_{0}, ϵ_{1}) collar decomposition. of section 2.4.

Theorem 5.1. Let S be a surface of finite type with a standard hyperbolic metric σ in which boundary lengths are at most ℓ_{0}, and let Q be the set of components of the (ϵ_{0}, ϵ_{1}) collar decomposition. Then, for any β ∈ C(S, ∂S) we have

λ_{S,σ}(β) ≍ max_{Q∈Q} λ_{Q,σ}(β)

where the constants for ≍ depend only on ϵ_{0}, ϵ_{1}, ℓ_{0} and the topological type of S.

Proof. Fix, for the remainder of the section, a homotopy class β ∈ C(S, ∂S). Throughout the rest of the section we shall make the abbreviations λ_{Q} = λ_{Q,σ}(β), t_{A} = t_{A,σ}(β), m_{A} = m(A, σ), and n(A) = n(β, A). On each annulus A ∈ A, let σ_{e} denote the unique standard Euclidean metric on A which is conformally equivalent to σ, and agrees with σ on ∂A. The upper bound. To bound λ_{S,σ}(β) from above, we shall need to construct a metric on S, in the conformal class of σ, which is sufficiently similar to the extremal metric for β that the spacing estimate in lemma 4.1 can be applied. We remark that it is of course known that an extremal metric exists, and in fact is given by a quadratic differential. Our construction, by contrast, is not extremal (in fact we make no explicit comparison to the extremal metric), but gives concrete estimates which are explicitly related to the hyperbolic geometry of (S, σ).

Let us make some new definitions. Let h_{1} : Q → [0, ∞) be given by

h_{1}(Q) = \{ \begin{matrix}\sqrt{λ_{Q}} & Q ∈ P \\ \sqrt{λ_{Q}/m_{Q}} & Q ∈ A. \end{matrix}  (5.1)

This quantity has the property that the scaled metrics h_{1}(P)σ on P ∈ P and h_{1}(A)σ_{e} on A ∈ A have areas comparable to λ_{P} and λ_{A}.

However, in order to scale the metric in a consistent way on the entire surface we shall need to adjust h_{1} to obtain a function h, as follows:

Lemma 5.2. There exists a positive function h : Q → R_{+} such that the following hold.

(1) h(Q) ≥ h_{1}(Q) for all Q ∈ Q,

(2) max_{Q} h(Q) = max_{Q} h_{1}(Q), and, whenever A ∈ A is adjacent to P ∈ P,

(3) h(A) ≤ h(P). Furthermore, for any internal annulus A ∈ A which is adjacent to components P_{1}, P_{2} ∈ P:

(4) h(A)^{2} = max (h_{1}(A)^{2}, h(P_{1})h(P_{2})e^{−πm_{A}}) , and if A is peripheral and adjacent to P ∈ P,

(5) h(A) = max (h_{1}(A), h(P)e^{−πm_{A}/2})

Remark: This function will be used to rescale σ on each P and σ_{e} on each annulus A, in such a way that we can interpolate between the different scaling factors in a controlled fashion. Conditions 3, 4 and 5 bound the ratios between scaling factors on adjacent components for this purpose.

Proof. Define h_{2}(A) = h_{1}(A) for A ∈ A. For P ∈ P, let A_{1}, …, A_{k} ∈ A be the annuli adjacent to P. Define

h_{2}(P) = max {h_{1}(P), h_{1}(A_{1}), …, h_{1}(A_{k})} .  (5.2)

This function satisfies conditions (1), (2) and (3) of the lemma, and note also that h_{2}(P) > 0 for at least one P. Next, let us adjust h_{2} to a function h_{n} that satisfies the following additional condition: for each internal A ∈ A adjacent to P_{1}, P_{2} ∈ P, we require that h_{n}(P_{1})h_{n}(P_{2}) > 0, and

|log \frac{h_{n}(P_{1})}{h_{n}(P_{2})}| ≤ πm_{A}.  (5.3)

If the condition fails for h_{2}, then it must fail with some A, P_{1}, P_{2} for which h_{2}(P_{1}) > 0 and h_{2}(P_{2}) < h_{2}(P_{1})e^{−πm_{A}}. Define

h_{3}(P_{2}) = h_{2}(P_{1})e^{−πm_{A}},  (5.4)

and h_{3} = h_{2} elsewhere. Clearly h_{3} ≥ h_{2}, and the maximal value of h_{2} hasn’t been changed. We may repeat this process to obtain a sequence of functions h_{1}, h_{2}, h_{3}, …, which clearly terminates after a finite number of steps in a function h_{n} that satisfies conditions (1),(2),(3) and (5.3). Note also that h_{n} is positive on P, since if h_{n}(P) = 0 after these adjustments, then P must be adjacent only to peripheral annuli on which h_{1} = 0, and in that case β is empty (we are assuming that S is connected).

At this point we may set h = h_{n} for every P ∈ P. For each internal annulus A ∈ A, we set h(A) to the maximum given in condition (4). For each peripheral annulus A we set h(A) according to condition (5). Note that h still satisfies conditions (1)- (3). In particular, condition (3) is satisfied by virtue of the inequality (5.3). This concludes the proof of lemma 5.2.

The next step is to define a new metric σ′ on S, conformally equivalent to σ. For P ∈ P we have simply

σ′|_{P} = h(P)σ.  (5.5)

For an internal annulus A, let P_{1} and P_{2} be the components of P which adjoin A. For each P_{i}, if h(P_{i}) > h(A) we must interpolate between the natural scalings for P_{i} and A. Let m = m_{A} and let

μ_{i} = \frac{1}{2π} log(h(P_{i})/h(A)).  (5.6)

This is the modulus of an annular region in the plane between two concentric circles whose radii are in the ratio h(P_{i})/h(A). We would like to reserve subannuli on A of appropriate modulus on which σ′ is chosen isometric to such concentric plane annuli. In addition we want to reserve a sufficiently large (modulus at least m/2) middle section of A for other purposes. It is for these reasons that condition (4) was imposed in lemma 5.2.

