STANDARDLY EMBEDDED TRAIN TRACKS AND PSEUDO-ANOSOV MAPS WITH MINIMUM EXPANSION FACTOR ERIKO HIRONAKA AND CHI CHEUK TSANG

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STANDARDLY EMBEDDED TRAIN TRACKS AND PSEUDO-ANOSOV

MAPS WITH MINIMUM EXPANSION FACTOR

ERIKO HIRONAKA AND CHI CHEUK TSANG

Abstract.

We show that given a fully-punctured pseudo-Anosov map

S→S

whose

punctures lie in at least two orbits under the action of

, the expansion factor

(

) satisﬁes

the inequality

λ(f)|χ(S)|≥µ4≈6.85408,

where

1+√5

2≈

61803 is the golden ratio. The proof involves a study of standardly

embedded train tracks, and the Thurston symplectic form deﬁned on their weight space.

1. Introduction

Let S=Sg,s be an oriented surface of ﬁnite type with genus gand spunctures, such that

(

)=2

−

g−s

is negative, and let

S→S

be a homeomorphism of

. By the

Nielsen-Thurston classiﬁcation of mapping classes, up to isotopy,

either preserves an

essential multicurve (and is periodic or reducible), or it is pseudo-Anosov. In the latter case,

there exists a transverse pair of measured, singular stable and unstable foliations

ℓs, ℓu

such

that

stretches the measure of

ℓu

(

) and contracts the measure of

ℓs

λ(f)

. The

number λ(f)>1 here is known as the expansion factor of f.

Thurston’s train track theory links the dynamics of pseudo-Anosov maps with that of

Perron-Frobenius matrices, and implies that

(

) is a bi-Perron algebraic unit with degree

bounded in terms of

(

) [

FLP12

]. In particular, for ﬁxed (

g, s

(

) attains a minimum

λg,s >

1. Furthermore, each pseudo-Anosov map determines a closed geodesic on the moduli

space

(

) with respect to the Teichm¨uller metric, whose length equals

log

(

)). Thus

log

(

λg,s

) is the length of the shortest geodesics on

(

Sg,s

) [

Abi80

]. However, so far

λg,s

known only for small gand s, see [HS07], [CH08], [LT11a].

In this paper, we focus on the normalized expansion factor

L(S, f) = λ(f)|χ(S)|,

and consider pseudo-Anosov maps that are fully-punctured, meaning the singularities of

ℓs

and

ℓu

lie on the punctures of

. Let

be the golden ratio

1+√5

2≈

61803. Our main

result is the following.

Theorem 1.1. Let

S→S

be a fully-punctured pseudo-Anosov map with at least two

puncture orbits, then L(S, f)satisﬁes the sharp inequality

L(S, f)≥µ4≈6.85408.

The ﬁrst author was partially supported by a grant from the Simons Foundation #426722.

The second author was partially supported by a grant from the Simons Foundation #376200.

arXiv:2210.13418v3 [math.GT] 3 Jan 2024

2 ERIKO HIRONAKA AND CHI CHEUK TSANG

Here by a puncture orbit, we mean an orbit of the action of

on the punctures of

. If we

replace the punctures of

with boundary components, then the number of puncture orbits

equals the number of boundary components of the mapping torus of

. We remark that

the assumption of

having at least two puncture orbits is necessary. See Section 7.4 for

explicit counterexamples.

For the rest of this introduction, we will ﬁrst give some background on what is known about

the pattern of minimum expansion factors and its relations to the topology and geometry

of the mapping torus. Then we will state a more precise version of Theorem 1.1, discuss

some special cases, and ﬁnally explain the tools that are used in the proof.

1.1. Background on small expansion factors and mapping tori. The expansion

factor is a measure of the dynamical complexity of a pseudo-Anosov map. More concretely,

it is equal to the exponential of the entropy, which measures how much the map mixes

points on the surface. Thus, the minimum expansion factor is related to the complexity of

the underlying surface and of the mapping torus, with the latter being a ﬁbered hyperbolic

3-manifold [

Thu88

]. In the following, we describe what is known about this relation.

1.1.1. Minimum expansion factor as a function on the (

g, s

)plane. Applying properties of

Perron-Frobenius matrices, Penner [

Pen91

] observed that

log λg,s

is bounded below by the

reciprocal of a linear function in gand s

log λg,s ≥log 2

12g−12 + 4s.

Furthermore, for

= 0, Penner constructed a sequence of examples giving an upper bound,

yielding

log λg,0≤C

for some C > 0, which together with the lower bound implies

log λg,0≍1

|χ(Sg,0)|.

