arxiv:2008.12436v2 [math.gt] 23 mar 2021

PERIODS OF CONTINUED FRACTIONS AND VOLUMES OF

MODULAR KNOTS COMPLEMENTS

JOSE ANDRES RODRIGUEZ-MIGUELES

Abstract. Every oriented closed geodesic on the modular surface has a canonically asso-ciated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow.We study the volume of the associated knot complement with respect to its unique completehyperbolic metric. We show that there exist sequences of closed geodesics for which thisvolume is bounded linearly in terms of the period of the geodesic’s continued fraction ex-pansion. Consequently, we give a volume’s upper bound for some sequences of Lorenz knotscomplements, linearly in terms of the corresponding braid index.

Also, for a large family hyperbolic surfaces we give volume’s bounds for some sequencesof filling finite sets of closed geodesics in terms of the geodesic length of the finite set ofgeodesics.

1. Introduction

Let Σ be a complete, orientable hyperbolic surface or 2-orbifold of finite area. An orientedclosed geodesic γ on Σ has a canonical lift γ in its unit tangent bundle T 1Σ, namely thecorresponding periodic orbit of the geodesic flow. Let Mγ denote the complement of a regular

neighborhood of γ in T 1Σ. As a consequence of the Hyperbolization Theorem, Mγ admits afinite volume complete hyperbolic metric if and only if γ fills Σ [16]. Such metric is uniqueup to isometry, by Mostow’s Rigidity Theorem, meaning that any geometric invariant is atopological invaraint. Recently, there has been interest in relating the volume of Mγ in termsof properties of the close geodesic γ.

Bergeron, Pinsky and Silberman have already studied in [4] the problem of finding an upperbound for the volume of Mγ , by giving one which is linear in the length of the geodesic. Herewe improve their upper bound for some sequences of closed geodesics in infinitely manypunctured hyperbolic surfaces:

Corollary 1.1. Given a hyperbolic metric ρ on a punctured hyperbolic surface Σ, of genusg with k punctures (k ≡ 2 (mod g) if g ≥ 2) and let dΣ := max6gk, 6(k− 3), 6. Then thereexist a constant Cρ > 0 and a sequence γn of filling finite sets of closed geodesics on Σ withat most dΣ elements in each set γn, and `ρ(γn)∞, such that

Vol(Mγn) ≤ 8dΣv3

Cρ`ρ(γn)

W(`ρ(γn)Cρ− 2) + 2

,

where v3 is the volume of a regular ideal tetrahedron and we use the Lambert W function.

Nevertheless, it is easy to construct sequences of closed geodesics with length approachingto infinity but whose associated canonical lift complements are homeomorphic . For example,the iterations under an infinite-order diffeomorphism of the surface, of a given filling closedgeodesic. In ([23], Theorem 1.1) we constructed more interesting sequences of closed geodesicswhose associated canonical lift complements are not homeomorphic with each other and the

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sequence of the corresponding volumes is bounded. Also in ([23], Theorem 1.5) we gave atopological lower bound of the volume of Mγ in terms of the number homotopy classes ofarcs of γ in each pair of pants of a given a pants decomposition of Σ. This gave us a methodto construct sequences of closed geodesics on any hyperbolic surface where the correspondingvolumes are bounded from below in terms of the length of the geodesics ([23], Theorem 1.3).Here we improve the estimation of ([23], Theorem 1.3) and obtained a lower bound (seeTheorem 5.5) which is similar to the upper bound found in Corollary 1.1. Although thesequences of closed geodesics in Theorem 5.5 and Corollary 1.1 are different, the followingresult shows that this length bound is sharp for a sequence of filling finite sets of closedgeodesics in infinitely many punctured hyperbolic surfaces:

Corollary 1.2. Given a hyperbolic metric ρ on a punctured hyperbolic surface Σ, of genus gwith k punctures ( k ≡ 2 (mod g) if g ≥ 2) and let dΣ := max6gk, 6(k − 3), 6. Then thereexist a constant Cρ > 0 and a sequence γn of filling finite sets of closed geodesics on Σ,with at most dΣ elements in each set γn and `ρ(γn)∞ such that

dΣv3

12

Cρ`ρ(γn)− 32

W(`ρ(γn)Cρ

) − 3

2

≤ Vol(Mγn) ≤ 8dΣv3

Cρ`ρ(γn)

W(`ρ(γn)Cρ− 2) + 2

.


It is interesting to point out that in a collaboration with Tommaso Cremaschi and AndrewYarmola in [10] we study the same problem for large families of filling finite sets of simpleclosed geodesics, such as filling pairs of simple closed geodesics, and found bounds for thevolume of the corresponding link complement in terms of expressions involving distances inthe pants graph. As a consequence we constructed sequences of filling pairs of simple closedgeodesics in infinitely many hyperbolic punctured surfaces where the volume of the canonicallift complement is bounded by the logarithm of the length (see [10], Theorem C). Noticethat up to a subsequence in the sequences considered in Corollary 1.2, each element of thesequence is a set of non-simple closed geodesics, so they are not considered in [10].

1.1. Modular links and Lorenz links. We focus here mainly in the case of the modularsurface Σmod = H2/PSL2(Z). This hyperbolic 2-orbifold is particularly interesting since itsunit tangent bundle is homeomorphic to the complement of the trefoil knot in S3. Therefore,in this particular case, Mγ can be considered as link complement in S3. Moreover, after

trivially Dehn filling the trefoil cusp of T 1Σmod, [20] Ghys observed that the periodic orbitsof the geodesic flow over the modular surface are the Lorenz links in S3.

On the topological side, Lorenz links are prime, fibered, positive, hence amphicherical,also the link genus and braid index are determined combinatorially (see [6],[28],[17] and[13]). Furthermore, Birman and Kofman proved in [5] that Lorenz links and T -links coincide.Unfortunately, we do not know if this topological properties are preserve after drilling thetrefoil knot, meaning for modular knots in T 1Σmod. However Ghys in [20] proved that linkingbetween the trefoil knot, cusp of T 1Σmod, with the modular knot turns out to be related tothe Rademacher function.

