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THE L1 EHRENPREIS CONJECTURE

T.M. GENDRON

ABSTRACT. Let Σ be the algebraic universal cover of a closed surface of genusg > 1,T (Σ) its Teichmuller space,M(Σ,ℓ) the group of mapping classes stabilizing a fixed leafℓ. TheL1 Ehrenpreis conjecture asserts thatM(Σ,ℓ) acts onT (Σ) with dense orbits in theL1 topology (the topology coming from theL1 norm on quadratic differentials). We give aproof of this weaker version of the Ehrenpreis conjecture, announced first in [6].

1. INTRODUCTION

Let Σ,Σ′ be closed Riemann surfaces of genus greater than 1. The most succinct for-mulation of the Ehrenpreis conjecture (EC) uses the fact thatΣ,Σ′ may be regarded asriemannian manifolds with metrics of curvature−1. While it is an elementary fact that theriemannian universal coversΣ, Σ′ are isometric toH2, theEC asserts a similar, asymptoticphenomenon for the family of finite riemannian covers:

Ehrenpreis Conjecture (Hyperbolic). For any ε > 0, there are finite degree isometriccovers Z→ Σ, Z′ → Σ′ whose total spaces are(1+ ε)-quasiisometric.

In [6], we announced the solution of anL1-version of this conjecture. In this paper, weprovide the proof.

The traditional or conformal version of theEC [2] can be described in terms of Te-ichmuller theory. LetΣ be a fixed compact surface of genus greater than 1,T (Σ) itsTeichmuller space,dT (Σ) the Teichmuller metric. Then givenµ ,ν ∈ T (Σ), the conformalversion of theEC states

Ehrenpreis Conjecture (Conformal). For anyε > 0, there exists a surface Z and finitecoversρ ,σ : Z → Σ such that

dT (Z)(ρ∗µ ,σ∗ν) < ε.In the above,ρ∗,σ∗ : T (Σ) → T (Z) are the isometric inclusions induced byρ ,σ . We

remark that this version of theEC makes sense in genus 1, where it is not difficult to verify[6].

However, it is the genus independent or solenoidal version of the EC that will be mostimportant for us. LetΣ be the algebraic universal cover of the closed surfaceΣ, by def-inition the inverse limit of the total spaces of finite coversρ : Z → Σ (one cover for eachhomotopy class of cover). The algebraic universal cover is asurface solenoid (a surfacelamination with Cantor transversals), and as such has a Teichmuller spaceT (Σ) of markedconformal structures [18]. The mapping class groupM(Σ, ℓ) of homotopy classes of home-omorphisms ofΣ fixing a base leafℓ may be identified with the group of homotopy classesof lifts of correspondencesσ ◦ρ−1 : Σ → Σ.

Date: 12 June 2005.2000Mathematics Subject Classification.Primary, 32G15, 57R30.Key words and phrases.Ehrenpreis conjecture, solenoids, moduli space.

1

2 T.M. GENDRON

Ehrenpreis Conjecture (Solenoidal). M(Σ, ℓ) acts onT (Σ) with dense orbits.

The proof that all of these versions of theEC are equivalent can be found in [6].The genus independent version ofEC says that the ”universal moduli space”

M (Σ) = T (Σ)/M(Σ, ℓ),

although an uncountable set, has the topology of a point (thediscrete topology). It hasthe virtue of giving a certain explanation of the “moduli-rigidity gap” that separates thetheory of compact hyperbolic surfaces from that of compact hyperbolic manifolds in di-mension three or greater. From a practical point of view, working with Σ allows us toregard Teichmuller theory of closed hyperbolic surfaces as concerning complex structureson a single topological type (as in the genus 1 case). In this way, we may isolate geometricproperties of closed Riemann surfaces of hyperbolic type that do not depend on genera.

In this paper, we shall formulate and prove anL1 version of theEC. On T (Σ) thereare – in addition to the Teichmuller metric – three other metrics coming from theL1, theL∞ and theL2 structures on the cotangent bundle ofT (Σ). TheL1 version of theEC isobtained by asking thatM(Σ, ℓ) act densely onT (Σ) with regard to theL1 geometry.

The proof of theL1 EC is as follows. A dense set of pairsµ, ν ∈ T (Σ) (dense in theTeichmuller geometry) lie along the axisA of a pseudo-Anosov homeomorphismΦ : Σ →Σ. A is a Teichmuller geodesic and the action ofΦ onT (Σ) stabilizesA, translating pointsalongA a distance of12 logλ , whereλ is the entropy ofΦ. Givenn∈ N, by anL1 nth rootof Φ we mean a sequence of pseudo Anosov homeomorphisms{Ψm} in which

• The entropiesλm of Ψm converge toλ 1/n.• The axesAm of Ψm converge toA in theL1 Hausdorff topology.

We shall show thatL1 nth roots exist for every (lifted) pseudo Anosov homeomorphism ofΣ. This will then imply theL1 version ofEC.

Acknowledgements: This work benefited greatly from conversations with Dennis Sullivanand Yair Minksy.

2. TOPOLOGY OF THEALGEBRAIC UNIVERSAL COVER

Let Σ be a fixed compact surface of genus at least two. We describe inthis sectionthe topology of the algebraic universal coverΣ. Unless otherwise noted, all proofs ofstatements in this section can be found in [6].