Condition (4) implies that μ_{1} + μ_{2} ≤ m/2. Thus, let B_{1} and B_{2} be annulus neighorhoods of the boundary components of A, whose radii in the Euclidean metric σ_{e} are ϵ_{0}μ_{1}, ϵ_{0}μ_{2}. Let C denote the closure of the complementary annulus A − B_{1} − B_{2}; the modulus of C is at least m/2.

On B_{1} and B_{2} we can make σ′ the metric conformally equivalent to σ such that (B_{i}, σ′) is a concentric planar annulus as above, with boundary lengths ϵ_{0}h(P_{i}) (for the boundary between A and P_{i}) and ϵ_{0}h(A) on the inner boundary. On the remaining annulus C between B_{1} and B_{2} let σ′ = h(A)σ_{e}. Note that this makes σ′ a continous metric on S. (See figure 8.)

Figure 8. The metric σ′ on the annulus A decomposed into B_{1}, C and B_{2}.

Similarly if A is a peripheral annulus adjacent to P, we can apply condition (5) of lemma 5.2 which implies that μ = \frac{1}{2π} log(h(P)/h(A)) ≤ m/4. Thus we place a planar annulus metric on the annulus neighborhood B of P ∩ A, of σ_{e}-radius ϵ_{0}μ. On A − B we again let σ′ = h(A)σ_{e}.

Finding a good representative. Having defined σ′, we shall now construct a representative of β whose spacing in σ′ is as uniform as possible.

Lemma 5.3. There is a representative β_{2} of β whose spacing in the metric σ′ is bounded by

ν_{S,σ′}(β_{2}) ≥ c.

The constant c depends only on ϵ_{0}, ϵ_{1}, ℓ_{0} and χ(S).

Proof. We begin by applying lemma 4.4 to deform β ∩ P (if non-empty) on each P ∈ P to a curve system β_{1} whose intersection with each boundary component of P is evenly spaced, and whose spacing on the interior satisfies ν_{P,σ} ≥ c/ℓ_{σ}(β^{σ} ∩ P).

(The portions of β in the annuli A ∈ A can be moved along with the homotopy on P ∈ P, and we will examine these separately). By lemma 4.3, this implies

ν_{P,σ}(β_{1}) ≥ c′/\sqrt{λ_{P}} = c′/h_{1}(P)

and therefore

ν_{P,σ′}(β_{1}) ≥ c′h(P)/h_{1}(P) ≥ c′.

Now consider the case of an internal annulus A ∈ A, partitioned as before into boundary annuli B_{1}, B_{2} and a central annulus C. Suppose first that i(β, A) > 0. Deform β_{1} ∩ A, fixing its intersection with ∂A, so that each resulting arc intersects (B_{i}, σ′) in a radial segment, and (C, σ′) in a σ′-geodesic arc that twists several times around C. Call this new curve β_{2}. We have

t_{C,σ′}(β_{2} ∩ C) = t_{A} + δ  (5.7)

where the error satisfies |δ| ≤ 1, since by lemma 4.4 the endpoints have moved at most halfway around the boundary in the homotopy from β to β_{1}. It follows that the length of β_{2} ∩ C is, if i(β, A) > 0,

ℓ_{σ′}(β_{2} ∩ C) = n(A)h(A)ϵ_{0}\sqrt{m(C)^{2} + (t_{A} + δ)^{2}}
≤ cϵ_{0}h(A)\sqrt{m_{A}λ_{A}}  (5.8)

Here we are using the formula (4.3) for λ_{A} = λ_{A,σ}(β). Note that the error δ can be replaced by the multiplicative factor c because m_{A} ≥ 1 (by choice of ϵ_{0}, ϵ_{1}).

If i(β, A) = 0 and n(A) > 0 we can make β_{2} ∩ A a disjoint union of closed σ′-geodesics uniformly spaced in C. Then we have:

ℓ_{σ′}(β_{2} ∩ C) = n(A)h(A)ϵ_{0}
= ϵ_{0}h(A)\sqrt{m_{A}λ_{A}}  (5.9)

again using (4.3). Now, since β_{2} is uniformly spaced in C with respect to the standard Euclidean metric σ′, its spacing is equal to the area divided by its length. Applying either (5.8) or (5.9), and the fact that m(C) ≥ m_{A}/2, we obtain:

ν_{C,σ′}(β_{2} ∩ C) = \frac{Area(C, σ′)}{ℓ_{σ′}(β_{2} ∩ C)}
≥ c\frac{ϵ_{0}^{2}h(A)^{2}m_{A}/2}{ϵ_{0}h(A)\sqrt{m_{A}λ_{A}}}
= c′ \frac{h(A)}{h_{1}(A)}.  (5.10)

The spacing in the boundary annuli B_{1}, B_{2} is no smaller than this, and h(A) ≥ h_{1}(A) by lemma 5.2, so we conclude

ν_{A} ≥ c′.  (5.11)

The case when A is peripheral is dealt with in a similar way, and (5.11) still holds. In fact it is slightly easier since t_{A} = 0 in this case.

Thus we have a curve β_{2} with a uniform lower bound on ν_{Q,σ′} for all Q ∈ Q. We can now make the same observation as in the proof of lemma 4.4 to conclude that

ν_{S,σ′}(β_{2}) ≥ c min_{Q∈Q} ν_{Q,σ′}(β_{2}).

