QUASI-PROJECTIONS IN TEICHMÜLLER SPACE

YAIR N. MINSKY

October 22, 1996

Abstract. We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls disjoint from the geodesic project to sets of bounded diameter. (This property always holds in sufficiently negatively curved spaces.) It is shown here to hold if and only if the geodesic is precompact, i.e. its image in the moduli space is contained in a compact set. Some applications are given, e.g. to stability properties of certain quasigeodesics in Teichmüller space, and to estimates of translation distance for pseudo-Anosov maps.

Many parallels have been drawn between geometric properties of the Teichmüller space of a Riemann surface, and those of complete, negatively curved spaces (see for example Bers [2], Kerckhoff [8], Masur [12], Wolpert [17]). This paper investigates one such parallel – the contracting properties of certain projections to geodesics.

We will use the Teichmüller metric throughout the paper (the other famous metric on Teichmüller space, the Weil-Petersson metric, is negatively curved although it is not complete – see Wolpert [16]. The Teichmüller metric is complete, but not negatively curved – see Masur [10].)

Every closed subset C of a metric space X determines a “closest-points” projection, defined as a map π_{C} : X → P(C), where P(C) is the set of closed subsets of C. Namely,

π_{C}(x) = {y ∈ C : d(x, y) = d(x, C)}

where d(x, C) = inf_{y∈C} d(x, y). (Provided X is proper, i.e. R-balls are compact, π_{C}(x) is never empty.) Suppose now that X is a simply-connected Riemannian manifold with non-positive sectional curvatures, and that C is a geodesic segment, ray or line. In this case π_{C}(x) always consists of a single point. If the sectional curvatures are bounded above by a negative constant, then ||dπ_{C}|| → 0 as d(x, C) goes to ∞ (where dπ_{C} is the differential of π_{C}). In other words, π is contracting at large distances.

If X = T (S), the Teichmüller space of a surface S of finite type, this contraction property fails in general, but a coarse form of it can be shown to hold for a special class of geodesics. Call a geodesic (segment, ray, or line) L in T (S) precompact if its projection to the moduli space M(S) lies in a compact set. We say L is ϵ-precompact if ϵ > 0 is a lower bound on the extremal length of any non-peripheral simple closed curve in S, for any conformal structure in L (see section 1 for definitions). Our main theorem is the following:

Contraction Theorem.

(1) (Contraction property) Given ϵ there is a constant b depending on ϵ, χ(S) such that for any ϵ-precompact geodesic L and σ ∈ T (S),

diam(π_{L}(N_{d(σ,L)}(σ))) ≤ b.

(2) (Failure of contraction) Conversely, if L is a non-precompact geodesic then the contraction property does not hold for any b.

Here N_{r} denotes an r-neighborhood in the Teichmüller metric, and we have taken the liberty of denoting by π_{L}(A) the union ∪_{a∈A}π_{L}(a).

Note that, in particular, part (1) implies a uniform bound

diam(π_{L}(σ)) ≤ b  (0.1)

for an ϵ-precompact geodesic.

The contraction theorem is a consequence of a characterization of π_{L} in terms of an extremal-length ratio problem. Minimizing the distance of σ ∈ T (S) to a geodesic L is equivalent to solving the problem of finding

inf_{τ∈L} sup_{λ} \frac{E_{τ}(λ)}{E_{σ}(λ)}

where the supremum is over λ in PMF(S), the set of projectivized measured foliations of compact support in S. This is because the supremum of extremal length ratios is just exp 2d(σ, τ), by a theorem of Kerckhoff. In section 2 we examine the relation of this problem to the “dual” problem

sup_{λ} inf_{τ∈L} \frac{E_{τ}(λ)}{E_{σ}(λ)},

in a slightly generalized situation of families of functions satisfying certain conditions. We prove, in Proposition 2.1, that under these conditions the sup inf and inf sup problems have closely related solutions. In Proposition 2.2 we show that the solutions are also robust under rescalings of the functions.

In section 3 we prove part (1) of the Contraction Theorem by applying the ideas of section 2. In particular, the dependence of π_{L}(σ) on σ corresponds to the variation of the inf sup problem under rescalings of the function family.

In section 4 we give some consequences of the contraction property, which are directly analogous to well-known properties of hyperbolic space. Theorem 4.2 is a “stability” property for a class of quasi-geodesics: a quasi-geodesic whose endpoints are connected by an ϵ-precompact geodesic must remain in a bounded neighborhood of the geodesic. Theorem 4.3 bounds the projected images of regions in Teichmüller space which are analogous to horoballs in hyperbolic space. Theorem 4.4 is a lower bound for the translation distance of a pseudo-Anosov automorphism of T (S), in terms of distance from its axis; a similar estimate holds for arbitrary loxodromic isometries in H^{n}.

In section 5 we prove Theorem 5.2, which is just part (2) of the Contraction Theorem. This relies on a characterization of the “thin parts” of Teichmüller space as approximately isometric to products of metric spaces, for which a contraction property cannot hold. Acknowledgements: The author is grateful to the Technion and the University of Michigan, where much of this paper was written.