Let π = π1Σ. For every finite index normal subgroupH < π , choose a pointed coverρ : (Z,xZ)→ (Σ,x) for whichρ∗π1Z = H. By adding to this collection of covers all coversτ : Z → Z′ between total spaces for whichρ ′ ◦ τ = ρ , we obtain an inverse system ofsurfaces. The limit of this systemΣ is called thealgebraic universal coverof Σ, a compacttopological space. Its topological type is independent of the choice of covers.

If we denote byπ the profinite completion ofπ , thenΣ is homeomorphic to the quotient

(1)(

Σ× π)/

π ,

whereπ acts diagonally, and so has the structure of a surface solenoid: a surface laminationwhose model transversals are Cantor sets. It also follows from (1) thatΣ is connected, itspath components are its leaves, and each leaf is homeomorphic toR2 and dense inΣ. Thepoint x = (xZ) ∈ Σ – defined by the string of basepoints of the surfaces in the definingsystem – is contained in a leafℓ which we call the base leaf. The Haar measure onπ

THE L1 EHRENPREIS CONJECTURE 3

induces a transverse invariant measureη on Σ which gives measure 1/degZ to the fibersof the natural projectionΣ → Z.

Any pointed finite coverσ : (Y,y) → (Σ,x) lifts to a base leaf preserving homeomor-phism σ : Y → Σ, whereY is the algebraic universal cover ofY. If ρ : (Y,y) → (Σ,x)is another such cover, the correspondenceσ ◦ ρ−1 lifts to the homeomorphismσ ◦ ρ−1

of Σ preservingℓ. Let M(Σ, ℓ) denote the the group of homotopy classes of orientationpreserving homeomorphisms ofΣ preservingℓ. See [7], [13] for a proof of the following

Theorem 1. Every class[h] ∈M(Σ, ℓ) contains an element of the formσ ◦ ρ−1.

3. MEASUREDLAMINATIONS ON Σ

We begin by recalling a few facts about measured laminationson Σ, a closed Riemannsurface of genusg> 1. See [1], [3], [8], [20] for further discussion. Ameasured laminationf on Σ is a closed 1-dimensional lamination smoothly embedded inΣ and possessing atransverse invariant measuremf. Two measured laminations areequivalentif they areisotopic through an isotopy taking one measure to the other.The set of equivalence classesof measured laminations is denotedML (Σ). Let C (Σ) denote the set of isotopy classesof simple closed curves inΣ. Givenf ∈ ML (Σ) andc∈ C (Σ), theintersection pairingisdefined

I(f,c) = inf∫

f∩cdmf,

where the infimum is taken over representatives of the classes of f andc. The intersectiontopology onML (Σ) is the weak topology with respect to the intersection pairing. Thespace of projective classes of measured laminations is denotedPL (Σ) and is homeomor-phic to a sphere of dimension 6g−7. We haveC (Σ) ⊂ PL (Σ) with dense image. Theintersection pairing extends to a mapML (Σ)×ML (Σ)→ R via the formula [10]

I(f,g) = inf∫

f∩gdmf⊗dmg.

A word is in order here regarding the allied concept of ameasured foliation, a singu-lar foliation F of Σ equipped with a transverse invariant measure: these typically ariseas trajectories of holomorphic quadratic differentials onΣ [17]. Two measured foliationsare equivalent if after a finite number of Whitehead moves areapplied to their singularleaves, they are isotopic through an isotopy taking one measure to the other. There is abijective correpondence between classes of measured foliations and classes of measuredlaminations [8]. For example, to obtain a measured lamination starting with a measuredfoliation F , one chooses a nonsingular leaf from each minimal componentof F , pullseach such leaf geodesic (with respect to the hyperbolic metric of Σ) and completes theresulting space. Our default will be to work with measured laminations, and – with theexception of the proof of Theorem 4, where we revert back to measured foliations – when-ever a measured foliation happens to arise, we will assume ithas been converted into itsassociated measured lamination.

A (homotopy class of) homeomorphismΦ : Σ→Σ induces a homeomorphism ofML (Σ)via pullback of measures, in particular inducing a homeomorphism ofPL (Σ). Accordingto the classification of surface diffeomorphisms, [1], [3],[19], Φ is calledpseudo Anosovif its induced action onPL (Σ) fixes precisely two classes[fu] and[fs]. If λ is the entropyof Φ, thenλ > 1; and if fu ∈ [fu] (fs ∈ [fs]) is a representative inML (Σ), then there isa representative diffeomorphism in the class ofΦ (also denotedΦ) such thatΦ(fu) = λ fu(Φ(fs) = λ−1fs).

4 T.M. GENDRON

In this paper, we will be interested in the following class ofpseudo Anosov homeo-morphisms. LetC , D be families of pairwise nonisotopic simple closed curves, for whichelements ofC intersect minimally in their isotopy classes with elementsof D , and forwhich Σ \ (C ∪D) consists of a union of disks (the families are then said to befilling).For c ∈ C , d ∈ D , let Fc resp.Gd denote the right Dehn twist aboutc resp.d. Then ahomeomorphism of the form

Φ = G−Nkdk

◦ · · · ◦G−N1d1

◦FM jcj ◦ · · · ◦FM1

c1,

where the exponentsM1, . . . ,M j ,N1, . . . ,Nk are positive and where all curves inC , D occur,is pseudo Anosov, [15], [20]. We call these pseudo Anosovs ofThurston-Penner type.