This asymptotic behavior has been shown to persist along other lines in the (

g, s

)-plane,

namely,

where

m >

0 [

Val12

];

= 0 [

HK06

];

= 1 [

Tsa09

]; and

for ﬁxed

s0>0 [Yaz20]. For lines g=g0with ﬁxed g0≥2, however, [Tsa09] shows that

log λg,s ≍log |χ(Sg,s)|

|χ(Sg,s)|.

We also mention a result of Agol, Leininger, and Margalit [

ALM16

] that concerns the

minimum expansion factor among pseudo-Anosov maps on an oriented closed surface with

ﬁxed genus

whose action on the ﬁrst homology preserves a

-dimensional subspace. We

denote this minimum expansion factor as λg,0,k. Their result is that

log λg,0,k ≍k+ 1

See [BBKT22] for a partial generalization of this result to punctured surfaces.

STANDARD EMBEDDINGS AND EXPANSION FACTORS 3

Let

λK

be the minimum expansion factor for pseudo-Anosov maps

S→S

satisfying

|χ(S)|=K.

Question 1.2 (McMullen [

McM00

]).Does the sequence (

λK

)

converge, and if so does it

converge to µ4?

As we will see in this paper, it is possible to use the theory of digraphs and standardly

embedded train tracks to get information about minimum expansion factors in the fully-

punctured case. To translate results from the fully-punctured case to the general case, the

following question is of interest.

Question 1.3. For ﬁxed

, where, on the level sets in the (

g, s

)-plane where 2

s−

2 =

is the minimum

λK

realized? Is there a bound

, such that

λK

is always achieved by

λg,s

for s≤s0?

A positive answer to Question 1.3 would suggest that the minimum expansion factors

λ◦

for fully-punctured pseudo-Anosov maps

S→S

satisfying

|χ

(

)

should have the

same asymptotic behavior as λK.

We note that Theorem 1.9 below answers Question 1.3 in the aﬃrmative if we restrict to

fully-punctured pseudo-Anosov mapping classes with even Euler characteristic and at least

two puncture orbits.

1.1.2. Mapping tori. Pseudo-Anosov mapping classes (

S, f

) can be partitioned into ﬂow-

equivalence classes, that is, collections of pseudo-Anosov mapping classes whose mapping

tori and suspension ﬂows are equivalent. Thurston’s ﬁbered face theory gives a way to

parameterize elements of a ﬂow-equivalence class by primitive integral points on a cone

over a top-dimensional face

, called a ﬁbered face, of the Thurston norm ball in

(

;

Here, the Thurston norm of a primitive integral element

a∈cone

(

) associated to (

Sa, fa

)

equals |χ(Sa)|, and hence its projection to Fis given by a=1

|χ(Sa)|a, see [Thu86].

It follows that the rational elements on

are in one-to-one correspondence with elements of

the ﬂow equivalence class. By a result of Fried, the normalized expansion factor

(

Sa, fa

considered as a function of integral classes

, extends to a continuous concave function

deﬁned for all

a∈F

that goes to inﬁnity towards the boundary of

[

Fri82

]. Thus, in

particular, for any compact subset of

, the corresponding pseudo-Anosov maps have

bounded normalized expansion factor.

For a hyperbolic 3-manifold

with ﬁbered face

, let

CM,F

be the inﬁmum of

(

S, f

), where

(

S, f

) is associated to an element of

cone

(

). One consequence of Fried and Thurston’s

ﬁbered face theory is the following.

Theorem 1.4 (Fried-Thurston [

Fri82

] [

Fri85

] [

Thu86

]).If

(

)

≥

2, then for any

ε >

there are inﬁnitely many ﬁbrations M→S1whose monodromy (S, f)satisﬁes

L(S, f)< CM,F +ε.

4 ERIKO HIRONAKA AND CHI CHEUK TSANG

Given

C >

1, the pseudo-Anosov mapping classes (

S, f

) that satisfy

(

S, f

)

< C

must

have mapping tori

and associated ﬁbered face

with

CM,F ≤C

. The following result

is often referred to as the universal ﬁniteness property and states that after removing the

singular orbits of the suspension pseudo-Anosov ﬂow, the number of such pairs (

M, F

) is

ﬁnite.

Theorem 1.5 (Farb-Leininger-Margalit [

FLM11

]).Given any

, there is a ﬁnite set of

ﬁbered, hyperbolic 3-manifolds Ω

such that if (

S, f

)is a fully-punctured pseudo-Anosov

map, and L(S, f)< C, then the mapping torus of (S, f)lies in ΩC.

(See also [Ago11] and [AT22].)

The above theorem also implies ﬁniteness for families of deﬁning polynomials that can arise

for small expansion factors. Given a polynomial

(

), let

|p|

be the complex norm of the

largest root of p(t).