On the geometric side, with respect Thurston‘s geometrization of prime knots complementsin S3 [27], we have that torus knots occur among Lorenz knots. Nevertheless, more than half ofthe ‘simplest hyperbolic knots‘ (whose complements are in the census of hyperbolic manifoldswith seven or fewer tetrahedra) are Lorenz knots (see [5], Section 5). Moreover, there are

PERIODS OF CONTINUED FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 3

combinatorial upper bounds for the volume of Lorenz links complements (see [9]). From thefact that simplicial volume is non-increasing under Dehn filling [26], here we prove a sharperupper bound for some sequences of Lorenz knots:

Corollary 1.3. There exist a sequence Kn of Lorenz knots in S3 such that n is the braidindex of Kn, and

Vol(S3 \Kn) ≤ 8v3(5n+ 2),

where v3 is the volume of a regular ideal tetrahedron. If Kn is not hyperbolic, Vol(S3 \Kn) isthe sum of the volumes of the hyperbolic pieces of S3 \Kn.

By comparing this result with the general upper bounds for volumes of Lorenz links com-plements, obtained by Champanerkar, Futer, Kofman, Neumann and Purcell in [9], we noticethat their upper bound ([9], Theorem 1.7) is sharper than ours for the subsequences of Lorenzknots used in Corollary 1.3 whose braid index is at most 33. On the other hand, our upperbound is sharper that theirs ([9], Theorem 1.7) for those Lorenz knots in Corollary 1.3 whosebraid index is bigger than 33, because their upper bounds are at least quadratically in termsof the braid index and not linear.

Returning to the modular surface case, in ([4], Section 3) Bergeron, Pinsky and Silbermanalso gave an upper bound for the volume of Mγ which is proportional to the period of thegeodesic’s continued fraction expansion plus the sum of the logarithms of its correspondingcoefficients. Nevertheless, in ([23], Corollary 1.2) we constructed sequences of closed geodesicswith the period approaching to infinity, but whose sequence of the corresponding volumes isuniformly bounded. In this paper, we prove that there exist sequences of closed geodesics forwhich the volume of the canonical lift complement has an upper bound linearly in terms ofthe period:

Theorem 1.4. For the modular surface Σmod, there exist a sequence γn of closed geodesicson Σmod such that n is half the period of the continued fraction expansion of γn, and

Vol(Mγn) ≤ 8v3(5n+ 2),

where v3 is the volume of a regular ideal tetrahedron.

As a consequence of Theorem 1.4 in [23], we have that up to a constant, Theorem 1.4 issharp.


v3n

12≤ Vol(Mγn) ≤ 8v3(5n+ 2),


Dehornoy and Pinsky showed in [14] a description of the geodesic flow and templates inthe particular case of the unit tangent bundle of any sphere with three cone points. Wewonder if after removing the exceptional fibers of the Seifert fibration of the correspondingunit tangent bundle, the periodic orbits of the geodesic flow are isotopic to periodic orbitsinside a Lorenz template embedded in the unit tangent bundle of the thrice-punctured sphere.If this is true for the sequence of Lorenz knots considered in Corollary 1.3 then one couldgeneralized Theorem 1.4 for any sphere with three cone points.


1.2. The thrice-punctured sphere case. It is interesting to point out that for a thrice-punctured sphere, denoted by Σ0,3, we proved Theorem 1.2 for sequences of closed geodesics(see Corollary 6.5). The main motivation to give this different proof of Theorem 1.2 forΣ0,3, is the next result which estimates a lower bound for the volume of the canonical liftcomplement of figure-eight type closed geodesics (see Definition 2.8) on Σ0,3, in terms ofcombinatorial data of the reduced word representing the conjugacy class of the geodesic inπ1(Σ0,3).

Theorem 1.6. Given a thrice-punctured sphere Σ0,3, and γ a figure-eight type closed geodesicwith respect to X and Y (two free-homotopy classes of distinct punctures in Σ0,3), we havethat:

Vol(Mγ) ≥ v3

2(]exponents of X in ωγ+ ]exponents of Y in ωγ − 2),

where v3 is the volume of the regular ideal tetrahedron, ωγ is the cyclically reduced wordrepresenting the conjugacy class of γ in 〈X,Y 〉 ⊂ π1(Σ0,3).

Theorem 1.6 uses a result due to Agol, Storm and Thurston [3] giving a lower bound forthe volume of Mγ in terms of the simplicial volume of the double of the manifold constructedby cutting Mγ along an incompressible surface. Here we apply it to the incompressible

surface coming from the pre-image under the map T 1(Σ0,3) → Σ0,3 of a simple geodesic arcwhose vertices points belong to the same puncture. Moreover, Theorem 1.6 is an analogueof a lower bound for the volumes of canonical lift complements of geodesics on hyperbolicsurfaces admitting a non-trivial pants decomposition ([23], Theorem 1.5).

1.3. The sequences of collections of closed geodesics. Recall that every closed geodesicon the modular surface is represented by a primitive element in the semi-group generated by

two parabolic elements X =

(1 10 1

)and Y =

(1 01 1

)in PSL2(Z) (see [7], Section 3).

These type of closed geodesics can also be found in any hyperbolic surfaces, and are encodedby primitive elements in the semi-group generated by the free homotopy class of two disjointand distinct simply closed curves denoted by X and Y. In this case we say that the closedgeodesics is a figure-eight type closed geodesics with respect X and Y. The sequences of closedgeodesics appearing in this paper are of this kind and can be found explicitly encoded in Table1.3.

Before presenting the structure of this paper, an important question is the distributionof the geodesics obtained here. In the case of the modular surface it will be interestingto calculate volume bounds for the geodesics which come from the ideal class group of thefields Q(

√d) with d a square free positive integer bigger than 1. The interest of these closed

geodesics is that they are uniformly distributed on T 1Σmod (see [15]). This approach wasoriginally made by Brandts, Pinsky and Silberman in [7] but for only a fixed finite collection ofclosed geodesics. Unfortunately, we ignore if the sequences of closed geodesics on Theorem 1.5are uniformly distributed. Nevertheless, it will be very interesting to understand the behaviorof volumes of canonical lift complements for random closed geodesics on any hyperbolic 2-orbifold, in terms of its length. In subsequent works, we intend to prove that for randomclosed geodesics on the modular surface, the volumes of the canonical lift complements arecomparable to period of the geodesic’s continued fraction expansion.

Outline: In Section 2 we review the coding of geodesics of the modular surface by positivewords in the alphabet X,Y . In Section 3 we review the William’s algorithm, giving a


2-Orbifold Σ Geodesics γn #γk Vol(Mγn) Bounds Refernce

All hyperbolic All Any ≤ Cρ`ρ(γk)where Cρ is positive constant that only dependson the hyperbolic metric ρ.