1. Preliminaries

We begin with a brief summary of results and notation. Let S be a surface of finite genus with finitely many punctures, and let T (S) be the Teichmüller space of conformal structures of finite type on S, where two structures are considered equivalent if there is a conformal isomorphism of one to the other, which is isotopic to the identity on S. By “finite type” we mean that each puncture has a neighborhood conformally equivalent to a punctured disk.

T (S) has a natural topology, and is homeomorphic to a finite-dimensional Euclidean space. The Teichmüller distance between two points σ, τ ∈ T (S) is defined as

d(σ, τ) = \frac{1}{2} log K(σ, τ),

where K(σ, τ) = K(τ, σ) is the smallest possible quasi-conformal dilatation of a homeomorphism from (S, σ) to (S, τ) isotopic to the identity.

Let α be a homotopy class of simple closed curves in S. Given σ ∈ T (S), one may define the extremal length of α in σ, as the inverse of the conformal modulus of the thickest embedded annulus homotopic to α in S:

E_{σ}(α) = inf_{A∼α} \frac{1}{Mod(A)}.

This number is positive provided α is non-peripheral – that is, not deformable into a puncture. We will make heavy use of Kerckhoff’s theorem (see [7]), which states that

K(σ, τ) = sup_{α} \frac{E_{τ}(α)}{E_{σ}(α)}.  (1.1)

Kerckhoff in fact obtains this supremum as a maximum, by using a completion of the space of homotopy classes of non-peripheral simple closed curves known as the space of “measured foliations with compact support”, MF(S) (see [3, 9]). Elements of MF(S) are equivalence classes of foliations of S with saddle singularities (negative index) in the interior, and singularities of index at most 1/2 at the punctures, equipped with transverse measures. The equivalence is via isotopy, and saddle-collapsing (Whitehead) moves. MF(S) carries a natural topology, and is homeomorphic to a Euclidean space of the same dimension as T (S). Multiplication of the measure by positive scalars gives a ray structure to MF(S), and the quotient of MF(S) by this multiplication is the projectivized space PMF(S), which is a sphere.

A homotopy class of simple closed curves, equipped with a real number giving the measure, is represented in MF(S) as a foliation whose non-singular leaves are in the homotopy class and fit together in a cylinder whose height is the measure. These special foliations are dense in MF(S), and Kerckhoff shows that the extremal length function extends to a function E_{σ}(λ) which is continuous in both σ and λ, and scales quadratically:

E_{σ}(aλ) = a^{2}E_{σ}(λ).

Moreover, the supremum in (1.1) is realized by a unique projective class in PMF(S).

Fixing σ ∈ T (S), let QD(S, σ) denote the space of integrable holomorphic quadratic differentials on (S, σ) (see [4]). This space (minus the zero differential) may be identified with MF(S) by associating to a quadratic differential Φ its horizontal foliation, which we call Φ_{h}. Hubbard-Masur [6] showed that this correspondence is a homeomorphism. From [7] we also have the equality

E_{σ}(Φ_{h}) = ||Φ||  (1.2)

where ||Φ|| = ∫_{S} |Φ|. We note also that the vertical foliation Φ_{v}, which is the orthogonal foliation to Φ_{h}, is just (−Φ)_{h}.

Geodesics in T (S) are determined by holomorphic quadratic differentials (and hence measured foliations) as follows: Fix σ ∈ T (S) and Φ ∈ QD(S, σ). The family {σ_{t}}_{t≥0} of conformal structures obtained by contracting along the leaves of Φ_{h} by e^{t} and expanding along Φ_{v} by e^{−t}, forms a geodesic ray in the Teichmüller metric, where t gives arclength. Geodesics exist and are unique between any two points in T (S). We also have

E_{σ_{t}}(Φ_{h}) = e^{−2t}E_{σ}(Φ_{h}),  (1.3)
E_{σ_{t}}(Φ_{v}) = e^{2t}E_{σ}(Φ_{v}).  (1.4)

Our last group of definitions has to do with intersection number. The standard geometric (unoriented) intersection number i(·, ·) between two homotopy classes of simple closed curves may be extended to a continuous function on MF(S)×MF(S), homogenous in both arguments. One may best visualize this in the case when one of the arguments is a simple closed curve and the other is any measured foliation. The intersection number is the smallest possible transverse measure deposited on a homotopic representative of the simple closed curve.

We say a foliation λ is complete if for any other foliation μ, i(λ, μ) = 0 implies that the underlying leaf structures of λ and μ are the same (up to isotopy and Whitehead moves).

Let L denote a geodesic segment, ray or line in T (S). We say that L is ϵprecompact, for ϵ > 0, if for all τ ∈ L and all non-peripheral simple closed curves γ in S, E_{τ}(γ) ≥ ϵ. Equivalently, let M(S) denote the moduli space of S, which is the quotient of T (S) by the induced action of the group of homeomorphisms of S. A geodesic is precompact exactly when its image in M(S) is contained in a compact set.