We now extend the above considerations toΣ. A measured laminationf on Σ is acollection of measured laminations{fℓ}, one on each leafℓ of Σ, which have the sametransversal modelT and which vary in the following way with respect to the transversalsof Σ. Let O ≈ D× T be a flowbox forΣ, such thatft := f|D×{t} is a flowbox forfℓ ifD×{t}⊂ ℓ. Then

(1) The family of flowboxesft varies continuously int. (Thusf gives rise to a smooth1-dimensional sublamination ofΣ with transversal modelsT ×T.)

(2) Given any continuous family of test transversalsTt ⊂ D×{t}, the function ob-tained by pairing with the measures off is continuous int.

The space of equivalence classes of measured laminations isdenotedML (Σ). If c is asimple closed curve occurring in some surfaceZ in the defining system ofΣ, its preimagec in Σ is a 1-dimensional solenoid which we call asimple closed solenoid(abusively, sincesuch a ˆc always has infinitely many connected components). The set ofsuch is denotedC (Σ).

The intersection pairingI(f, c) between a measured lamination and a simple closedsolenoid is defined using the transverse invariant measureη ,

I(f, c) =

∫

f∩cdmf dη .

We equipML (Σ) with the resulting weak topology. With its induced topology, PL (Σ)is precompact (being essentially a space of probability measures), but owing to its infinite-dimensionality,PL (Σ) is not compact. The simple closed solenoidsC (Σ) are dense inPL (Σ).

If Z is a surface occurring in the defining limit ofΣ, then we may pullback measuredlaminations onZ to measured laminations onΣ. The result is a direct system of inclu-sionsML (Z) → ML (Σ) whose limitML

∞ := lim−→ML (Z) has dense image inML (Σ). In this paper we will work exclusively withML

∞.A pseudo Anosov diffeomorphismΦ : Z → Z lifts to a diffeomorphism ofΣ fixing

precisely the lifted projective classes[fs] and[fu]. In what follows, the terminology “pseudoAnosov homeomorphism ofΣ” will always mean such a lift.

4. TRAIN TRACKS

We recall first some facts about train tracks onΣ: details may be found in [14]. Letτ ⊂ Σ be a smooth 1-dimensional branched manifold: thusτ is a 1-dimensional CW-complex in which the interiors of edges are smooth curves, and the field of tangent linesTxτ, x ∈ τ \ {vertices}, extends to a continuous line field onτ. We say thatτ is a traintrack if it satisfies the following additional properties:

THE L1 EHRENPREIS CONJECTURE 5

(1) The valency of any vertex is at least 3, except for simple closed curve components,which have a single vertex of valence 2.

(2) If D(S) is the double of a componentS⊂ Σ \ τ, then the Euler characteristic ofD(S) is negative.

We shall follow the custom of referring to the vertices of a train track asswitches.A bigon trackis a smooth 1-dimensional branched manifoldτ ⊂ Σ satisfying item (1)

and which satisfies (2) after collapsing bigon complementary regions to curves. Bigontracks arise naturally from a pairC , D of transverse filling curves, by turning each inter-section of aC curve with aD curve into a pair of 3-valent vertices as in Figure 1. Sincesuch bigon tracks will be the only ones appearing in this article, we will assume from nowon that all switches in bigon tracks have valency no more thanthree.

FIGURE 1. Creating a bigon track from a filling pair of curves.

Denote byE the set of edges of the bigon trackτ. In a small disk neighborhood ofa switchv, the ends of edges incident tov may be divided into two classes, which forconvenience we refer to as ”incoming” and ”outgoing”: each class consists of ends thatare asymptotic to one another, and the decision of naming oneclass incoming, the otheroutgoing, is arbitrary. We writee∈ in(v) or e∈ out(v) if e has an end belonging to theappropriate class. See Figure 2. (Note: it can happen thate belongs to bothin(v) andout(v).)

FIGURE 2. Incoming and outgoing ends.

A switch-additive measureon τ is a functionm : E →R+ for which

∑e∈in(v)

m(e) = ∑e∈out(v)

m(e)

for all switchesv. The set of all switch-additive measures forms a linear coneCτ in RE.Let N(τ) be a tubular neighborhood ofτ equipped with a (singular) foliation by line

segments transverse toτ. See Figure 3.A measured laminationf⊂ Σ is said to becarried by τ if it may by isotoped intoN(τ)

transverse to its foliation. We write in this casef < τ. The subspace of isotopy classes of

6 T.M. GENDRON

FIGURE 3. The tubular neighborhoodN(τ).

measured laminations carried byτ is denotedML τ(Σ). There is an open surjection

Cτ −→ ML τ(Σ)

which is a homeomorphism ifτ is a train track.Let Φ : Σ → Σ be a pseudo Anosov diffeomorphism. We say thatΦ actson τ if Φ(τ)

may be isotoped intoN(τ) transverse to its foliation. We write thenΦ(τ) < τ. Fix a leafti ⊂ N(τ) through each edgeei of τ. The carrying matrix ofΦ is by definitionMΦ = (ai j )where

ai j = |Φ(ei)∩ t j |.MΦ induces an inclusionCΦ(τ) → Cτ which when precomposed with the pushforward mapCτ → CΦ(τ) defines a linear map

MΦ : Cτ −→ Cτ .