[4],Thm.1.1

All hyperbolicexcept sphereswith 3 conepoints

All Any ≥ v32

∑P∈Π

(]homotopy Cl. of γk-arcs in P − 3)

where Π is any pants decomposition of Σ and v3

the volume of a regular ideal tetrahedron.

[23],Thm.1.5

All hyperbolicsurfaces

The concate-nation of αn(a fixed closedcurve in a pair ofpants with all thehomotopy classesof arcs with wordlength at most n)and η0 (a fixedfilling geodesic).

1 ≥ 2v33

(Cρ`ρ(γn)−δρ

W

(`ρ(γn)

Cρ

) − 9

)where Cρ and δρ are positive constant that onlydepend on the hyperbolic metric ρ and η0.

Thm.5.5

Σ0,3

n∏i=1

(XkiY mi)

where ki,mi ∈ N

Any ≥ v32

(]ki+ ]mi − 2) Thm.1.6

Σmodn∏i=1

(XkiY ) and

n∏i=1

(XY mi) with

ki < ki+1 andmi < mi+1

1 ≤ 8v3(5n+ 2) Thm.1.4

k-puncturedhyperbolicsurfaces ofgenus g (k ≡ 2 (mod g)if g > 1)

Lifts ofn∏i=1

(XkiY ) and

n∏i=1

(XY mi) with

ki < ki+1 andmi < mi+1

≤ dΣ ≤ 8dΣv3

(Cρ`ρ(γn)

W

(`ρ(γn)

Cρ−2

) + 2

)where dΣ := max6gk, 6(k − 3), 6.

Coro.1.1

Σmodn∏i=1

(X6ki+1Y )

with ki < ki+1

1 nwhere means equality up to constant multi-plicative and additive error.

Thm.1.5

k-puncturedhyperbolicsurfaces ofgenus g (k ≡ 2 (mod g)if g > 1)

Lifts ofn∏i=1

(X6i+1Y )

≤ dΣ dΣv3

(Cρ`ρ(γk)

W

(`ρ(γk)

Cρ

))

Coro.1.4

Σ0,3

n∏i=1

(Xmi+rY )

andn∏i=1

(XY mi+r)

with m, r ∈ Nand 0 ≤ r < m

1 v3

(`ρ(γn)

Cρ−δρ

W (Cρ`ρ(γn)+δρ)

)where Cρ and δρ are positive constant that onlydepend on the hyperbolic metric ρ and γ0.

Coro.6.5

Table 1. Sequences of filling finite sets of closed geodesics.


combinatorial description of the canonical lift of figure-eight type closed geodesics. In Section4 we prove Theorem 1.4, Theorem 1.5 and Corollary 1.3. In Section 5 we prove Corollary1.1 and Corollary 1.2. And in Section 6 we give a new prove of Theorem 1.4 for the thrice-punctured sphere and prove Theorem 1.6.

Acknowledgments: I would like to thank Pierre Dehornoy and Ilya Kofman for some in-teresting discussions on this and other topics. I gratefully acknowledge the support of PekkaPankka and from the Academy of Finland project 297258 “Topological Geometric FunctionTheory”.

2. Coding of closed geodesics

2.1. Modular surface case. We know that the conjugation classes in PSL2(Z) are in cor-respondence with the closed geodesics on the modular surface. The following application ofthe Euclidean Algorithm allows us to code the conjugation classes in PSL2(Z) in a uniqueway:

Lemma 2.1. Let A be an element of SL2(Z) which has two distinct eigenvalues in R∗+. Theconjugation class of A in SL2(Z) contains a representative of the form:

nA∏i=1

XkiY mi

where X =

(1 10 1

), Y =

(1 01 1

)and nA, ki,mi ∈ N. In addition, the representation is

unique up to cyclic permutation of the factors XkiY mi . Conversely, any product not emptyof such factors is an element of PSL2(Z) with two distinct eigenvalues in R∗+.

Consider the model of the upper half-plane for the hyperbolic space H2, provided with aFarey’s triangulation F (the ideal triangle with the vertices 0, 1 and ∞, and all its images bysuccessive reflections with respect to its sides). We know that the group of oriented isometriesof H2, which preserves F , is identified with PSL2(Z).

Let α be the axis of A oriented towards the attractive fixed point. Then α crosses aninfinity of ideal triangles (..., t−1, t0, t1, ...) of F . You can formally write a bi-infinite word:

ωA := ...LRRRLLRR...

where the kth letter is R (resp. L) if and only if the line α comes out from tk by the rightside (resp. on the left) with respect the side where it enters, in this case we will say that αturns right (resp. turns left) at tk. The word ωA contains at least one R and one L, becausethe ends of α are distinct. The image of t0 by A is a certain tm (m > 1) and ωA is periodical.

We associate the matrix X to R and Y to L, which are parabolic transformations of H2

which fix the points 0 and∞, respectively. Let B be any subword of ωA of length m, we lookat B as a product of the matrices X,Y and therefore as an element of SL2(Z). By studyingthe action of X and Y on F we can easily see that A and B are conjugated in PSL2(Z) sinceboth have strictly positive trace.

We check the uniqueness in the following way: on one hand, if A and B are conjugate, thereis an element of PSL2(Z) (preserving F) which sends the axis of A on the axis of B, thereforeA and B define the same word ωA up to translation. On the other hand, by considering the


action of X and Y on H2, we see that a product of the matrices X,Y as in the statement of

Lemma 2.1 always defines the word ωA =nA∏i=1

XkiY mi , which repeats infinitely.

Definition 2.2. We denote nA as the period of ωA, which is the same as the number of(cyclic) subwords of the form XY in ωA.

2.1.1. The continued fraction expansion of a geodesic in the modular surface. In this part, wewill specify how the sequence (k1,m1, ..., knA ,mnA) of Lemma 2.1 is related to the continuedfraction expansion of the fixed points of A. For more details on the proofs of the results onthis see [12].