We will have need of this observation, which appears in [10] and again in [14]:

Lemma 1.1. If Φ_{h} ∈ MF(S) determines an ϵ-precompact Teichmüller ray, then Φ_{h} is complete.

2. Two optimization problems

In this section we are going to present the basic line of argument in a mildly abstracted form. Let F denote a family of positive functions with domain a closed interval I ⊂ R (possibly I = R or I is a half-line). Consider the “dual” pair of optimizations

sup_{f∈F} inf_{t∈I} f(t)  (2.1)

and

inf_{t∈I} sup_{f∈F} f(t).  (2.2)

We will study the relation between solutions of these for families satisfying particular constraints. Assume that the functions in F are continuous, and that F is compact in the compact-open topology. Say that (g, s) is a solution pair for problem (2.1) if g(s) = inf g (or if s = ±∞ and inf g is realized by points in I tending to s) and inf g = sup_{f} inf f. Let Supinf(F) denote the set of solution pairs. Similarly define Infsup(F) to be the set of solution pairs of problem (2.2). Note that under our assumptions both sets are non-empty.

Define m(f) ⊂ I ∪ {±∞} to be the set where f takes on its infimum. From now on we allow f(∞) and f(−∞) to denote the appropriate limit, where defined.

Suppose further that F satisfies the following conditions, for fixed constants ρ, a_{0}, a_{1}, ϵ_{0}, D_{0}, c_{0} > 0.

F1. (Exponential growth) For each f there are constants b_{−}, b_{+} ≥ 0 such that ∀t

a_{0}(b_{−}ρ^{−t} + b_{+}ρ^{t}) ≤ f(t) ≤ a_{1}(b_{−}ρ^{−t} + b_{+}ρ^{t}).

F2. (Definite supinf) sup_{f} inf f ≥ ϵ_{0}.

F3. (Non-interference) For any f, g ∈ F and t ∈ m(f), s ∈ m(g),

|s − t| ≥ D_{0} ⟹ f(t)g(t) ≤ c_{0}.

Remarks. It will be easy to see from the rest of this section that the growth condition (F1) could be loosened in various ways; but this form of it will suffice for us. Condition (F3) may seem mysterious, and in the next section we will relate it to intersection-number inequalities for measured foliations.

Proposition 2.1. Conditions (F1-3) on a family F imply the following restrictions on the solutions to the two optimization problems, where d_{0}, d_{1}, d_{2}, d_{3} > 0 depend only on the constants appearing in (F1-3).

(1) For all f ∈ F, diam m(f) ≤ d_{0}.

(2) If (f, t), (f′, t′) ∈ Supinf(F) then |t − t′| ≤ d_{1}.

(3) If (f, t) ∈ Supinf(F) then sup_{g∈F} g(t) ≤ d_{2}f(t).

(4) If (f, t) ∈ Supinf(F) and (g, s) ∈ Infsup(F) then |s − t| ≤ d_{3}.

In particular, note that (2) and (4) imply that t-coordinates of solutions of the Supinf and Infsup problems lie in an interval of bounded diameter. Part (3) implies that sup inf f(t) gives an upper bound for inf sup f(t).

Proof. Part (1) follows obviously from the growth condition (F1). Note that we adopt the convention that diam{∞} = diam{−∞} = 0.

Let M = sup_{f∈F} inf_{t∈I} f(t) ≥ ϵ_{0}. Continuing to part (2), let (f, t) and (f′, t′) be in Supinf(F), so that f(t) = f′(t′) = M, and t ∈ m(f) and t′ ∈ m(f′). Note we may assume t, t′ ≠ ±∞ since an infimum occurring at ±∞ is always 0, again by (F1). Furthermore, (F1) implies that, for a fixed c > 0,

f′(t) ≥ cρ^{|t−t′|}f′(t′).

Now if |t − t′| ≥ D_{0}, condition (F3) implies that

f(t)f′(t) ≤ c_{0}

and we conclude that

cM^{2}ρ^{|t−t′|} ≤ c_{0}.

This gives a uniform upper bound on |t − t′|.

For part (3), again take (f, t) ∈ Supinf(F) and let g ∈ F be any other function. We must bound g(t) in terms of f(t) = M. Choose s ∈ m(g). If |s − t| ≥ D_{0} (including the possibility that s = ±∞), we have again f(t)g(t) ≤ c_{0}, which bounds g(t) by c_{0}/f(t). In particular since f(t) ≥ ϵ_{0} this implies g(t) ≤ (c_{0}/ϵ_{0}^{2})f(t). On the other hand if |s − t| ≤ D_{0} then by (F1) g(t) ≤ c′ρ^{D_{0}}g(s) ≤ c′ρ^{D_{0}}f(t), and again we have the desired bound.

For part (4), let (f, t) ∈ Supinf(F) and (g, s) ∈ Infsup(F). By definition of Infsup(F), we have

g(s) ≤ sup_{h∈F} h(t)

and by part (3)

sup_{h∈F} h(t) ≤ d_{2}f(t).