Note that the carrying matrixMΦ is non-negative. Such a matrix has a unique eigenvalueof greatest modulus, which is positive-real and simple [4].This eigenvalue is called thePerron root. A corresponding eigenvector may be taken non-negative, and is called aPerron vector. ForMΦ, the Perron root coincides with the entropyλ of Φ, and the Perronvector parametrizes in track coordinates the unstable measured laminationfu of Φ.

When Φ is a pseudo Anosov of Thurston-Penner type, one can recover the carryingmatrix and all of its Perron data from a simpler matrix which records the action ofΦ onthe curves in the familiesC andD . Indeed, letτ be the bigon track formed fromC ∪D .If ei ,ej ∈ Eτ are edges contained in sayc, d resp. then givenei′ ⊂ c another edge, thereexists a uniqueej ′ ⊂ d with ai′ j ′ = ai j . Conversely, forej ′ ⊂ d, there existsei′ ⊂ c withai′ j ′ = ai j . It follows that the carrying matrix ofΦ can be subdivided into blocks indexedby pairs(c,d), which are of the formac,dI whereI is a square matrix in which each columnand row has exactly one non zero entry = 1. The matrix(ac,d) whose columns and rows areindexed byC ∪D has exactly the same Perron root asMΦ, and its Perron vector gives thatof MΦ in the obvious way. We shall call this matrix thecurve matrixof the Thurston-Pennertype pseudo AnosovΦ, and we shall denote itMΦ as well.

For example, ifC = {c}, D = {d} with |c∩d|= r andΦ = G−Nd ◦FN

c , then

MΦ =

(1 rNrN (rN)2+1

).

We note that using the quadratic formula, it is easy to see that the eigenvaluenot equal tothe Perron root is< 1.

We now discuss tracks on the solenoidΣ: in fact, we will only require tracks pulledback from surfaces appearing in its defining inverse system.Thus, if τ is a train track onsuch a surfaceZ, its preimageτ is a smooth 1-dimensional branched solenoid with edge set

THE L1 EHRENPREIS CONJECTURE 7

E≈E× TZ, whereE is the edge set ofτ andTZ is the fiber over a point ofZ, homeomorphicto the Cantor groupπ1Z. With respect to this decomposition we define a measure onE bythe formula

µE = µE ×ηTZ

whereµE(e) = 1 for each edge ofE and ηTZis the restriction toTZ of the transverse

invariant measure ofΣ. In addition, ifτ is equipped with a switch additive measureυ , thepullbackυ is a transversally continuous switch-additive measure onτ ; the cone of suchmeasures onτ is denotedCτ . The relationf < τ has exactly the same meaning as in thecase of a surface.

5. INTERSECTIONFORMULAS

Let τ,κ be bigon tracks inΣ that intersect transversally and minimally with edge setsEτ andEκ ; let f,g be measured laminations carried by them, parametrized by weightsυ ,ω .The intersection pairing may be calculated by the followingformula [14]:

(2) I(f,g) = ∑e∈Eτ , e′∈Eκ

υ(e)ω(e′).

It is useful to re-express (2) as a sum over edges inEτ only. Thus if we write

ω(e) = ∑e′∈Eκ

ω(e′)|e∩e′|

then

(3) I(f,g) = ∑e∈Eτ

υ(e)ω(e).

Suppose now thatf, g are measured laminations obtained as preimages of measuredlaminationsf ⊂ Y andg ⊂ Z, surfaces occurring in the defining system ofΣ. Let W be asurface finitely covering each ofY,Z, and letf, g be the preimages inW of f, g. Let deg(W)be the degree of the coveringW → Σ.

Proposition 1. I(f, g) = I(f,g)deg(W) .

Proof. The intersection locus off andg is of the form

(f∩ g)× TW

whereTW is a fiber ofΣ →W. SinceTW hasη-measure 1/deg(W), the result follows. �

Let f, g be as in the previous paragraphs. Suppose now thatf is parametrized byυ : E →R+ a weight on a bigon trackτ ⊂Y andg is parametrized byω : E′ → R+ a weight on abigon trackτ ′ ⊂ Z. The preimagesf, g are parametrized by the pullback weightsυ , ω onthe preimagesτ, τ ′. Rewritingω as above as a function of the edge setE, we have

I(f, g) = ∑e∈E

υ(e)ω(e).

Let τ be the preimage ofτ in Σ, and letυ , ω be the pullbacks along the projectionE → E.

Proposition 2. I(f, g) =∫

E υω dµE whereµE is the edge measure onE.

8 T.M. GENDRON

Proof. A calculation:

I(f, g) =

(

∑e∈E

υ(e)ω(e)

)· 1deg(W)

=

(

∑e∈E

υ(e)ω(e)

)|TW| =

∫

Eυω dµE.