The continued fraction associated with x ∈ R∗+ can be read in the Farey tiling F asfollows. We join x to a point of the imaginary axis of the upper half-plane by a hyperbolicsemi-geodesic ray. This arc crosses a sequence of triangles of F . We label this arc as beforewith L and R. In the exceptional case where the arc goes through a vertex of the triangle,we choose one or the other of the labels. The resulting sequence Ln0Rn1Ln2 ... with ni ∈ N,is called the cut sequence of x. If x > 1 the sequence begins with L, while if 0 < x < 1 thesequence begins with R. Note that the cutting sequence is independent of the initial pointon the imaginary axis. The key observation is:

Lemma 2.3. Let x > 1 with a cutting sequence Ln0Rn1Ln2 ... with ni ∈ N. Then

x = [n0;n1, n2, ...].

By the same reason 0 < x < 1 has a cutting sequence Rn1Ln2Rn3 ... with ni ∈ N, then

x = [0;n1, n2, ...].

To manage negative numbers, simply replace the negative number x with −1x = [b0; b1, b2, ...]

with b0 ≥ 0, by using an element of the modular group.

Example 2.4. Let A =

(1 1−2 1

)the matrix associated with the semi-geodesic ray α which

connects the points i and 1+i2 . Then α intersects the real axis at

√5−12 . Following α, we notice

that α has the cutting sequence LRLRLR.... Then√

5−12 = [0; 1, 1, 1, 1, ...].

Definition 2.5. We say that two numbers x = [a0; a1, ...] and y = [b0; b1, ...] have the sametails if there is p, q ∈ N such that ap+r = bq+r for all r ≥ 1. We say that they have the sametails (mod 2) if in addition p+ q is even.

To understand this definition, we need the following lemma:

Lemma 2.6. Let α and α′ be oriented geodesics lines in H2 with the same positive end pointx, then the cutting sequences of α and α′ coincide from a certain rank. In addition, if theend point of α is x = [a0; a1, ...], and for α′ is y = [b0; b1, ...], then x and y have the sametails mod 2, if and only if, there is a matrix g ∈ SL2(Z) such that g(α) = α′.

It is not difficult to see algebraically that any number whose continued fraction is almostperiodic is quadratic, meaning a solution of an equation of the form:

ax2 + bx+ c = 0 with a ∈ N∗ et b, c ∈ Z.Conversely, any quadratic number has an almost periodic continued fraction expantion. Thefollowing result makes possible to establish a relation between the fixed points of a hyperbolicisometry of SL2(Z) and the quadratic numbers:


Lemma 2.7. Let x be irrational. The following properties are equivalent:

(i) x is fixed by a hyperbolic isometry of PSL2(Z);(ii) x is quadratic.

In conclusion, an element A of PSL2(Z) is hyperbolic if and only if it is conjugated in

PSL2(Z) to an isometry of the form ωA =n∏i=1

XkiY mi , with ki,mi ∈ N and its fixed point

has the same tail (mod 2) as the quadratic number x = [0; k1,m1, k2,m2, ..., kn,mn]. Noticethat the period of the word ωA (see Definition2.2) is exactly half the period of the continuedfraction of x.

2.2. Coding figure-eight type closed geodesics on hyperbolic surfaces.

Definition 2.8. Given a hyperbolic surface Σ, we say that a closed geodesic γ in Σ is afigure-eight type closed geodesic if there exist X and Y in π1(Σ) representing distinct freehomotopy class of two disjoint simple closed curves α and β, such that γ represents theelement:

ωγ :=

nγ∏i=1

XkiY mi ,

where ki,mi ∈ N, nγ is the period of γ and XY represents the free homotopy class of anon-simple closed curve.

Notice that all figure-eight type closed geodesic relative to X,Y are contained in a subsur-face of Σ which is homeomorphic to a thrice-punctured sphere.

3. Parametrization of the canonical lift

The path we follow to study the geometry of Mγ , is by constructing a representant in thesame isotopy class as γ, just by using the coding of the word ωγ (see Lemma 2.1). Theserepresentants will be embedded into a particular branched surface. For more details on thealgorithm see [5].

Figure 1. The figure on the left is the Lorenz template T . The figure on theright shows the splitting of the template T to obtain the braid.


A template [6] is an embedded branched surface made of several ribbons and equippedwith a semi-flow. A template is characterized by its embedding in the ambient manifold andby the way its ribbons are glued (see Figure 1).

The template for the modular surface case comes with a symbolic dynamics given by thesymbols X and Y that correspond to passing through the left or through the right ear (orequivalently through the left or the right half of the branch line). There is not starting pointfor the orbit, then the words in the alphabet X,Y used to describe them are primitive upto cyclic permutation. Ghys proves in [20], relying on a theorem of Birman and Williams[6], that symbolic dynamics of the representation of a periodic orbit in T is equivalent to itsrepresentation as a word in the generator X,Y for PSL2(Z).

Figure 2. The template T inside the unit tangent bundle of the modularsurface (trefoil knot complement in S3) with the canonical lift correspondingto the word X3Y 2.

Theorem 3.1 (Ghys). The set of closed geodesics on the modular surface is in bijectivecorrespondence with the set of periodic orbits on the template T , embedded in the trefoil knotcomplement on the 3-sphere, excluding the boundary curves of T . On any finite subset, thecorrespondence is by an ambient isotopy.

Williams in [29] constructed an algorithm to find the periodic orbits inside the templateT , just from the representing word ω. In the right of Figure 1, the template has been cutopen to give a related template for braids, which inherit an orientation from the template,top to bottom.

We will ilustrate Williams’ algorithm in the Figure 3, by showing how to recover theperiodic orbit from the word ω := X4Y 3XY 2. We start by writing the 10 cyclic permutationsω = ω1, ω2, ..., ω10, in the natural order. We reorder lexicographically these 10 words usingthe rule X < Y. The new position µi is given after each ωi :


X4Y 3XY 2 1 Y 2XY 2X4Y 9X3Y 3XY 2X 2 Y XY 2X4Y 2 7X2Y 3XY 2X2 3 XY 2X4Y 3 4XY 3XY 2X3 5 Y 2X4Y 3X 8Y 3XY 2X4 10 Y X4Y 3XY 6

This determines a new cyclic order (1, 2, 3, 5, 10, 9, 7, 4, 8, 6), and induces a permutation braid,where the strand i begins with µi and ends ends with µi+1. The ith strand of the braid isan overcrossing strand if and only if µi < µi+1, otherwise it is an undercrossing strand. Inthe example, there are 5 overcrossing strands and 5 undercrossing strands, so 5 strands turnaround the left ear and 5 around the right ear. Beginning with the permutation braid andconnecting the end points of the strands with the same index, as in a closed braid, we recoverthe periodic orbit associated to the cyclic word X4Y 3XY 2 (see Figure 3). The braid obtainedis called the Lorenz braid associated with X4Y 3XY 2.