On the other hand (again by definition, and by (F1))

g(s) ≥ f(s) ≥ cρ^{|s−t|}f(t).

We conclude that cρ^{|s−t|} ≤ d_{2}, and obtain the desired bound on |s − t|.

Now we consider the way in which solutions of problems (2.1,2.2) vary when the family of functions is varied by rescaling. Let u : F → R_{+} be any positive function and let uF denote the family of functions {u(f)f : f ∈ F}. The effect of this change on the Infsup problem is not readily apparent, but the Supinf problem is less affected since the position of the minima of u(f)f and f are the same.

Proposition 2.2. If F satisfies (F1-3) and u : F → R_{+} is a rescaling function, such that

\frac{sup u}{inf u} ≤ Q(sup_{f} inf_{t} f(t))^{2},

then if (f, t) ∈ Supinf(F) and (u(g)g, s) ∈ Supinf(uF), we have

|s − t| ≤ Δ,

where Δ depends only on Q and the constants in (F1-3).

Proof. If |s − t| < D_{0} we are done, so suppose |s − t| ≥ D_{0}. In this case (F3) implies f(t)g(t) ≤ c_{0}. The growth condition (F1) implies g(t) ≥ cρ^{|s−t|}g(s). Finally, by definition,

u(g)g(s) = sup_{h∈F} inf_{t∈I} u(h)h(t)

≥ ( inf_{h∈F} u(h)) sup_{h∈F} inf_{t∈I} h(t)

= ( inf_{h∈F} u(h)) f(t).

Putting these together we have

cρ^{|s−t|} \frac{inf u}{u(g)}f(t)^{2} ≤ c_{0}

and by the assumption on sup u/ inf u we have

|s − t| ≤ log_{ρ} \frac{c_{0}Q}{c}

which is the desired bound.

3. Proof of the contraction property

In this section we will prove part (1) of the Contraction Theorem, using the results of section 2. The family of functions for which we consider the Supinf and Infsup problems will be ratios of extremal lengths, obtained as follows.

Let L be an ϵ-precompact geodesic segment, ray, or line and let \overline{L} denote the complete geodesic line containing L. Choose an arbitrary origin and orientation on \overline{L} and let the parameter t denote signed distance from the origin. Then L corresponds to an interval I = [x, y], where −∞ ≤ x < y ≤ ∞. Denote by \overline{L}(t) the conformal structure on S given by the point parametrized by t; or by L(t) if t ∈ [x, y].

For each α ∈ MF(S) and t ∈ R, let E_{t}(α) be short for the extremal length E_{\overline{L}(t)}(α). Now fixing a point σ ∈ T (S), we obtain the ratio

R_{σ,α}(t) = \frac{E_{t}(α)}{E_{σ}(α)}.

Note that this quantity is invariant under scaling of α, so it only depends on the class [α] ∈ PMF(S). Thus the family of functions

R_{σ} = {R_{σ,α}}_{[α]∈PMF(S)}

is compact.

The problem of computing π_{L}(σ) now reduces to solving Infsup(R), by virtue of Kerckhoff’s theorem (1.1). Let us therefore show that R_{σ} satisfies conditions (F1-3). Condition F1. Let Φ be the holomorphic quadratic differential on \overline{L}(0) that determines the geodesic. The extremal lengths of the horizontal and vertical foliations of Φ vary exponentially with t according to (1.3) and (1.4). We also have E_{0}(Φ_{h}) = E_{0}(Φ_{v}) = ||Φ|| by (1.2), and we may normalize so that ||Φ|| = 1.

We will need to know (see e.g. Gardiner-Masur [5]) that Φ_{h} and Φ_{v} fill up S, in the sense that every non-zero element of MF(S) has positive intersection with at least one of them. Thus we may define the strictly positive function

e_{t}(α) = \frac{1}{2} (\frac{i(α, Φ_{h})^{2}}{E_{t}(Φ_{h})} + \frac{i(α, Φ_{v})^{2}}{E_{t}(Φ_{v})} )  (3.1)

which, by virtue of (1.3,1.4) can be written

e_{t}(α) = \frac{i(α, Φ_{h})^{2}}{2} e^{2t} + \frac{i(α, Φ_{v})^{2}}{2} e^{−2t}.  (3.2)

This will serve as an approximation for E_{t}(α), by the following:

Lemma 3.1. There is a fixed c_{0} = c_{0}(ϵ, χ(S)) such that if L is ϵ-precompact then for any α ∈ MF(S) and t ∈ [x, y],

e_{t}(α) ≤ E_{t}(α) ≤ c_{0}e_{t}(α).

Proof. The left side actually does not depend on the precompactness of L – it follows directly from lemma 3.2 (stated shortly), applied to α and each of Φ_{h} and Φ_{v}.

The right side follows from a compactness argument. Let Φ^{t} denote the image of Φ in L(t) under the Teichmuller map from \overline{L}(0), scaled so that ||Φ^{t}|| = 1. Then as measured foliations we have Φ_{h}^{t} = ρ^{t}Φ_{h} and Φ_{v}^{t} = ρ^{−t}Φ_{v}. Thus, by (3.2),

e_{t}(α) = \frac{1}{2} (i(α, Φ_{h}^{t})^{2} + i(α, Φ_{v}^{t})^{2}) .