Namely, since on a uniformly sized neighborhood of an interface between two components the situation is uniformly bilipschitz equivalent to the Euclidean situation depicted in figure 6, the spacing of the combined curve is no less than a constant times the minimum of the spacings on either side. This concludes the proof of lemma 5.3.

We can now apply lemma 4.1 to obtain:

λ_{S,σ}(β) = λ_{S,σ′}(β)
≤ c Area(S, σ′) = c ∑_{Q∈Q} Area(Q, σ′).  (5.12)

The desired upper bound on λ_{S,σ} will be obtained if we show that

max_{Q} Area(Q, σ′) ≤ c max_{Q} λ_{Q}  (5.13)

for a uniform constant c, since the number of terms in the sum (5.12) is bounded in terms of the topology of S.

For any P ∈ P, Area(P, σ′) ≤ ch^{2}(P) for a constant c, by (5.5) and the fact that Area(P, σ) is bounded. An internal annulus A ∈ A is divided into three subannuli B_{1}, B_{2} and C whose areas are Area(C, σ′) ≤ ϵ_{0}h^{2}(A)m_{A}, and Area(B_{i}, σ′) = ϵ_{0}^{2}(h^{2}(P_{i}) − h^{2}(A))/4π, where P_{1}, P_{2} are the neighboring components of P. A peripheral annulus is treated similarly. Thus, it suffices to bound the quantities h^{2}(P) and h^{2}(A)m_{A}.

Note first that, by definition (5.1), h_{1}^{2}(Q) ≤ λ_{Q}, since m_{A} ≥ 1 for all A ∈ A. Thus we have

max_{Q∈Q} h_{1}^{2}(Q) ≤ max_{Q} λ_{Q}.  (5.14)

It follows from this and part (2) of lemma 5.2 that, for any P ∈ P,

h^{2}(P) ≤ max_{Q} λ_{Q}.  (5.15)

Now for A ∈ A, suppose first that h(A) = h_{1}(A). In this case

h^{2}(A)m_{A} = h_{1}^{2}(A)m_{A}
= λ_{A} ≤ max_{Q} λ_{Q}  (5.16)

If h(A) > h_{1}(A) then, if A is internal we use parts (4) and (2) of lemma 5.2 to obtain

h^{2}(A)m_{A} = h(P_{1})h(P_{2})m_{A}e^{−πm_{A}}

≤ max_{Q} h_{1}^{2}(Q)m_{A}e^{−πm_{A}}

≤ \frac{1}{πe} max_{Q} λ_{Q}  (5.17)

where the last line follows from (5.14) and the fact that xe^{−πx} ≤ 1/πe for all x. In case A is peripheral we apply parts (2) and (5), to similar effect. The bounds (5.15-5.17) together imply (5.13), so that with (5.12) we have the desired upper bound

λ_{S,σ}(β) ≤ c max_{Q∈Q} λ_{Q,σ}(β).

The lower bound. For each Q ∈ Q, we shall find a metric σ′′ conformal to σ such that ℓ_{σ′′}^{2}([β])/ Area(S, σ′′) is approximately λ_{Q}. This will give the desired lower bound.

If Q = P ∈ P, let σ′′ = σ. Note that ℓ_{σ}([β]) ≥ ℓ_{σ}(β^{σ} ∩ P) ≍ \sqrt{λ_{Q}}, and that Area(S, σ) is a constant, 2π|χ(S)|. Thus:

λ_{S,σ}(β) ≥ \frac{ℓ_{σ}^{2}(β)}{Area(S, σ)} ≻ λ_{Q}  (5.18)

If Q = A ∈ A, define the metric σ^{A} to be

σ^{A} = \{ \begin{matrix}σ & in S − A \\ σ_{e} & in A \end{matrix}  (5.19)

where σ_{e} is the Euclidean metric defined as in the beginning of this section.

If i(β, A) = 0 but n(A) > 0, there are n(A) copies in β of the core curve γ of A. It is easy to see that ℓ_{σ^{A}}(γ) = ϵ_{0} in this case, since σ^{A} is non-positively curved so that the σ^{A}-geodesic cores of A are the shortest representatives of γ. Thus,

λ_{S,σ}(β) ≥ \frac{n^{2}(A)ϵ_{0}^{2}}{ϵ_{0}^{2}m_{A} + 2π|χ(S)|} ≻ λ_{A}.

where for the last step we use the fact that m_{A} ≥ 1 to replace an additive constant with a multiplicative one.

Now consider the case where i(β, A) > 0. Let β^{A} be a representative of β of minimal σ^{A}-length (in fact β^{A} is unique, but we do not need to use this), and note that each arc of β^{A} ∩ (S − A) is geodesic, and homotopic with endpoints on ∂A to a corresponding arc of β^{σ} ∩ (S − A) (this is most clearly seen in the universal cover). Thus by lemma 4.5, the homotopy between β^{σ} and β^{A} only moves the intersection with ∂A a bounded amount c. In particular we may conclude that

|t_{A,σ}(β^{A}) − t_{A}| ≤ 2c/ϵ_{0},

and therefore

ℓ_{σ^{A}}(β^{A} ∩ A) ≥ n(A)ϵ_{0}\sqrt{m^{2}(A) + (t_{A} − 2c/ϵ_{0})^{2}}
≻ n(A)ϵ_{0}\sqrt{m^{2}(A) + t_{A}^{2}}
≻ \sqrt{λ_{A}m_{A}}.  (5.20)

(Note that we have again used the fact that m_{A} ≥ 1.)