Figure 3. The Lorenz braid associatied with X4Y 3XY 2

In a Lorenz braid, two strands of overcrossing (or undercrossing) never intersect, so thepermutation associated with the overcrossing stands determine uniquely the rest of the per-mutation.

To give a general parameterization of the Lorenz braids, suppose that there are p > 1overcrossing strands. On each overcrossing strand, the position of the end will always begreater than that of the initial point. Suppose that the ithstrand begins at i and ends ati+ di. Since two undercrossing strands of never cross, we have the following series of positiveintegers:

d1 ≤ d2 ≤ ... ≤ dp−1 ≤ dp.We collect this data in the following vector:

v = 〈d1, ..., dp〉X , 1 ≤ d1 and di ≤ di+1.

The vector v determines the positions of the strands starting from X (overcrossing). Thestrands starting from Y (undercrossing) fill the remaining positions, so that all the crossingsare formed between the overcrossing strands and the undercrossing strands. In Figure 3, themiddle arrows separate the left and right strands. Each di with i = 1, ..., p is the difference


between the initial and final positions of the ith overcrossing strand. The integer di is alsothe number of strands that pass under the ith braid strand. The vector v determines theLorenz braid with n = (p+ dp) strands. All periodic orbits on the template T appear in thisway.

The overcrossing strands travel in groups of parallel strands, which are strands of the sameslope, or equivalent strands whose associated di coincide. If dµj = dµj+1 = ... = dµj+sj+1,

where sj is the number of strands in the jth group then let rj = dµj . Thus, we can write von the form:

v = 〈ds1µ1, ..., dskµk〉X = 〈rs11 , ..., r

skk 〉X , 1 ≤ si and ri < ri+1.

Note thatp = s1 + ...+ sk, d1 = r1, dp = rk.

The period of the word ω is found in terms of the braid representation by the followingnumber:

t = ]i | i+ di > p where 1 ≤ i ≤ p.It is known that the braid index of a Lorenz knot is the period of ω, a concept that was firstencountered in the study of Lorenz knots from the point of view of symbolic dynamics (see[6]).

In the example of Figure 3, 〈1, 1, 2, 4, 5〉X = 〈12, 21, 41, 51〉X . Also, p = 5, k = 4, rk = 5,n = p+ rk = 10 and the braid index is 2.

In the following result we show explicitly the coefficients of the Lorenz braid vector forinfinitely many words in X,Y . This will be used in almost every result in this paper.

Lemma 3.2. Let (ki)ni=1 ∈ Nn such that k1 + 1 < k2, ki < ki+1 for 2 ≤ i ≤ n − 1, and

ω =n∏i=1

(XkiY ). Then the associated Lorenz braid to ω is:

〈1s1 , 2s2 , ..., (n− 1)sn−1 , nsn〉X ,where si = i(kn+1−i − kn−i) for 1 ≤ i ≤ n− 2, sn−1 = (n− 1)(k2 − k1 − 1), and

sn = n(k1 + 1)− 1.

Proof: By induction over n. For the base of induction consider n = 3, it determines thefollowing permutation, obtained by reordering lexicographically the k3 + k2 + k1 + 3 wordsinduced by ωγ under cyclic permutation:

(1, ..., k3−k2, (k3−k2)+1+2j, (k3−k2)+2(k2−k1)+2+3l, (k3−k2)+2+2j, (k3−k2)+2(k2−k1)+3+3l, (k3−k2)+2(k2−k1)+1+3l),

where 0 ≤ j ≤ k2 − k1 − 1, and 0 ≤ l ≤ k1.

(1) The number of overcorssing strands shifted one place are the first k3 − k2 then s1 =k3 − k2.

(2) The number of overcorssing strands shifted two places are the ones whose ends havea j > 0 index then s2 = 2(k2 − k1 − 1).

(3) The number of overcorssing strands shifted three places are the ones whose ends hasa l index, with the exception of (k3 − k2) + 2(k2 − k1) + 1 which correspond to anundercorssing strand, then s3 = 3(k1 + 1)− 1.

If our statement is true for n = m, then after multiplyingm∏i=1

(XkiY ) with Xkm+1Y we will

modify the Lorenz braid by adding km+1 overcorssing strands:


(1) km+1 − km at the begining of the braid, then sm+1 = km+1 − km.(2) km+1−i − km−i are added in the ith collection of parallel strand for 1 ≤ i ≤ m − 2,

because one new overcrossing strand enters each collection of parallel strands of thepreviouse Lorenz braid. This implies that:

si+1 = i(km+1−i − km−i) + km+1−i − km−i = (i+ 1)(km+1−i − km−i).(3) For the penultimate and last collection of parallel overcorssing strands we will add

k2 − k1 − 1 and k1 + 1 respectively because of the new strand entering into eachcollection of parallel overcrossing strands. This implies that:

sm = (m− 1)(k2 − k1 − 1) + k2 − k1 − 1 and sm+1 = (m+ 1)(k1 + 1)− 1.

Remark 3.3. Notice that by turning over the Lorenz template we have the analogue result to

Lemma 3.2 for the wordsn∏i=1

(XY mi), such that m1+1 < m2, and mi < mi+1 for 2 ≤ i ≤ n−1.

3.1. Canonical lifts of figure-eight type closed geodesics. Given γ a figure-eight typeclosed geodesic on a hyperbolic surface, we can restrict to the pair of pants Σ0,3 where γ isfilling. We are going to construct an explicit representant of the isotopy class of γ in T 1(Σ0,3).

Lemma 3.4. The set of canonical lifts relative to figure-eight type closed geodesics on Σ0,3

with respect two different fixed punctures, is in bijective correspondence with the set of periodicorbits on the template T .Proof. Let γ be a figure-eight type closed geodesic on Σ0,3 with respect two different fixedpunctures.

First we will fix a vector field on Σ0,3, to do so consider the corresponding Lorenz braid inT , induced by the word ωγ . Notice that there is a natural projection of T to a pair of pantsΣ0,3 which is obtained by the overlaping the ears of the template T (see Figure 4).