The ratio E_{t}(α)/e_{t}(α) is therefore a continuous positive function of the projectivized measured foliation [α] ∈ PMF(S) and the pair (L(t), Φ^{t}) in the total space of Riemann surfaces equipped with holomorphic quadratic differentials. Fixing the Riemann surface L(t), the space PMF(S) × P QD(L(t)) is compact; and the function is invariant under the action of the mapping class group. Further, the image of L(t) in the moduli space is restricted to a compact set by our definition of precompactness. It follows that E_{t}(α)/e_{t}(α) is bounded.

This lemma and equation (3.2) for e_{t}(α) establish condition (F1). Condition (F2). To get a lower bound on sup_{R∈R} inf R, recall first that there is a constant ℓ_{0} depending on χ(S) such that the shortest non-peripheral simple closed curve α on S has E_{σ}(α) ≤ ℓ_{0}. Since L is ϵ-precompact, R_{σ,α}(t) ≥ ϵ/ℓ_{0} for all t ∈ [x, y]. This gives a uniform lower bound for inf R_{σ,α}, and thus on the sup over all functions in R_{σ}. Condition (F3). This condition is really the central geometric idea of the paper. We must show that if the minima of two functions R_{σ,α} and R_{σ,β} are sufficiently distant, then the product of the two is bounded above at either minimum.

Note first that the minima m(R_{σ,α}) lie in a uniformly bounded neighborhood of the minimum s_{α} of e_{t}(α). This is a direct consequence of lemma 3.1.

The minimum point s_{α} ∈ [x, y] can be calculated directly from (3.2); in particular it takes on the value +∞ (resp. −∞) exactly when y = ∞ and i(α, Φ_{h}) = 0 (resp. x = −∞ and i(α, Φ_{v}) = 0).

We will control the product of length ratios by the intermediary of intersection numbers. The following basic lemma appears for example in [15]:

Lemma 3.2. If α, β are any two measured foliations on any Riemann surface X, then

E_{X}(α)E_{X}(β) ≥ i(α, β)^{2}.

The following lemma obtains bounds in the opposite direction. Together they will allow us to bound the appropriate products.

Lemma 3.3. There are constants D_{1} and c_{1}, depending only on ϵ and χ(S), such that for any α, β ∈ MF(S),

|s_{α} − s_{β}| ≥ D_{1} ⟹ i(α, β)^{2} ≥ c_{1}E_{s_{α}}(α)E_{s_{α}}(β).

Remark. This lemma is similar to (and was motivated by) lemma 3.2 from [15], in which the role of L is played by a hyperbolic 3-manifold N homeomorphic to S ×R, and the role of the minimum point s_{α} is played by the geodesic representative of α in N.

Proof. Denote by I_{t}(α, β) the normalized intersection number i(α, β)^{2}/E_{t}(α)E_{t}(β). Suppose the lemma is false; then there exists a sequence of ϵ-precompact geodesics L_{i} parametrized by [x_{i}, y_{i}], with foliations α_{i}, β_{i} such that |s_{α_{i}} − s_{β_{i}}| → ∞ while I_{s_{α}}(α, β) → 0.

Adjusting if necessary by automorphisms of S, we may assume that the points L_{i}(s_{α_{i}}) lie in a fixed compact subset of a fundamental domain of the action of the mapping class group on T (S). Further, the quantity I_{t} is invariant under scaling of the measures, and so is defined on the compact space PMF(S) × PMF(S). Thus we may take a convergent subsequence and obtain a limit example (L, α, β) in which I_{s_{α}}(α, β) = 0, and |s_{β} − s_{α}| = ∞. Without loss of generality we assume s_{β} = +∞, so that L must contain an ϵ-precompact ray [s_{α}, ∞).

However, by lemma 1.1, this implies that the foliation Φ_{h} is complete. Since s_{β} = +∞, we have i(β, Φ_{h}) = 0, so that β must be topologically equivalent to Φ_{h}. On the other hand, the fact that s_{α} ≠ +∞ implies that i(α, Φ_{h}) > 0, and therefore that i(α, β) > 0, a contradiction.

Combining lemma 3.3 with lemma 3.2 applied to X = (S, σ), we immediately obtain

|s_{α} − s_{β}| ≥ D_{1} ⟹ R_{σ,α}(s_{α})R_{σ,β}(s_{α}) ≤ 1/c_{1}.  (3.3)

Condition (F3) is an immediate consequence. Completion of the proof. The projection π_{L}(σ) consists exactly of those points L(t) where, for some α, (R_{σ,α}, t) ∈ Infsup(R_{σ}), and furthermore

inf_{t} sup_{α} R_{σ,α}(t) = e^{2d(σ,L)}

By Kerckhoff’s theorem.