�

6. TEICHMULLER THEORY OF Σ

References for material in this section are [6], [12], [18].The definition of the Teichmuller spaceT (Σ) and of its metricd

T (Σ) copies that of asurface. In particular,

(1) A conformal structure onΣ is determined by a conformal structure on each leaf.These structures are required to vary continuously in the transverse direction.

(2) Elements ofT (Σ) are represented by marked solenoidsi.e. by homeomorphismsµ : Σ → Σµ , whereΣµ is presumed to have a conformal structure.

(3) The marked solenoidµ ′ : Σ → Σµ ′ is equivalent toµ if there exists an isomorphism

σ : Σµ → Σµ ′ such thatσ ◦ µ ≃ µ ′.

T (Σ) has the structure of a separable Banach manifold. The canonical projectionp : Σ → Z onto any surfaceZ in the defining inverse system induces a direct system ofisometric inclusions ˆp∗ : T (Z) → T (Σ).

Theorem 2 ([12]). The induced inclusion

i : lim−→

T (Z) → T (Σ)

is isometric with dense image.

For most of our purposes, it will be sufficient to work with thedense subspace

T∞ := i(lim

−→T (Z)),

which is an incomplete metric space with respect to the direct limit of the Teichmullermetrics and a pre-Banach manifold. Unless otherwise said, all structuresµ consideredbelow will be assumed to be inT ∞.

Let Σµ be as above. By a holomorphic quadratic differential ˆq on Σµ , we shall alwaysmean the pull-back of a holomorphic quadratic differentialq occurring on some surfaceZµ ,whereµ is the pull-back ofµ . Thus,q is a choice of holomorphic quadratic differential oneach leaf, constant along the fiber transversalsTZ overZ. The tangent space toT ∞ at µmay be identified with the direct limit

Q∞µ = lim

−→Qµ(Z).

The tangent bundle ofT ∞ is then identified withQ∞ = lim→Q(Z) whereQ(Z) is thespace of holomorphic quadratic differentials onZ (with respect to all possible complexstructures).

TheL1 norm onQ∞µ is defined

‖q‖ =∫

Σµ|q|dη .

From the pre-Finsler norm‖ · ‖ we induce a path metricdL1 on T ∞ that defines 1) theL1

topology onT ∞ and 2) along with‖ · ‖, theL1 topology onQ∞.

THE L1 EHRENPREIS CONJECTURE 9

To any quadratic diferential ˆq one associates two transverse, measured laminationsfh

andfv (those that correspond to the horizontal and vertical trajectories ofq). We have thefollowing generalization toΣ of a well-known formula for surfaces:

Lemma 1. I(fh, fv) = ‖q‖ .

Proof. q is the lift of a holomorphic quadratic differential on some surfaceZµ , whereµ isthe lift of µ . The result now follows from the classical formula and the fact that the lift ofan area measure onZµ scales by 1/(degZ) in Σ. �

In the same way, we may also avail ourselves of a direct limit version of the theorem ofHubbard and Masur [9]:

Theorem 3. Any pair of measured laminationsf, g ∈ ML∞ determines a unique qua-

dratic differentialq.

Theorem 4. Let q, qi ∈ Q∞, i = 1,2, . . . , be quadratic differentials. Iffhi → fh and fvi → fv

in the intersection topology, thenqi → q in the L1 topology.

Proof. We assume first that there existsµ ∈ T ∞ with qi, q∈ Q∞µ . Let fh, fv and fhi , fvi be

the pairs of measured foliations which are the horizontal and vertical line fields of ˆq resp.qi. By the comments of§3, the hypothesis on the convergence of measured laminations isequivalent to the corresponding statement for measured foliations. In particular we have,

(4) limI(fh, fhi ) = 0 and limI(fv, fvi ) = 0.

Consider smooth measured foliations ˆghi , gv

i equivalent tofhi , fvi whose heights with respectto fh, fv nearly give the intersectionsI(fh, fhi ), I(fv, fvi ). More precisely, forεi → 0,

(5)∫

ghi

|Im√

q| dη − I(fh, fhi ) < εi and∫

gvi

|Re√

q| dη − I(fv, fvi ) < εi .

Let q′i denote the smooth quatratic differential whose horizontaland vertical foliations aregh

i , gvi . Givenδ > 0, let Ai be the set of points for which|q− q′i| is uniformlyδ small. Let

Bi = Σ\ Ai.We begin by showing that

(6)∫

Bi

|q− q′i| dη → 0

asi → 0. By (4) and (5),∫

Bi|q|dη → 0. If for i large, there is am> 0 with

0 < m ≤∫

Bi

|q− q′i| dη

we must also have

(7) 0 < m0 ≤∫

Bi

|q′i | dη

for somem0 > 0. The fact that∫

Bi|q|dη → 0 whereas

∫Bi|q′i |dη does not would violate

(4) and (5) as well. Thus∫

Bi|qi| dη → 0, proving (6). In particular, we have

lim∫

|q− q′i| dη = lim∫

Ai

|q− q′i|.

Now q′i is measure equivalent to ˆqi , hence ˆq− qi is measure equivalent to ˆq− q′i. By thesecond minimal norm property [5], it follows that

‖q− qi‖ ≤ ‖q− q′i‖.