We conclude that

λ_{S,σ}(β) ≻ \frac{λ_{A}m_{A}}{ϵ_{0}m_{A} + 2π|χ(S)|} ≻ λ_{A}.  (5.21)

This completes the proof of the lower bound and therefore of theorem 5.1.

We can generalize theorem 5.1 slightly as follows. Suppose that A_{1}, …, A_{k} are a subset of the annuli of the (ϵ_{0}, ϵ_{1}) collar decomposition of (S, σ) and P_{1}, …, P_{m} are the components of S − ∪_{i}A_{i}. We call Q = {A_{1}, …, A_{k}, P_{1}, …, P_{m}} a partial (ϵ_{0}, ϵ_{1}) decomposition. We can then state:

Corollary 5.4. If Q is a partial (ϵ_{0}, ϵ_{1}) collar decomposition with ϵ_{0}, ϵ_{1} chosen as above, then for any β ∈ C(S, ∂S)

λ_{S,σ} ≍ max_{Q∈Q} λ_{Q,σ}(β).

Proof. Let Q̄ = P̄ ∪ Ā be the full (ϵ_{0}, ϵ_{1}) collar decomposition. Let Q̄_{i} = P̄_{i} ∪ Ā_{i} be the subset of Q̄ consisting of components that lie in P_{i}. Let σ̂_{i} be the standard hyperbolic metric on P_{i} which is conformally equivalent to the restriction of σ. We first show that, for any Q̄ ∈ Q̄_{i},

λ_{Q̄,σ}(β) ≍ λ_{Q̄,σ̂}(R_{P_{i}}(β)).  (5.22)

For a hyperbolic component Q̄ ∈ P̄_{i}, these two quantities are the same by definition (4.2), since only the conformal class of the metric matters for extremal length.

For an annulus Q̄ = Ā ∈ Ā_{i}, the definition (4.3) depends in particular on the measure of twisting t_{Ā,σ}, which depends on the actual geometry. However, lemma 3.4 assures us that the difference between twisting in σ and σ̂ is no more than a constant. It follows that (5.22) holds for annuli as well, with uniform constants that depend only on ϵ_{0}, ϵ_{1} and the topology of S.

The corollary now follows by applying theorem 5.1 separately to each P_{i}.

6. The product region theorem

In this section S denotes an oriented finite genus surface without boundary, but possibly with finitely many punctures.

Let γ be an element of C(S) all of whose components γ_{1}, …, γ_{k} are homotopically distinct, and fix an orientation on each component of γ. For any ϵ > 0, denote by T hin_{ϵ}(S, γ) the subset of T (S) consisting of all [σ] such that ℓ_{σ}(γ_{i}) ≤ ϵ.

Let γ_{k+1}, …, γ_{n} denote a completion of γ to an oriented pair-of-pants decomposition γ̂. Fix also a curve system μ composed of seam arcs, giving a marking of S as in §3. This gives rise to length and twist coordinates on T (S), denoted in §3 by the map F : T (S) → R_{+}^{n} × R^{n}. Let us write the coordinates of F as F_{i}, where F_{1}, …, F_{n} are length coordinates, and F_{n+1}, …, F_{2n} are twist coordinates. Separating out the coordinates of the first k curves from the rest decomposes T (S) as a product.

Let S_{γ} denote the punctured (or “noded”) surface obtained from S by deleting γ_{1}, …, γ_{k} and replacing each curve by a pair of punctures. Let T (S_{γ}) be the space of (analytically finite) conformal structures on S_{γ}. Let H denote the upper halfplane {(x, y) : y > 0} endowed with the hyperbolic metric d_{H}. Let X_{γ} denote the product space X_{γ} = T (S_{γ}) × H_{1} × ⋯ × H_{k} where each H_{i} is a copy of H, endowed with the sup metric d_{X} = max(d_{T (S_{γ})}, d_{H_{1}}, …, d_{H_{k}}).

The marking by γ̂ and μ induces a marking on each component of S_{γ}, and a map G : T (S_{γ}) → R_{+}^{n−k}×R^{n−k}, given by G(ρ) = (ℓ_{ρ}(γ_{k+1}), …, ℓ_{ρ}(γ_{n}), s_{k+1}(ρ), …, s_{n}(ρ)) for any complete hyperbolic metric ρ on S_{γ}.

We can therefore define a map

Π : T (S) → T (S_{γ}) × H_{1} × ⋯ × H_{k}  (6.1)

as follows. Let Π_{0} denote the component of Π mapping to T (S_{γ}), and for i > 0 let Π_{i} denote the component mapping to H_{i}. We define Π_{0} via

Π_{0}(σ) = G^{−1}(F_{k+1}(σ), …, F_{n}(σ), F_{n+k+1}(σ), …, F_{2n}(σ)).  (6.2)

In other words, we build a punctured surface by forgetting the length and twist coordinates of the first k curves, and using the remaining coordinates to build new pairs of pants. We then use the forgotten coordinates for the rest of Π, by defining

Π_{i}(σ) = (s_{i}(σ), 1/ℓ_{σ}(γ_{i}))  (6.3)

for i = 1, …, k.

Let ϵ_{0} and ϵ_{1} be chosen as in the previous section. Our main theorem is the following.

Theorem 6.1. Let γ ∈ C(S) be an oriented curve system as above, extended to a marking γ̂, μ of S. Given ϵ ∈ (0, ϵ_{1}), there is a contant a_{0} depending on ϵ_{0}, ϵ_{1} and the topological type of S, such that the homeomorphism Π : T (S) → X_{γ} defined above, when restricted to T hin_{ϵ}(S, γ), satisfies

|d_{T (S)}(σ, τ) − d_{X}(Π(σ), Π(τ))| ≤ a_{0}

for any [σ], [τ] ∈ T hin_{ϵ}(S, γ).