Having established conditions (F1-3) for the family R_{σ}, Proposition 2.1 applies to show the following: By part (2), the set of t-values appearing in Supinf(R_{σ}) has a bounded diameter. By part (4), the t-values appearing in Infsup(R_{σ}) are a bounded distance from those appearing in Supinf(R_{σ}). Together these imply a uniform bound on diam(π_{L}(σ)). Furthermore, by part (3), if (R_{σ,β}, s) ∈ Supinf(R_{σ}) then

d_{2}R_{σ,β}(s) ≥ sup_{α} R_{σ,α}(s)

≥ inf_{t} sup_{α} R_{σ,α}(t)

= e^{2d(σ,L)}.  (3.4)

Now consider a second point τ ∈ T (S), with d(σ, τ) ≤ d(σ, L). The family R_{τ} is a rescaling of R_{σ} by the function u(α) = E_{σ}(α)/E_{τ}(α). It follows from Kerckhoff’s theorem that

sup_{α} u(α) = e^{2d(σ,τ)} = \frac{1}{inf_{α} u(α)}

(where the second equality follows from the symmetry of d(σ, τ).) Thus we have

\frac{sup u}{inf u} ≤ e^{4d(σ,τ)} ≤ e^{4d(σ,L)}.

Now applying (3.4), we obtain

\frac{sup u}{inf u} ≤ d_{2}^{2} (sup_{α} inf_{t} R_{σ,α}(t))^{2} .

Therefore we may apply proposition 2.2 to compare Supinf(R_{τ}) with Supinf(R_{σ}): the t values occurring in each differ by at most Δ, which depends only on the previous constants.

The conclusion, since solutions of Supinf and Infsup are within bounded distance, is a uniform bound on diam(π_{L}(σ) ∪ π_{L}(τ)). Applying this to all τ ∈ N_{d(σ,L)}(σ), we obtain the desired bound on diam(π_{L}(N_{d(σ,L)}(σ))). This concludes the proof of the contraction property.

4. Applications

We shall need the following immediate consequence of the contraction theorem, which states that paths a definite distance from a precompact geodesic have projections which are shorter by a definite factor.

Corollary 4.1. Let L be an ϵ-precompact geodesic in T (S) and let b be the constant given in the Contraction Theorem. If R > 0 and points x, y ∈ T (S) are connected by a path of length T which remains outside an R-neighborhood of L, then

diam(π_{L}(x) ∪ π_{L}(y)) ≤ \frac{b}{R}T + b.  (4.1)

Furthermore for any x, y ∈ T (S),

diam(π_{L}(x) ∪ π_{L}(y)) ≤ d(x, y) + 4b.  (4.2)

Proof. Dividing the path from x to y into at most 1 + T/R pieces of size R or less, and applying the Contraction Theorem to each one, yields inequality (4.1). To obtain (4.2), let R = b and consider the geodesic G connecting x to y. If G stays outside of an R-neighborhood of L then apply (4.1). If not, let x′, y′ be the closest points in G to x, y (respectively) such that d(x′, L) = d(y′, L) = R. The triangle inequality together with (4.1) yields

diam(π_{L}(x) ∪ π_{L}(y)) ≤ d(x, x′) + 2R + d(x′, y′) + d(y, y′) + 2b

which suffices.

Define a (K, δ)-quasi-geodesic in T (S) as a path Γ : [a, b] → T (S), parametrized by arclength, such that for any s, t ∈ [a, b],

|s − t| ≤ Kd(Γ(s), Γ(t)) + δ.

Theorem 4.2. Let Γ be a (K, δ)-quasi-geodesic path in T (S), whose endpoints are connected by an ϵ-precompact Teichmüller geodesic L. Then Γ remains in a B(K, δ, ϵ)-neighborhood of L.

Proof. The idea of the argument is standard: any segment of Γ which is outside a sufficiently large neighborhood of L is, because of the contractive properties of π_{L}, much less efficient than the path obtained by going back to L and moving along the projection. Thus large excursions from L violate the quasi-geodesic property.

Fix R = 2Kb. Let [s, t] be a maximal interval for which Γ((s, t)) is outside an R-neighborhood of L. Applying corollary 4.1, we have

diam(π_{L}(Γ(s)) ∪ π_{L}(Γ(t))) ≤ \frac{b}{R}|s − t| + b.

The maximality of [s, t] implies that d(Γ(s), π_{L}(Γ(s))) = R, and similarly for t. Thus

d(Γ(s), Γ(t)) ≤ 2R + \frac{b}{R}|s − t| + b

On the other hand, the quasi-geodesic property of Γ gives a lower bound on the distance d(Γ(s), Γ(t)), and together we obtain (using the definition of R), a bound |s − t| ≤ 4KR + 2Kb + 2δ. It follows that Γ cannot exit a (2K + 1)R + Kb + δ neighborhood of L.

Let α be a measured foliation in MF(S). The set

T hin(α, δ) = {σ ∈ T (S) : E_{σ}(α) ≤ δ}

is somewhat analogous to a horoball in hyperbolic space. As in hyperbolic space, it has infinite diameter, but its projection to a precompact geodesic is nevertheless bounded, bar one exceptional case:

Theorem 4.3. Given ϵ, δ there is a constant B such that, if L is a complete ϵprecompact geodesic and α ∈ MF(S), then

diam(π_{L}(T hin(α, δ))) ≤ B + diam(L ∩ T hin(α, δ)).