Figure 4. The projection map from T to Σ0,3

Moreover, the periodic orbit is mapped under this projection to a figure-eight type closedcurve in minimal position and representing the same homotopy class as γ in Σ0,3. As thereis only one minimal configuration of γ up to isotopy.


The vector field ξ is obtained by extending the the oriented foliation given by the orientedarcs of γ in Σ0,3 outside the overlapping triangle of the projection map from the templateT to Σ0,3. The oriented foliation is obtain first by enclosing each arc to a family of disjointconcentric circles to the corresponding puncture, and then doing parallel copies of the closedsimple loops which converges from each side to the boundary of the piece of Σ0,3 obtainedafter splitting along a mid edge connecting the third boundary component of Σ0,3 (see Figure5).

Figure 5. The oriented foliation induced by the γ-arcs outside the non-injective piece of the projection map.

The vector field ξ induces a global section inside T 1(Σ0,3) and the canonical lift γ is isotopicto the embedding of the Lorenz braid in T associeted to the word ωγ .

4. Sequences of geodesics on the modular surface whose canonical liftcomplement volume is bounded linearly by the period

In this Section we prove, in Theorems 1.4 and 1.5, a bound for the volume of the com-plement of canonical lifts for some sequences of geodesics in the modular surface linearly interms the period of the geodesic’s continued fraction expansion . Also we give in Corollary1.3 an upper bound for volumes of some sequences of Lorenz knots complements.

To get to imagine the canonical lifts in Theorems 1.4 and 1.5, we will exemplify theresults by the canonical lift associated to the closed geodesic X11Y X10Y X8Y X5Y XY onthe modular surface. As we saw in Section 3 we can also associate it with the Lorenz braid〈11, 24, 39, 412, 59〉X (see Figure 6). In order to simplify the figures in this Section, we willremove the trefoil component from the link associated to the cusp of T 1Σmod (see Figure 2),and just focus on the Lorenz braid.


Vol(Mγn) < 8v3(5n+ 2),



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

X Y

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Figure 6. X11Y X10Y X8Y X5Y XY.

We will use the fact [26] that if a compact orientable hyperbolic 3-manifold M is obtainedby Dehn filling another hyperbolic 3-manifold N, then the volume of M is less than thevolume of N. So the key idea to give the upper bound is to construct a link Lγ associatedwith γ in S3 such that by Dehn filling along some components of Lγ , we get Mγ , and also

by noticing that S3 \ Lγ is homeomorphic to the link complement on the circle bundle overa punctured sphere whose projection to the base surface is a pair of closed curves and thevolume can be bounded from above by the self-intersection number of the two closed curves.

Before stating the prove of Theorem 1.4, we recall some tools that will be used in the proofof this result.

4.1. Vertical rings. Here we will construct some parallel rings around the ribbons of theLorenz template, that will help us to reduce the complexity of the Lorenz braid under someDehn filling surgery.

Remember that in general the Lorenz braid is associated to a vector 〈rs11 , ..., rskk 〉X . Equiv-

alently if we change the setting of the braid using the rule Y < X, we also have the vector ofthe same size 〈lq11 , ..., l

qkk 〉Y .

Let mX be the larger index such that there is a strand coming out of the interval(mX−1∑k=0

sk,

mX∑k=0

sk

)

which remains in the X-band. So let mY be the analogous index but for the Y -band.

We begin by adding the following mX + 1 vertical rings AXi delimited by unknots parallelto the X-band, which enclose the intervals(

i−1∑k=0

sk + 3/4,i∑

k=0

sk + 1/4

)where s0 = 0 and i < mX ,


and for i = mX we have two cases:(mX−1∑k=0

sk + 3/4,

mX∑k=0

sk + 1/4

), if smX ≤ rmX .

Or: (mX−1∑k=0

sk + 3/4,

mX−1∑k=0

sk +

⌊smXrmX

⌋rmX + 1/4

), if smX > rmX .

Finally, for i = mX + 1 we add a last vertical ring which encloses the interval:(mX∑k=0

sk + 3/4, p+ 1/4

), if smX ≤ rmX .

Or: (mX−1∑k=0

sk +

⌊smXrmX

⌋rmX + 3/4, p+ 1/4

), if smX > rmX .

In case there is no such mX we consider only the interval (0, p+ 1/4).

Note that the strands coming out of the same ring AXi , for i ≤ mX , are parallel.

We will do the same construction for the Y -band this time by considering the vector〈lq11 , ..., l

qkk 〉Y . So we will have the family vertical rings AYj

mY +1j=1 in the Y -band.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

X Y

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Figure 7. The vertical rings of X11Y X10Y X8Y X5Y XY.

Lemma 4.1. The total number of vertical rings is at most two times the braid index plustwo of the Lorenz braid.

Proof: As each si is associated to a family of parallel strands, then we know that thereare at least mX free spaces for arcs coming from the Y -band which fall into the X-band.As the period of the X-vector is the same as the Y -vector associated to γ we know thatthere are exactly the period number of strands comming from Y -band to the X-band, somX ≤ t. Finally, the number of vertical ring is mX + 1 +mY + 1, then we obtain the wantedestimate.


In the following Subsection we explain the Dehn surgery that we will apply along theboundary components of the vertical rings, to relate γ with a simplified knot.

4.2. Annular Dehn surgery. Given A an embedded ring into an oriented 3-manifold N,let ∂A = L+1 t L−1. Let us orient L+1 and L−1 with a compatible orientation of N . Letmi, li, i = ±1, be a basis where mi is the meridian over ∂N(Li) and li a longitud over∂N(Li) induced by A for i = ±1 (i.e. li = ∂N(Li) ∩A).

With these basis, the Dehn filling along the slope 1/n in L+1 and −1/n in L−1 givesan homeomorphism N ∼= N(L+∪L−)

(1n ,−1n

). This homeomorphism is obtained by cutting

N ∼= N(L+∪L−) along A then wind n-times, and trivialy filling the boundary components ofA. For example, see the Figure 8 for the case n = −2 an its effect on the curve that passesthrough A. Notice that this homeomorphism is the identity outside a normal neighborhoodof A.

Figure 8. The annular Dehn surgery surgery makes the green curve to twistalong A.

If an oriented surface S passes through the ring A along a simple closed curve c which isessential in A, then the homoemorphism

N → N(L+∪L−)

(1

n,−1

n

)restricted to S is a composition of a n-Dehn twist. If n > 0 and if we look S on the sameside as L+, then the Dehn twist is on the opposite sense.