Proof. For a given [σ] ∈ T hin_{ϵ}(S, γ), let σ denote its hyperbolic representative. Let A_{1}, …, A_{k} be the annuli of the (ϵ_{0}, ϵ_{1}) collar decomposition which correspond to the curves γ_{1}, …, γ_{k}. The complement S − ∪A_{i} is also naturally identified with S_{γ}. Let P_{1}, …, P_{m} denote the connected components of S_{γ}, and let Q = {A_{j}} ∪ {P_{i}}.

Note that for another point [τ] ∈ T hin_{ϵ}(S, γ) we obtain a similar decomposition, which we also call Q.

Our strategy will be to estimate distances using Kerckhoff’s theorem, so we shall need to be able to control extremal length ratios of the type λ_{Q,σ}(α)/λ_{Q,τ}(α). Note that, for Q a subsurface of S, if λ_{Q,σ}(α) = 0 then λ_{Q,τ}(α) = 0 as well. Thus we adopt the convention that, if λ_{Q,σ}(α) = 0 then the expression λ_{Q,σ}(α)/λ_{Q,τ}(α) is defined to be 1.

The first step is the following:

Lemma 6.2. Let Q be any hyperbolic surface of finite type with boundary, and λ_{1} > λ_{0} > 0 two given constants. Let [σ], [τ] ∈ T (Q) be two conformal structures such that λ_{Q,σ}(γ) ∈ [λ_{0}, λ_{1}] for any boundary component γ, and similarly for τ. Then

sup_{α∈C_{0}(Q)} \frac{λ_{Q,σ}(α)}{λ_{Q,τ}(α)} ≍ sup_{α∈C(Q,∂Q)} \frac{λ_{Q,σ}(α)}{λ_{Q,τ}(α)}  (6.4)

where the constants for “≍” depend only on χ(S), λ_{0} and λ_{1}.

(Recall that C_{0}(Q) is the set of homotopy classes of closed curve systems in Q with no peripheral components.)

Proof. The left side is clearly no larger than the right, since C_{0}(Q) ⊂ C(Q, ∂Q).

The supremum on the right side is equal to exp 2d_{T (Q)}(σ, τ), by Kerckhoff’s theorem 2.1, where d_{T (Q)} is the distance in the Teichmüller space of Q. In particular, this distance is symmetric, so that the supremum of the reciprocal ratio is the same quantity. Choose β ∈ C(Q, ∂Q) such that λ_{Q,τ}(β)/λ_{Q,σ}(β) is nearly equal to the supremum on the right side of (6.4).

By lemma 6.3, whose proof appears below, there exists α ∈ C_{0}(S) such that, for a uniform c > 0,

i(α, β)^{2} ≥ cλ_{Q,τ}(α)λ_{Q,τ}(β).  (6.5)

(One should think of α as approximately “orthogonal” to β.)

On the other hand, the following inequality is elementary for any two curves α, β ∈ C(Q) and [σ] ∈ T (Q) (see e.g. [26, Lemma 5.1]).

λ_{Q,σ}(α)λ_{Q,σ}(β) ≥ i(α, β)^{2}.  (6.6)

Combining (6.5) with(6.6), we obtain

\frac{λ_{Q,σ}(α)}{λ_{Q,τ}(α)} ≥ c\frac{λ_{Q,τ}(β)}{λ_{Q,σ}(β)}

which gives the remaining direction of lemma 6.2.

To complete lemma 6.2 we need to prove the following:

Lemma 6.3. Let Q be a hyperbolic surface of finite type, with a conformal structure [τ]. Suppose that for 0 < λ_{0} < λ_{1} we have λ_{Q,τ}(γ) ∈ [λ_{0}, λ_{1}] for any boundary component γ of Q. Then for any β ∈ C(Q, ∂Q) there exists α ∈ C_{0}(Q) for which

i(α, β)^{2} ≥ cλ_{Q,τ}(α)λ_{Q,τ}(β)

where c depends only on λ_{0}, λ_{1} and χ(Q).

Proof. Represent [τ] by a standard hyperbolic metric τ, and let T_{1}, …, T_{j} be the (ϵ_{0}, ϵ_{1}) collar decomposition of Q, with respect to τ. Note that ϵ_{1} may be chosen (depending on λ_{0}) so that no annuli in the decomposition are peripheral in Q. Applying theorem 5.1, there exists one component, say T_{1}, such that λ_{Q,τ}(β) ≍ λ_{T_{1},τ}(β).

Suppose first that T_{1} is a non-annular component. We claim that there exist constants c_{0}, λ_{2} such that there is an α ∈ C_{0}(T_{1}) with λ_{T_{1},τ}(α) ≤ λ_{2}, and i(α, β)^{2} ≥ c_{0}λ_{T_{1},τ}(β). We can see this as follows: Since T_{1} has no thin parts we can cut it along moderate-length curves to obtain bounded-diameter pieces. More precisely, note that the components of ∂T_{1} are either boundaries of thin parts of length ϵ_{0}, or boundaries of Q, which have τ-length in some interval [ℓ_{0}, ℓ_{1}] (depending on λ_{0}, λ_{1}). We can find a pair-of-pants decomposition for T_{1} of minimal total boundary lengths, and then add sufficiently many simple closed geodesics, as short as possible, to cut T_{1} into disks and boundary-parallel annuli. The result is a collection of non-peripheral simple closed geodesics α_{1}, …, α_{q} in T_{1} (q dependent only on χ(T_{1})) such that ℓ_{τ}(α_{i}) ≤ ℓ_{2} (depending on ϵ_{0}, λ_{0}, λ_{1}, χ(T_{1})) and which cut T_{1} up into components that are either disks, or annuli which share a boundary with T_{1}, or annular neighborhoods of cusps. In the first two cases the diameters of the components are uniformly bounded. In the third case we may remove a neighborhood of the cusp avoided by β (since β is simple) and so that what remains has bounded diameter. Furthermore note that any component of the intersection of the geodesic β^{τ} with one of these regions has length at most some δ_{0}, since β^{τ} meets ∂Q orthogonally, and meets the other components of ∂T_{1} nearly orthogonally (compare lemma 4.5).