Remarks: (1) A special case of this is when α is a non-peripheral simple closed curve, with weight 1. Then L∩T hin(α, δ) is uniformly bounded for all ϵ-precompact L, so we have a uniform bound on diam(π_{L}(T hin(α, δ))). (2) If a complete geodesic L is determined by a quadratic differential Φ, let us call [Φ_{h}] and [Φ_{v}] in PMF(S) the endpoints at infinity for L. Another special case occurs when [α] is equal in PMF(S) to an endpoint, and then both sides of the inequality are infinite.

Proof. In view of remark (2), let us assume that [α] ≠ [Φ_{h}], [Φ_{v}] in PMF(S). Masur proved in [11] that, if L is ϵ-precompact, then its endpoints at infinity are uniquely ergodic – they support a unique projective class of transverse measures. Thus the assumption that [α] is not an endpoint implies that α is topologically distinct from both foliations, and therefore (by lemma 1.1) it intersects both of them non-trivially. It follows that E_{t}(α) has a positive infimum, E_{0}, obtained at some t_{α} ≠ ±∞.

Let ϵ_{0}, c_{0}, D_{0} be the constants for which conditions (F2,F3) hold for the families R_{σ}, as in section 3. Fix δ_{0} = \frac{1}{2}E_{0}ϵ_{0}/c_{0}. We will show that π_{L}(T hin(α, δ_{0})) lies in a bounded neighborhood of L(t_{α}).

Given σ ∈ T hin(α, δ_{0}), let (λ, s) be a solution pair in Supinf(R_{σ}). By Proposition 2.1, it will suffice to bound |s − t_{α}|.

Suppose |s − t_{α}| ≥ D_{0}. Then by condition (F3) we have

R_{σ,α}(t_{α})R_{σ,λ}(t_{α}) ≤ c_{0}.

On the other hand, R_{σ,α}(t_{α}) ≥ E_{0}/δ_{0} by assumption, and R_{σ,λ}(t_{α}) ≥ R_{σ,λ}(s) ≥ ϵ_{0} by condition (F2). Thus we obtain E_{0}ϵ_{0}/δ_{0} ≤ c_{0}, which contradicts the choice of δ_{0}. It follows that |s − t_{α}| < D_{0}.

To prove the theorem for general δ > δ_{0}, we will first bound diam(π_{L}(T hin(α, δ))) above by a constant plus log δ/δ_{0}, and then bound diam(L∩T hin(α, δ)) from below by a (different) constant plus log δ/δ_{0}.

We first observe that T hin(α, δ) lies in a neighborhood of T hin(α, δ_{0}) of radius \frac{1}{2} log δ/δ_{0} (since one can move along a Teichmüller geodesic defined by Φ such that Φ_{h} = α, and apply (1.3)).

Applying corollary 4.1 (inequality 4.2) we conclude immediately that the projection π_{L}(T hin(α, δ)) lies in a (B + \frac{1}{2} log δ/δ_{0})-neighborhood of π_{L}(T hin(α, δ_{0})). Applying the estimate of the previous paragraph this is in a (C + \frac{1}{2} log δ/δ_{0})- neighborhood of L(t_{α}), for an appropriate C(ϵ), so that we have the upper bound.

On the other hand, applying the growth condition, there is a constant c_{2} for which E_{t}(α) ≤ c_{2}E_{0} exp 2|t − t_{α}|. Thus, if

|t − t_{α}| ≤ \frac{1}{2} log \frac{ϵ_{0}}{2c_{0}c_{2}} \frac{δ}{δ_{0}}

we can conclude (using the definition of δ_{0}) that E_{t}(α) ≤ δ. This bounds diam(L ∩ T hin(α, δ)) from below by a constant plus log δ/δ_{0}. The statement of the theorem follows.

Our last application is a geometric property of pseudo-Anosov homeomorphisms. Let f : S → S be pseudo-Anosov, and let f_{∗} denote the induced action on T (S) (see [3] or [1]). Bers showed in [2] that the infimum inf_{σ∈T (S)} d(σ, f_{∗}σ) of translation distance is achieved in T (S) on a geodesic, namely the axis of f_{∗}, on which f_{∗} acts by translation. The contraction theorem can be used to give lower bounds on how fast the translation distance grows as we consider points far away from the axis (this question was suggested by Feng Luo).

Theorem 4.4. Let f : S → S be a pseudo-Anosov homeomorphism, and L ⊂ T (S) the axis of f_{∗}. There exists k_{0} > 0 such that, for all x ∈ T (S),

d(x, f_{∗}x) ≥ k_{0}d(x, L).

Remark: it is easy to see that d(x, f_{∗}x) is at most linear with d(x, L), and so in an asymptotic sense the theorem gives maximal growth. However the constant k_{0} is completely non-constructive.