10 T.M. GENDRON

Lettingε → 0, we obtain‖q− qi‖→ 0. This proves the theorem in the special case whereµ = µi for i large.

Now we suppose that ˆq∈ Q∞µ , qi ∈ Q∞

µiwith µi 6= µ . Let Ci be the set of points where

the foliationsfhi , fvi are uniformlyε-close tofh, fv in the q-metric. Then fori large, inCi the complex structures defined by ˆqi, q are uniformally nearly conformal. On the otherhand, inDi = Σ \ Ci they are not, but the|q|-volume of this set limits to zero. Therefore,if pi ∈ Q∞

µ generates the Teichmuller geodesic connectingµ to µi in time 1, it follows that

‖pi‖ → 0 so thatµi converges toµ in theL1 path metric. Moreover, the induced flow ofquadratic differentials takes ˆq L1 close to ˆqi so that ˆqi converges to ˆq in theL1 topology onQ∞. �

7. THE L1 EHRENPREISCONJECTURE

TheL1 EC is the following statement:

L1 Ehrenpreis Conjecture. The mapping class groupM(Σ, ℓ) acts with L1 dense orbitsonT (Σ).

SinceT ∞ is dense inT (Σ) andM(Σ, ℓ) stabilizesT ∞ [6], it will be enough to demon-strate thatM(Σ, ℓ) acts withL1 dense orbits onT ∞.

In [11] it is shown that for a closed surfaceZ, a Teichmuller dense subset of pairsµ ,ν ∈ T (Z) lie on the axes of pseudo Anosov homeomorphisms. By its definition asan isometric direct limit,T ∞ enjoys the same property. Fix a pairµ, ν ∈ T ∞; withoutloss of generality, we may then assume thatµ , ν lie on the axisA of a pseudo AnosovdiffeomorphismΦ which is the lift of a pseudo AnosovΦ : Z → Z, for some surfaceZoccurring in the defining system ofΣ.

By anL1 nth rootof Φ we mean a sequence{Ψm} of pseudo Anosov homeomorphismsfor which

(1) If λm is the entropy ofΨm then limλm = λ 1/n.(2) If Am is the axis ofΨm thenAm → A converges in the Hausdorff topology induced

from theL1 metric.

Theorem 5. If for every pseudo AnosovΦ and every n> 0, Φ has an L1 nth root, then theL1 EC is true.

Proof. Suppose thatµ , ν lie on the axisA of Φ and let{Ψm} be anL1 nth root,n large.Since theAm → A in theL1 Hausdorff topology, there existsµ ′, ν ′ lying on some axisAm

with dL1(µ , µ ′) < ε, dL1(ν, ν ′) < ε. On the axisAm, we may move via a power ofΨm µ ′

close toν ′, which implies thatµ is moved close toν by the same power ofΨm as well. �

Note1. The existence ofTeichmuller rootsfor all Φ and alln implies the classicalEC.

8. DIRECTIONAL DENSITY

A family P of pseudo Anosov homeomorphisms is said to bedirectionally dense(inQ∞) if the set of quadratic differentials tangent to axes of elements ofP is Teichmullerdense inQ∞. By [11], the family of all pseudo Anosov maps is directionally dense. Infact, it follows easily from the arguments in [11] that the family of lifts of pseudo AnosovsΦ of the typeΦ = G−2N ◦ F2N, whereF, G are right Dehn twists about simple closedcurvesc,d that fill Z, whereZ ranges over all surfaces in the defining system ofΣ, is

THE L1 EHRENPREIS CONJECTURE 11

directionally dense. By Corollary 2.6 in [11], the subfamily obtained by demanding thatcis nonseparating is also directionally dense.

Now given a nonseparating simple closed curveγ ⊂ Z, let ργ : Zγ → Z be the degree 2cover obtained by cutting two copies ofZ alongγ and gluing ends. We say that a pair(c,d)of filling, simple closed curves isinterlacing if there exists a pair of nonseparating simpleclosed curvesα,β such thatρ−1

α (c) is connected whereasρ−1α (d) is not andρ−1

β (d) is

connected whereasρ−1β (c) is not. See Figure 4.

FIGURE 4. A filling pair c,d and a pairα,β interlacing them.

Let P be the family of pseudo Anosov homeomorphisms ofΣ which are lifts of pseudoAnosovs of the form

Φ = G−2N ◦F2N : Z → Z,

where

(1) F , G are right Dehn twists aboutc,d ⊂ Z.(2) c is nonseparating and(c,d) is an interlacing pair.(3) Z ranges over all surfaces in the defining system ofΣ.

Lemma 2. P is directionally dense.