It follows, adding over the pieces of β^{τ} in these components, that ∑_{j} δ_{0}i(β, α_{j}) ≥ ℓ_{τ}(β^{τ} ∩T_{1}). By lemma 4.3, ℓ_{τ}^{2}(β^{τ} ∩T_{1}) ≍ λ_{T_{1},τ}(β). Thus, there must be some α = α_{j} such that

i(α, β)^{2} ≥ Cλ_{T_{1},τ}(β).

We also know that there is an upper bound

λ_{Q,τ}(α) ≤ λ_{3}

for a uniform λ_{3}, since there is an upper bound on the hyperbolic length of α by construction. The statement of the lemma follows in this case.

Now let T_{1} = A be an annular component with modulus m = m(A, τ), and consider first the case when i(β, A) > 0, and t_{1} = t_{A,τ}(β) ≠ 0. Let t_{2} = −m^{2}/t_{1} – we choose this number because two geodesics with twists t_{1}, t_{2} in a standard Euclidean annulus of modulus m are orthogonal when t_{1}t_{2} = −m^{2}. Lemma 3.3 guarantees the existence of an element α ∈ C_{0}(Q) with |t_{A,τ}(α) − t_{2}| ≤ c_{1} for a uniform c_{1}.

Now using the definition of λ_{A,τ} in (4.3), we can see that

\begin{aligned}λ_{A,τ}(α)λ_{A,τ}(β) & ≍ n(α, A)n(β, A)(m + t_{1}^{2}/m)(m + t_{2}^{2}/m) \\ & = n(α, A)n(β, A)(t_{1} − t_{2})^{2}. \end{aligned}  (6.7)

where the additive error c_{1} becomes a multiplicative error since m ≥ 1.

On the other hand it is easy to see that, if two properly embedded arcs α_{1}, α_{2} in A have twisting numbers t_{i} = t_{A,τ}(α_{i}), then their intersection number is estimated by

|t_{1} − t_{2}| − 1 ≤ i(α_{1}, α_{2}) ≤ |t_{1} − t_{2}| + 1.  (6.8)

Combining (6.8) with (6.7) we obtain

λ_{A,τ}(α)λ_{A,τ}(β) ≻ i(α, β)^{2}  (6.9)

(again an additive error has become multiplicative). By lemma 3.3, α may be chosen to have bounded length outside A, and thus by theorem 5.1, λ_{A,τ}(α) ≍ λ_{Q,τ}(α). Since also λ_{A,τ}(β) ≍ λ_{Q,τ}(β) by choice of A, the statement of lemma 6.3 follows.

We have left the case where t_{1} = 0, and the case where i(β, A) = 0 but n(β, A) > 0. In the former case, We can pick α to be the core of A (essentially this is t_{2} = ∞), and note that λ_{Q,σ}(α) ≍ 1/m, so that the lemma holds automatically. In the latter case we may choose α with t_{2} = 0, and the roles of α and β are reversed.

We now resume the proof of theorem 6.1. Let us establish the following for [σ], [τ] ∈ T hin_{ϵ}(S, γ):

sup_{α∈C(S)} \frac{λ_{S,σ}(α)}{λ_{S,τ}(α)} ≍ max_{Q∈Q} sup_{α∈C(S)} \frac{λ_{Q,σ}(α)}{λ_{Q,τ}(α)}.  (6.10)

One direction is a direct application of the main extremal length estimate. Note that Q is what we called a “partial (ϵ_{0}, ϵ_{1}) collar decomposition” in the previous section. Thus by corollary 5.4 of theorem 5.1, we have

λ_{S,σ}(α) ≍ max_{Q∈Q} λ_{Q,σ}(α)  (6.11)

for each α ∈ C(S). Letting Q_{α} maximize λ_{Q,σ}(α) over all Q ∈ Q, we apply (6.11) to obtain

sup_{α∈C(S)} \frac{λ_{S,σ}(α)}{λ_{S,τ}(α)} ≍ sup_{α∈C(S)} \frac{max_{Q∈Q} λ_{Q,σ}(α)}{max_{Q∈Q} λ_{Q,τ}(α)}
≤ sup_{α∈C(S)} \frac{λ_{Q_{α},σ}(α)}{λ_{Q_{α},τ}(α)}
≤ max_{Q∈Q} sup_{α∈C(S)} \frac{λ_{Q,σ}(α)}{λ_{Q,τ}(α)}  (6.12)

For the inequality in the other direction, consider first a non-annular component Q ∈ Q. By definition (4.2) of λ_{Q,σ} the supremum sup_{α∈C(S)} λ_{Q,σ}(α)/λ_{Q,τ}(α) may be obtained by letting α vary over C(Q, ∂Q). By lemma 6.2, we may further restrict to α ∈ C_{0}(Q), at the expense of a bounded factor. By lemma 4.2, λ_{Q,σ}(α) ≍ λ_{S,σ}(α) for α ∈ C_{0}(Q) and any σ ∈ T hin_{ϵ}(S, γ). Since C_{0}(Q) ⊂ C(S), the supremal ratio over all of C(S) can only be bigger, so in this case we have the desired inequality

sup_{α∈C(S)} \frac{λ_{S,σ}(α)}{λ_{S,τ}(α)} ≥ sup_{α∈C(S)} \frac{λ_{Q,σ}(α)}{λ_{Q,τ}(α)}.  (6.13)