We now prove the upper bound of Theorem 1.4:

Proof of Theorem 1.4: Let (ki)ni=1 ∈ Nn such that k1 + 1 < k2, ki < ki+1 for 2 ≤ i ≤

n − 1, and γ the closed geodesic on Σmod associated ton∏i=1

(XkiY ), where X =

(1 10 1

)and

Y =

(1 01 1

)in PSL2(Z) (see Lemma 2.1) .

By Lemma 3.2 the associated Lorenz braid of γ is:

〈1s1 , 2s2 , ..., (n− 1)sn−1 , nsn〉X ,

where si = i(kn+1−i − kn−i) for 1 ≤ i ≤ n− 2, sn−1 = (n− 1)(k2 − k1 − 1), and

sn = n(k1 + 1)− 1.

B

Figure 9. The link τ ∪B ∪ ∂LAXi , ∂RAXin+1i=1 ∪ ∂LAY1 , ∂RAY1

The link Lγ consist of 2n+ 7 components, where 2n+ 6 of them are:

(1) The trefoil knot corresponding to the boundary of T 1Σmod, denoted by τ ,(2) the boundary components of each vertical ring AXi and AY1 in T 1Σmod, denoted by∂LAXi , ∂RAXi

n+1i=1 and ∂LAY1 , ∂RAY1,

(3) an unknot B enclosing only the links in (2) (see Figure 9).

Before constructing the last component of Lγ notice that the link complement formed by(1), (2), and (3) is homeomorphic to the complement of the link τ on T 1Σ0,2n+5 and τ projectsinjectively to a closed curve τ on Σ0,2n+5 (see Figure 10).

We will construct the last knot component of Lγ , denoted by σγ inside a normal neigbor-hood of the punctured disk bounded by the knot B. So it is enough to draw its projectionclosed curve, denoted by σγ , on Σ0,2n+5 and specify the crossing information with itself.

We start by marking intervals IXi and IY1 on Σ0,2n+5 whose preimage under the projectionmaps are the corresponding vertical rings AXi and AY1 .

(1) From each IXi with 1 ≤ i ≤ n− 1 we draw an arc αi starting at xi ∈ IXi to yi ∈ IY1

passing one time through each interval IXj with i ≤ j ≤ n+ 1 (see Figure 12). Drawall arcs in a way that are disjoint from each other.


Figure 10. The closed curve τ on Σ0,2n+5.

x1

x1A

x2

x2A

x3

x3A

x4

x4A

x5

x5A x6

A

y1

Y1A

y2y3y4y5y

Figure 11. The marking of the intervals associated to X11Y X10Y X8Y X5Y XY.

x1

x1A

x2

x2A

x3

x3A

x4

x4A

x5

x5A x6

A

y1

Y1A

y2y3y4y5y

Figure 12. αi arcs with respect to Xk1Y Xk2Y Xk3Y Xk4Y Xk5Y.

(2) Construct an arc αn from a point xn ∈ IXn to a point yn ∈ IY1 disjoint from theprevious arcs and intervals.

(3) Connect the point yi with xi+1 with 1 ≤ i ≤ n − 1 with an arc βi disjoint from theintervals and parallel between them (see Figure 13).

(4) Draw an arc βn from yn to x1 intersecting once each of the other βi arcs.

x1

x1A

x2

x2A

x3

x3A

x4

x4A

x5

x5A x6

A

y1

Y1A

y2y3y4y5y

Figure 13. The closed curve σγ associated to Xk1Y Xk2Y Xk3Y Xk4Y Xk5Y.

Once we joint all arcs we will have the closed curve σγ . The crossing information of thelink σγ is given by the fact that the strand βn is under the other βi strands (see Figure 14).


Figure 14. The knot σγ associated to Xk1Y Xk2Y Xk3Y Xk4Y Xk5Y.

Remark 4.2. The self-intersection number of the closed curve σγ is n− 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

X Y

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Figure 15. Lγ and the vertical rings.

Claim 4.3. Mγ is obtained by making annular Dehn filling along the boundary of the verticalrings components of Lγ and trivial Dehn filling on B.

Proof of claim: First we isotope σγ such that the projection of it to a plane parallel to thevertical rings (see Figure 15) is in a suitable position for the later description of the Dehnsurgeries.

(1) For the vertical rings AXi with i < n − 1 we will do an annular Dehn filling of type1

kn+1−i−kn−i .

(2) For AXn−1 we will do an annular Dehn filling of type 1k2−k1−1 .

(3) For AXn we will do an annular Dehn filling of type 1k1.

(4) For AXn+1 and AY1 we will do an annular Dehn filling of type 1.

The effect of the annular Dehn fillings on the vertical rings is inducing parallel overcrossingstrands whose slope for each vertical ring is been shifted by one for each βi that intersects avertical ring. Notice that for each AXi is intersected by the arc βi.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

X Y

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Figure 16. After annular Dehn filling along all the vertical rings associatedwith X11Y X10Y X8Y X5Y XY.

(1) For the vertical rings AXi with i < n − 1 we will obtain exactly i(kn+1−i − kn−i)overcrossing strands with the same slope.

(2) For AXn−1 we will obtain exactly (n − 1)(k2 − k1 − 1) overcrossing strands with thesame slope.

(3) For AXn and AXn+1 we will obtain exactly n(k1 + 1) − 1 overcrossing strands withthe same slope.

Between each set of parallel overcrossing strands with the same slope there is one un-dercrossing. After isotopy of the knot in Figure 16 we get the Lorenz braid associated toX11Y X10Y X8Y X5Y XY (see Figure 6).

Finally, let Γ = τ ∪ σγ , by [26]

Vol(Mγ) < ‖T 1(Σ0,2n+5)Γ‖,and by ([11], Thm 1.5)

‖T 1(Σ0,2n+5)Γ‖ ≤ 8v3i(Γ,Γ) ≤ 8v3(5n+ 2),

giving us an upper bound of the volume depending linearly only on n.

Using again [26] to trivially Dehn fill τ in Theorem 1.4, we have the follow upper boundfor some sequences of Lorenz knots Kn in S3.