Proof. Since the axis L is invariant by f_{∗}, its image in the moduli space is a closed curve, so that L is precompact. Let b be the constant given by the Contraction Theorem for L.

For x ∈ T (S), let R = d(x, L) and let t = d(x, f_{∗}(x)). Let t_{0} denote the translation distance of f_{∗} on L, so that t ≥ t_{0}, and t = t_{0} only for x ∈ L.

For any n > 0 we can connect x to f_{∗}^{n}x by a chain of n geodesics of length t, which remains outside an R − t/2 neighborhood of L. By corollary 4.1 we have

diam(π_{L}(x) ∪ π_{L}(f_{∗}^{n}x)) ≤ \frac{b}{R − t/2}nt + b.

(Note that we may assume that R − t/2 > 0, because otherwise we would have t ≥ 2R, and we would be done.) It follows that

d(x, f_{∗}^{n}x) ≤ 2R + \frac{b}{R − t/2}nt + b.

On the other hand L is also the axis of f_{∗}^{n}, so that d(x, f_{∗}^{n}x) ≥ nt_{0}. Letting n go to infinity, we conclude that

t_{0} ≤ b\frac{t}{R − t/2}.

Solving for t, we obtain the desired inequality.

5. The case of non-precompact geodesics

In this section we prove part (2) of the Contraction Theorem. Let L be a geodesic which is not precompact. Then L has segments which enter, arbitrarily deeply, into the “thin parts” T hin(α, δ) defined in the previous section, where α is a simple closed curve with weight 1.

For small δ, the geometry of T hin(α, δ) is quite far from negatively curved. In [13] we showed that it is approximated by a product space in the following sense.

Let X_{α} denote H^{2}×T (S\α), endowed with the sup metric d_{X} = max(d_{H^{2}}, d_{T (S\α)}) of the metrics on the factors. There is a natural homeomorphism Π : T (S) → X_{α}, defined using Fenchel-Nielsen coordinates, so that the coordinates in the hyperbolic plane H^{2} encode the length and twist parameters associated to α. In particular, the y coordinate in H^{2} (in the upper half-plane model) for Π(σ) is 1/ℓ_{σ}(α) where ℓ_{σ} is hyperbolic length, and it follows easily that Π(T hin(α, δ)) is contained in {y > h_{1}}×T (S\α) and contains {y > h_{2}}×T (S\α), where h_{1}, h_{2} are approximately inversely proportional to δ. The main theorem of [13] says that

Theorem 5.1. (Theorem 6.1 of [13]) For sufficiently small δ, the map Π restricted to T hin(α, δ) has bounded additive distortion; that is,

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

for σ, τ ∈ T hin(α, δ), where c depends only on δ and the topological type of S.

This product geometry suffices to give part (2) of the Contraction Theorem, namely the failure of part (1) in the non-precompact case:

Theorem 5.2. If a Teichmüller geodesic L is not precompact, then there is no choice of constant b for which the contraction property holds for π_{L}.

Remark: This theorem does not rule out the diameter bound (0.1) for images of single points which is a consequence of the contraction property. In fact it is not ruled out that π_{L}(σ) is in all cases a single point; resolving this question seems to require finer techniques than we use here.

Proof. Using the product structure we can show that large segments of L can be replaced by quasi-geodesics that stray arbitrarily far from L. This will contradict theorem 4.2.

Since L is not precompact, for arbitrarily small δ > 0 and large T > 0 there exists a segment in L of length T contained in T hin(α, δ) for some simple closed curve α. (We are using here the fact that the Teichmüller metric on the moduli space is complete, or equivalently that the distance from the boundary of T hin(α, δ) to T hin(α, δ′) goes to infinity as δ/δ′ → ∞ – in fact it is \frac{1}{2} log(δ/δ′).)

Let σ_{1}, σ_{2} be the endpoints of such a segment, and let Π(σ_{i}) = q_{i} be their images in the product X_{α}, so that |d(q_{1}, q_{2}) − T | < c. Using the sup metric on X_{α} it is easy to construct two (2, 0)-quasi-geodesics m, m′ connecting q_{1} to q_{2}, so that neither quasi-geodesic lies in a (T − c)-neighborhood of the other. (Restricting to an R^{2} slice, construct an appropriate quadrilateral whose diagonal is the line [q_{1}, q_{2}], and let m, m′ be the two paths around the perimeter.) Furthermore m, m′ can be made to remain in Π(T hin(α, δ′)) for δ′ slightly bigger than δ. Then applying Π^{−1}, we obtain two (2, c)-quasi-geodesics connecting σ_{1} to σ_{2}, neither of which lies in a (T −2c)-neighborhood of the other. It follows that at least one of the quasi-geodesics cannot be in a bounded neighborhood of L, as T is chosen arbitrarily large. This violates the stability of quasi-geodesics proved in Theorem 4.2, and therefore π_{L} cannot have the contraction property.

Remark: For a product space with the sup metric the stability of quasi-geodesics is actually violated in a stronger way than used above. See the last section of [13] for a brief discussion of this.

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