Now consider an annulus A ∈ Q, and let α be a curve nearly maximizing the ratio λ_{A,σ}(α)/λ_{A,τ}(α). By lemma 3.3, there exists α′ ∈ C(S) such that

|t_{A,τ}(α′) − t_{A,τ}(α)| ≤ c_{1},  (6.14)

and such that (applying theorem 5.1)

λ_{S,τ}(α′) ≍ λ_{A,τ}(α′).  (6.15)

Applying lemmas 3.2 and 3.5, we see that t_{A,τ}(α′)−t_{A,σ}(α′) and t_{A,τ}(α)−t_{A,σ}(α) are both approximated, up to bounded additive error, by the twist-parameter difference s_{j}(τ) − s_{j}(σ) associated to A. This together with inequality (6.14) implies that |t_{A,σ}(α′)−t_{A,σ}(α)| is uniformly bounded as well. Since λ_{A,σ} and λ_{A,τ} are determined by the twisting numbers, we conclude that

\frac{λ_{A,σ}(α′)}{λ_{A,τ}(α′)} ≍ \frac{λ_{A,σ}(α)}{λ_{A,τ}(α)}.

Now we may apply (6.15) together with the inequality λ_{A,σ}(α′) ≺ λ_{S,σ}(α′), which is a consequence of theorem 5.1, to conclude that

\frac{λ_{S,σ}(α′)}{λ_{S,τ}(α′)} ≻ \frac{λ_{A,σ}(α′)}{λ_{A,τ}(α′)}.  (6.16)

Inequalities (6.12), (6.13) and (6.16) together imply (6.10).

The length ratio on the left side of (6.10) is, by Kerckhoff’s theorem 2.1, equal to exp 2d_{T (S)}(σ, τ). It remains to connect the length ratios on the right of (6.10) to the distances d_{i} (i = 0, …, k), so that we may complete the proof of theorem 6.1.

Consider first d_{0}, the Teichmüller distance on T (S_{γ}). Given a Q ∈ Q which is a component of S_{γ}, with the conformal structure Π_{0}(σ) or Π_{0}(τ) (restricted from S_{γ} to Q), we have by Kerckhoff’s theorem:

sup_{α∈C_{0}(Q)} \frac{λ_{Q,Π_{0}(σ)}(α)}{λ_{Q,Π_{0}(τ)}(α)} = exp 2d_{T (Q)}(Π_{0}(σ), Π_{0}(τ)).

We now claim that the extremal length ratio on the left can be approximated by a ratio taken with respect to σ and τ. In fact we claim that (Q, σ) embeds Kquasi-conformally, with uniform K, in (Q, Π_{0}(σ)) (note that the embedding will not be onto). Recall the construction: we measure the length and twist parameters for γ_{k+1}, …, γ_{n} for σ, and then construct Π_{0}(σ) with the same parameters, but with the boundary curves γ_{1}, …, γ_{k} replaced by punctures. Thus the embedding of (Q, σ) in (Q, Π_{0}(σ)) can be done separately on each pair of pants. After splitting each pair of pants along seam curves into right-angled hexagons, the problem reduces to the following: Let H_{1} be a hyperbolic right-angled hexagon with lengths ℓ_{1}, ℓ_{2}, ℓ_{3} on alternating sides, where ℓ_{1} ≤ ϵ_{1}, and let H_{2} be a hyperbolic right-angled hexagon with lengths 0, ℓ_{2}, ℓ_{3} (really a right-angled pentagon with one ideal vertex). There is a K-quasiconformal embedding (in fact bilipschitz) of H_{1} into H_{2} which is an isometry on sides 2 and 3, where K depends on ϵ_{1}, but not on the ℓ_{i}. This fact follows easily from the hexagon cosine law (see e.g. [3]). A similar computation applies when a pair of pants has more than one short boundary which is to be replaced by punctures. The maps on the pairs of pants fit together across the boundaries, because the marking and twist parameters are the same for both surfaces.

It now follows (applying also lemma 4.2) that λ_{Q,σ}(α) ≍ λ_{Q,Π_{0}(σ)}(α) uniformly for all σ ∈ T hin_{ϵ}(S, γ) and α ∈ C_{0}(Q). Thus

sup_{α∈C_{0}(Q)} \frac{λ_{Q,Π_{0}(σ)}(α)}{λ_{Q,Π_{0}(τ)}(α)} ≍ sup_{α∈C_{0}(Q)} \frac{λ_{Q,σ}(α)}{λ_{Q,τ}(α)}.

Applying again lemma 6.2, we conclude

exp 2d_{T (Q)}(Π_{0}(σ), Π_{0}(τ)) ≍ sup_{α∈C(Q)} \frac{λ_{Q,σ}(α)}{λ_{Q,τ}(α)}.

Since d_{0} = d_{T (S_{γ})} is the supremum of the distances in the Teichmüller spaces of the components, we have shown that the contribution to the right side of (6.10) coming from the non-annular components is, up to bounded factor, just exp 2d_{0}.