Proof. It is enough to show that for a fixed surfaceZ, the family of maps satisfying (1) and(2) is directionally dense inQ(Z). Assume thatc andd are filling, generating the pseudoAnosov homeomorphismΦ = G−2N ◦F2N. If (c,d) is not an interlacing pair, there exists asimple closed curveδ for which the pair(c,δ ) is interlacing, though not necessarily filling.Indeed, one may assume after a homeomorphism thatc is the curve appearing in Figure5; then takingδ ,α,β as indicated there,(c,δ ) is interlacing with respect to the pair(α,β ). Now for j large,δ j = G j(δ ) is close tod, hence(c,δ j ) is eventually filling. If j is inaddition even,G j lifts to the total space of any degree 2 cover ofΣ, thus the pair(c,δ j)

12 T.M. GENDRON

is interlacable with respect to the same curves interlacing(c,δ ). For j large, the pseudoAnosovΦ j = G−2N

δ j◦F2N has axis close to that ofΦ, and since the maps of the formΦ are

already directionally dense, we are done. �

FIGURE 5. Every nonseparatingc is a member of a (not necessarily fill-ing) interlacing pair.

9. NECKLACE ROOTS

Let n∈ N and letΦ ∈ P, the family appearing in Lemma 2, so that in particularΦ isthe lift of a pseudo Anosov of the formΦ = G−2N ◦F2N : Z → Z. Let ρmn : Zmn→ Z be thecover obtained by cutting 2mncopies ofZ along a pairα andβ interlacingc,d and gluingin a circular fashion. We callρmn thenecklace coverassociated to(c,d). In Figure 6, weillustrate the construction of the necklaceZmn and the formation of the lifts of the curvec.In Figure 7 we display the finished necklace.

There aremn lifts c1, . . . ,cmn andd1, . . . ,dmn of each ofc andd, each mapping withdegree two onto their ancestor. OnZmn, Φ lifts to

Φ = G−Nmn ◦ · · · ◦G−N

1 ◦FNmn◦ · · · ◦FN

1

whereFi , Gi is the right Dehn twist aboutci , di . Let χ denote the clockwise rotation ofZmn

by an angle of 2π/n, so that the pairci , di is taken toc j+m, d j+m (indices taken modmn).

We define thenecklacenth root to be the sequence of lifts of pseudo Anosovs{ n√

Φm} toΣ where

n√

Φm = χ ◦G−Nm ◦ · · · ◦G−N

1 ◦FNm ◦ · · · ◦FN

1 ,

m= 2,3, . . . . The necklacenth root is the basic contruction used in the formation ofL1

roots. The construction ofn√

Φm is a generalization of one that first appeared in [16], wherebranched covers were used.

Lemma 3. n√

Φm is pseudo Anosov for all m.

Proof. For i = 1, . . . ,n, let

Ti = G−Nim ◦ · · · ◦G−N

(i−1)m+1◦FNim◦ · · · ◦FN

(i−1)m+1.

Then it is easy to see that

(n√

Φm)n = T2 ◦ · · · ◦Tn◦T1,

which is of Thurston-Penner type, hence [14]( n√

Φm)n is pseudo Anosov, implyingn

√Φm

is pseudo Anosov as well. �

THE L1 EHRENPREIS CONJECTURE 13

FIGURE 6. Making the necklace.

10. EXISTENCE OFL1 ROOTS

Denote byCZ(Σ) the family of simple closed solenoids which are lifts of simple closedcurves onZ. We begin by constructing a family{Ψm} whose stable and unstable lamina-tions intersection converge to those ofΦ with respect to test solenoids inCZ(Σ).

For eachm= 2,3, . . . let mn√

Φm denote themth element in the sequence of pseudoAnosovs whose lifts define themnth necklace root ofΦ. Define the sequence{Ψm} as thelifts to Σ of the pseudo Anosov homeomorphisms

Ψm =( nm√

Φm)m

.

Observe that the stable and unstable foliations ofΨm and nm√

Φm are equal.

Note2. Ψm is not the same asn√

Φm. In fact, if we lift n√

Φm to Zm2n – whereΨm is defined– we see that this lift twists alongmdisjoint “blocks” of curves, each block consisting of asuccession ofm lifts of c andd. On the other hand,Ψm consists of twists along one blockconsisting of a succession ofm2 lifts of c andd. As we shall see, the stable and unstable

laminations of the family{Ψm} have better convergence properties than those of{ n√

Φm}.

Denote byfum, fsm and byfu, fs the unstable and stable laminations ofΨm andΦ.

Lemma 4. For all c∈ CZ(Σ),

I(fum, c)−→ I(fu, c) and I(fsm, c)−→ I(fs, c).

14 T.M. GENDRON

FIGURE 7. The finished necklace.

Proof. The proof will be through examination of curve matrices. We begin with Φ. Letr = I(c,d). The action ofΦ along the curvesc,d is given by the matrix

MΦ =

(1 2rN

2rN (2rN)2+1

).

Let Zm2n be the surface whereΨm is defined. The curve familiesC = {c1, . . . ,cm2n},D = {d1, . . .dm2n} are filling, and the action ofΦ on C ∪D is prescribed schematicallyby the matrix in Figure 8, where all entries not contained in the boxed vectors are zero,and where the “broken” vectors indicate that only that portion of the corresponding vectoris used. For example, in the upper right hand corner we have the entry rN, which isthe bottom half of theB-vector; in the lower right hand corner, we have the vector entry((rN)2 2(rN)2 +1

)T, which is the top two thirds of the vectorD, and so on. The curve

matrix of (Ψm)n = ( mn

√Φm)

mn is displayed in Figure 9. The black vectors indicate regions

THE L1 EHRENPREIS CONJECTURE 15

whereM(Ψm)n = (MΨm)n = (M mn√Φm

)mn differs from MΦ. Figure 10 contains the curve

matrix of mn√

Φm.