Theorem 1.6 (McMullen [

McM00

]).Let

be a ﬁbered hyperbolic 3-manifold, with

(

)

≥

2and let

F⊂H1

(

;

)be a ﬁbered face. Then there is a polynomial

∈Z

[

t1, . . . , tn

]with the property that for any primitive integral

= (

a1, . . . , an

)in the

ﬁbered cone over

Θ(

ta1, . . . , tan

)

is the expansion factor of the monodromy of (

S, f

)

corresponding to a.

The polynomial Θ is known as the Teichm¨uller polynomial of the ﬁbered face.

Deﬁne

to be the inﬁmum of

(

S, f

), where (

S, f

) is the monodromy of a ﬁbration

M→S1

. The topological invariant

of a ﬁbered hyperbolic 3-manifold

is related to

the geometry of Mas shown by the following theorem.

Theorem 1.7 (Kojima-McShane [

KM18

]).For any hyperbolic, ﬁbered 3-manifold

satisﬁes the following inequality

CM≥exp vol(M)

3π.

Remark 1.8. Theorem 1.7 implies, for example, that if (

S, f

) satisﬁes

(

S, f

) =

µ4

, the

volume of the mapping torus Mmust satisfy

vol(M)≤12πlog(µ)≈18.14123.

For comparison, the magic manifold, which realizes many of the smallest known expansion

factors, has volume

≈

33349 according to SnapPy [

CDGW

]. Further work comparing

known pseudo-Anosov maps with small expansion factor and the volume of their mapping

tori can be found in [AD10] and [KKT13].

1.2. Reﬁnement of the main theorem and some special cases. For positive integers

a, b with a < b, deﬁne

LTa,b(t) = t2b−tb+a−tb−tb−a+ 1,

STANDARD EMBEDDINGS AND EXPANSION FACTORS 5

This family of polynomials was ﬁrst noticed by Lanneau and Thiﬀeault [

LT11b

] to play a

role in the study of minimum expansion factors.

The more precise version of our main theorem is as follows.

Theorem 1.9. Let

S→S

be a fully-punctured pseudo-Anosov map with at least two

puncture orbits where |χ(S)|=K≥3. Then λ(f)satisﬁes the inequality

λ(f)≥ |LT1,K

2|,

for Keven, and

λ(f)K≥8,

for Kodd.

Moreover, for each even

, equality is achieved by a fully-punctured pseudo-Anosov map

with |χ(S)|=Kand s≤4.

Sharpness in Theorem 1.9 follows from computations in [Hir10], [AD10], [KKT13].

1.2.1. Orientable pseudo-Anosov mapping classes. A pseudo-Anosov mapping class (

S, f

) is

called orientable if its stable and unstable foliations are orientable. These have the property

that the expansion factor of

is equal to the largest eigenvalue of its action on the integral

homology of S.

In [

LT11b

], Lanneau and Thiﬀeault asked whether the minimum expansion factor

λ+

for

even genus gorientable pseudo-Anosov mapping classes is given by

λ+

g=|LT1,g|,

and showed that

|LT1,g|

is a lower bound for

λ+

for

= 2

10. For the case when

g≥

is even and not divisible by 3, |LT1,g|is an upper bound for λ+

g[Hir10].

Theorem 1.10. Let (

S, f

)be an orientable pseudo-Anosov mapping class on a genus

g≥

closed surface with exactly two singularities, each of which is ﬁxed by

. Then

(

)satisﬁes

the inequality

λ(f)≥ |LT1,g|

and this inequality is sharp when gis even and not divisible by 3.

1.2.2. Braid Monodromies. Consider (

S, f

) where

S0,n+1

, and one puncture

p∞

is ﬁxed

. A mapping class of this type is called a braid monodromy on

strands. Given a

standard braid

strands, there is an associated braid monodromy, and vice versa (see

e.g. [

Bir74

]). For convenience, we use the standard Artin braid group generators to specify

a braid and its associated monodromy.

Theorem 1.11. Let

be a fully-punctured pseudo-Anosov braid monodromy on

strands.

Then λ(β)satisﬁes the inequality

λ(β)n−1≥µ4.

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STANDARDLYEMBEDDEDTRAINTRACKSANDPSEUDO-ANOSOVMAPSWITHMINIMUMEXPANSIONFACTORERIKOHIRONAKAANDCHICHEUKTSANGAbstract.Weshowthatgivenafully-puncturedpseudo-Anosovmapf:S→Swhosepunctureslieinatleasttwoorbitsundertheactionoff,theexpansionfactorλ(f)satisfiestheinequalityλ(f)|χ(S)|≥µ4≈6.85408,whereµ=1+√52≈1.6...

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STANDARDLY EMBEDDED TRAIN TRACKS AND PSEUDO-ANOSOV MAPS WITH MINIMUM EXPANSION FACTOR ERIKO HIRONAKA AND CHI CHEUK TSANG

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