We next consider an annular component A_{j} in Q, and estimate its extremal length contribution λ_{A_{j},σ}(α), for α ∈ C(S) and σ ∈ T hin_{ϵ}(S, γ), in terms of the coordinates Π_{j}(σ) = (s_{j}(σ), 1/ℓ_{σ}(γ_{j})). Rewrite Π_{j}(σ) = (x_{j}(σ), y_{j}(σ)), and note first that (by (2.2)) y_{j}(σ) is closely approximated by m(A_{j}, σ)/π.

Fix some arbitrary σ_{0} ∈ T hin_{ϵ}(S, γ) and define b_{j}(α) = t_{γ_{j},σ_{0}}(α) − s_{j}(σ_{0}). By lemma 3.5, t_{γ_{j},σ}(α) is estimated by b_{j}(α) + s_{j}(σ), up to bounded additive error, for all σ and α. Applying also lemma 3.2, this gives an estimate for t_{A_{j},σ}(α). Now applying definition (4.3) when i(α, γ_{j}) > 0, we obtain this approximation for the length ratio contribution of the annulus:

\frac{λ_{A_{j},σ}(α)}{λ_{A_{j},τ}(α)} ≍ \frac{y_{j}(σ) + (b_{j}(α) + x_{j}(σ))^{2}/y_{j}(σ)}{y_{j}(τ) + (b_{j}(α) + x_{j}(τ))^{2}/y_{j}(τ)},  (6.17)

Note that the case where i(α, γ_{j}) = 0 and n(α, γ_{j}) > 0 can be obtained by the limit of (6.17) as b_{j} → ∞.

By lemma 3.3, we know that t_{γ_{j},σ_{0}}(α), and therefore b_{j}(α), can take on any value in R up to bounded difference. Since y_{j}(σ), y_{j}(τ) are at least 1/ϵ, we may once again convert the additive error to a multiplicative one. Thus, taking a supremum of (6.17) over all α ∈ C(S), and applying lemma 2.2, we obtain

sup_{α∈C(S)} \frac{λ_{A_{j},σ}(α)}{λ_{A_{j},τ}(α)} ≍ exp 2d_{H_{j}}(Π_{j}(σ), Π_{j}(τ)).

It follows that the right side of (6.10) is, up to bounded multiple,

max_{j=0,... ,k} exp 2d_{j}(Π_{j}(σ), Π_{j}(τ)).

Taking logarithms, we obtain the desired statement of theorem 6.1.

7. Geometric remarks

Having shown that regions of the form T hin_{ϵ}(S, γ) in T (S) are approximated up to bounded additive distortion by products with the sup metric, we discuss briefly here the implications to “non-hyperbolic” behavior of the Teichmüller space. Indeed, a product with the sup metric is not just non-negatively curved, but exhibits some definite positive-curvature behavior, as we shall see.

Let X be a metric space, and let x, y, z ∈ X denote three points. We say that z is “δ-between” x and y if

d(x, z) + d(z, y) − d(x, y) < δ

for δ > 0. In particular if X is a length space and [xy] denotes a shortest path (geodesic) between x and y, then when z is δ-between x and y, the path [xz] ∪ [zy] is a quasigeodesic with only additive distortion δ.

In a negatively curved length space in the sense of Gromov and Cannon, if z is δ-between x and y then it is contained in an R(δ)-neighborhood of [xy], where R(δ) is independent of x and y. This is called stability of quasi-geodesics.

Let us define an “instability function” s for X as follows: For δ, L > 0 define s(δ, L) by

s(δ, L) = sup{d(z, [xy]) : z is δ-between x and y and d(x, y) ≤ L}.

In order to consider only the large-scale behavior, let us denote by s̄(δ) the asymptotic growth rate of s(δ, L); that is,

s̄(δ) = lim sup_{L→∞} \frac{log s(δ, L)}{log L}.

It is easy to see that s̄(δ) ∈ [0, 1], and s̄ is a non-decreasing function of δ.

In a negatively curved space, s(δ, L) is bounded independently of L. Thus, s̄ ≡ 0. In Euclidean space it is not hard to see that s(δ, L) = \sqrt{2Lδ + δ^{2}}/2. In particular s̄(δ) ≡ 1/2. In a space which contains large positively curved regions such as hemispheres, s could be made to grow roughly proportionally to L, for fixed δ. Thus s̄(δ) can be made identically 1 in such a space.

Let X denote the product R × R with the sup metric. Then for any δ, s(δ, L) in this space is at least L/2, as one can see by considering x = (0, 0), y = (L, 0) and z = (L/2, L/2). Thus the growth rate s̄ ≡ 1, as in the case of positive curvature.

Let X, Y be two length spaces with instability functions s_{X}, s_{Y}. If f : X → Y is a map such that |d_{X} − d_{Y} ∘ f| ≤ c, it is easy to show that

s_{X}(δ, L) ≤ 3c + 4s_{Y} (δ + 3c, L + c),  (7.1)

and vice versa if f is invertible. Here is a sketch of the proof: given x, y, z ∈ X with d(x, y) ≤ L and z δ-between x and y, let [xy] be a geodesic and let z′ denote the point in [xy] for which d(x, z′) = d(x, z). Then both f(z) and f(z′) are , at worst, (δ + 3c)-between f(x) and f(y). This bounds the distance from f(z) and f(z′) to their nearest points ζ and ζ′ on a geodesic [f(x)f(y)]. We can also bound d(ζ, ζ′) using the choice of z′, and the desired bound on d(z, z′) follows.

Applying (7.1) to the homeomorphism Π of theorem 6.1, we may conclude that, for T (S), s̄ ≡ 1. This is the sense in which we say that the large scale behavior of T (S) has characteristics of positive curvature.

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Mathematics Department and IMS, SUNY Stony Brook, NY 11794 E-mail address: yair@math.sunysb.edu