FIGURE 8. The curve matrix forΦ.

Denote byλm the Perron root ofMΨm and by

υm = (a1 . . . ,am2n,b1, . . . ,bm2n)

the corresponding Perron vector, normalized to haveL1 norm 1i.e. so thatυm is a proba-bility vector. Letυavg

m be the vector

υavgm = (a1+ · · ·+am2n,b1+ · · ·+bm2n),

which is also a probability vector. In the case ofΦ and Φ, the Perron roots are iden-tical and will be denotedλ ; if υ = (a,b)T is theL1 norm 1 Perron vector ofMΦ, υ =(1/2m2n)(a, . . . ,a,b, . . . ,b)T is theL1 norm 1 Perron vector forMΦ. We recall that thisspectral data has the following interpretation:

(1) The Perron rootsλm, λ are equal to the entropies ofΨm, Φ.(2) Letτm be the bigon track formed from the curve familiesC , D . The measuresµm,

µ formed from the Perron vectorsυm, υ , parametrize the unstable laminationsfum,fu in the coneC(τm).

Note that the column sums ofM(Ψm)n have uniform upper and lower boundsB andb> 1.We thus obtain the bound [4]

1< b< (λm)n < B.

We may then assume, after passing to a subsequence if necessary, that theλm converge tosome valueλ ∗ > 1. We shall need to control the following entries ofυm:

Claim1. am,am+1,bm,bm+1 → 0 asm→ ∞.

16 T.M. GENDRON

FIGURE 9. The curve matrix forΨnm.

FIGURE 10. The curve matrix formn√

Φm.

THE L1 EHRENPREIS CONJECTURE 17

Proof of claim. Let ξm be the Perron root ofM nm√Φm: thus (ξm)

m = λm. If one of thefour entries listed in the statement does not converge to 0, then consideration of the matrixM(Ψm)n shows that none of them do. Thus, let us suppose thatam 6→ 0. It follows thateventuallya2m≥ δ for some positiveδ . Examination of the action ofM nm√Φm

onυm shows

that am+im = ξ i−1m a2m for i = 2, . . .mn. Howeverξ i−1

m > 1 for all i, and sinceυm is aprobability vector, this would imply thatmnδ < mna2m< 1, impossible sincem→ ∞. Thisproves the claim.

Let us shorten notation, writingf = fu andfm = fum for the unstable foliations ofΦ andΨm. Let c be any simple closed solenoid inCZ(Σ).

Then

I((λm)

nfm, c)

= I((Ψm)

nfm, c)

=

∫

EM(Ψm)nυm|τ ∩ c|,

where M(Ψm)nυm is the lift of the vectorM(Ψm)nυm to E. Now since ˆc is the lift of a simpleclosed curvec in Z, we have that

I(fm, c

)=

1deg(Z)

· I(favgm ,c

),

wherefavgm is the measured lamination inZ corresponding to the weightυavg

m . However anexamination of the matricesMΦ andM(Ψm)n yields

(λm)nυavg

m = MΦυavgm + εm

where

εm =((rN)2(am+am+1),−(rN)2bm+1+(rN)3am+((rN)2−(rN)+1)am+1+(rN)4bm

)T.

By Claim 1, it follows thatεm → (0,0)T or υavgm converges to an eigenvector ofMΦ of

eigenvalueλ n∗ . But sinceλ n

∗ > 1 and the second eigenvalue ofMΦ is strictly less than 1,we must have thatυavg

m → υ andλ∗ = λ 1/n. In particular,

limm→∞

I(favgm ,c

)= I

(f,c).

This takes care of the unstable part of the theorem; the stable part is handled by repeatingthe above argument forΦ−1 andΨ−1

m . �

Theorem 6. Every pseudo AnosovΦ has an L1 nth root for all n.

Proof. SinceP is directionally dense, there exists a sequence{Φ(g)} ⊂ P, whereΦ(g)

is the lift of a pseudo AnosovΦ(g) : Xg → Xg in which Xg is a surface of genusg → ∞,

and the axesAg → A = axis of Φ. For eachg, let {Ψ(g)m } be the sequence of pseudo

Anosovs constructed above. Then by Lemma 4 we may obtain anL1 nth root {Ψm} of

Φ by extracting a suitable diagonal subsequence of{Ψ(g)m }. Indeed, a suitable diagonal

subsequence{Ψm} yields a sequence of pseudo Anosov homeomorphisms whose stableand unstable laminations intersection converge to those ofΦ. By Theorems 3 and 4, thisgives rise to a sequence of quadratic differentials ˆqi along the associated axesAi whichL1-converge to the quadratic differential ˆq determined byfu andfs. �

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18 T.M. GENDRON

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INSTITUTO DE MATEMATICAS – UNIDAD CUERNAVACA , UNIVERSIDAD NACIONAL AUTONOMA DE

M EXICO, AV. UNIVERSIDAD S/N, C.P. 62210 CUERNAVACA , MORELOS, MEXICOE-mail address: tim@matcuer.unam.mx