Corollary 1.3. There exist a sequence Kn of Lorenz knots in S3 such that n is the braidindex of Kn, and

Vol(S3 \Kn) ≤ 8v3(5n+ 2),

where v3 is the volume of a regular ideal tetrahedron. If Kn is not hyperbolic, Vol(S3 \Kn) isthe sum of the volumes of the hyperbolic pieces of S3 \Kn.

Remark 4.4. Notice that Corollary 1.4 improves the upper bound for Lorenz knots, withbraid index n greater than 33, found in ([9] , Theorem 1.7) which is at least v3((n − 1)2 +9(n − 1) − 8). Nevertheless, in general the braid index of a knot or link gives no indication


of its volume’s complement. For example, in [9] they show there exist Lorenz knots witharbitrarily large braid index and yet bounded volume. The reverse result is also known, forexample, closed 3-braids can have unbounded volume [18].

Finally, as a consequence of Theorem 1.4 in [23], we have that up to a constant, Theorem1.4 is sharp for a sequence of closed geodesics on the modular surface.


v3n

12≤ Vol(Mγn) ≤ 8v3(5n+ 2),


Proof: The sequence can be obtained for any infinite subsequence of the following sequenceof closed geodesics:

n∏i=1

(X6ki+1Y ) where ki ∈ NN, and ki < ki+1.

The upper bound for the volume of the corresponding canonical lift complements is aconsequence of Theorem 1.4 and the lower bound is proven in ([23], Theorem 1.4).

5. Sequences of closed geodesics on hyperbolic surfaces whose volumecomplement is bounded by the length

In this Section we lift the sequences of closed geodesics on the modular surface found inTheorem 1.4 and Corollary 1.5 to infinitely many punctured hyperbolic surfaces and comparethe volume of the corresponding canonical lift complement with the geodesic length of thelift.

Remark 5.1. We briefly recall that in the case A ∈ PSL2(R) is a hyperbolic element of trace

t, the eigenvalues of A are −t±√t2−4

2 . Let λA be the eigenvalue satisfying |λA| > 1. Then, thelength of the closed geodesic determine by A is 2 ln |λA|.

5.1. Volume’s upper bound. Here we prove an upper bound for the volumes of canoni-cal lift complements relative to filling sets of closed geodesics on infinitely many puncturedhyperbolic surfaces.

Corollary 1.1. Given a hyperbolic metric ρ on a punctured hyperbolic surface Σ, of genus gwith k punctures (k ≡ 2 (mod g) if g ≥ 2) and let dΣ := max6gk, 6(k − 3), 6. Then thereexist a constant Cρ > 0 and a sequence γn of filling finite sets of closed geodesics on Σ withat most dΣ elements in each set γn, and `ρ(γn)∞, such that


Cρ`ρ(γn)

W(`ρ(γn)Cρ− 2) + 2

,


Proof: First we prove the result for the modular surface case. Let γn be the unique closedgeodesic on Σmod, whose corresponding matrix representant is:


An :=

n∏i=1

(XkiY ),

where (ki)ni=1 ∈ Nn such that k1 + 1 < k2, ki < ki+1 for 2 ≤ i ≤ n − 1, X =

(1 10 1

),

Y =

(1 01 1

), and denote the hyperbolic metric on the modular surface by ρ0.

Claim 5.2. For all n ∈ N we have that:

n ≤ e`ρ0(γn)

W(`ρ0 (γn)

2 − 2) .

Proof of claim: Let ηn be the unique closed geodesic on Σmod, whose corresponding matrix

representant isn∏i=1

(XiY ) = Bn :=

(an bncn dn

), then:(

an bncn dn

)=

((n+ 1)an−1 + ncn−1 (n+ 1)bn−1 + ndn−1

an−1 + cn−1 bn−1 + dn−1

).

Let us denote zn := an + bn + cn + dn then:

(n+ 1)zn−1 ≤ zn and zn−1 ≤ TraceBn.

Therefore,5

2n! ≤ TraceBn.

Notice that the eigenvalue of An whose absolute value is bigger than one, denoted as λAn , isbounded as follows:

TraceBn2

≤ |λBn |.

Finally, by Remark 5.1 we have that,

2 ln(n!) ≤ `ρ0(ηn) ≤ `ρ0(γn).

By using the inequality (see [22]),

√2πn

(ne

)n≤ n!,

and the Lambert W function, we have that:

n ≤ e`ρ0(γn)− e ln(2πe)

2W(`ρ0 (γn)−ln(2πe)

2

) ≤ e`ρ0(γn)

W(`ρ0 (γn)

2 − 2) .

By Theorem 1.4 and Claim 5.2 we have that:

Vol(Mγn) ≤ 8v3

(5e`ρ0(γn)

W (`ρ0 (γn)

2 − 2)+ 2

).

We can construct a finite covering map p from any punctured hyperbolic surface of genusg with k punctures Σ (the only restriction is that k = 2 (mod g), if g > 1) to the modularsurface and of degree dΣ := max6gk, 6(k − 3), 6 (see proof of Theorem 1.1 in [4]).


Let γn be the finite set of closed geodesics on Σ, obtained as the preimage under p of theclosed geodesic γn considered in the modular surface case (see Theorem 1.4). Moreover, ifwe pullback the hyperbolic metric to Σ under p, we have that `p∗(ρ0)(γn) = dΣ`ρ0(γn), anddenote the metric p∗(ρ0) as ρ.

Then by Theorem 1.4,

Vol(Mγn) = dΣ Vol((T 1Σmod) \ γn) ≤ dΣ8v3

(5e`ρ0(γn)

W (`ρ0 (γn)

2 − 2)+ 2

).

Finally,


5e`ρ(γn)dΣ

W(`ρ(γn)

2dΣ− 2) + 2

.

To proof this result for any hyperbolic metric on Σ, follows from the fact that any pair ofhyperbolic metrics on a hyperbolic surface are bi-Lipschitz (see for example [4], Lemma 4.1).

Remark 5.3. Notice that for large n the set γn in Corollary 1.1 consists of non-simplegeodesics by [19].

5.2. Volume’s lower bound. In this Subsection we improve the estimation of the lowerbound found in ([23], Theorem 1.3), but the proof follows the same steps. For the self-completeness of this paper, we recall the proof with the improved calculation.

Lemma 5.4. Given a hyperbolic metric ρ on the once -punctured torus Σ1,1, there exist asequence γn of filling closed geodesics on Σ with `ρ(γn)∞ such that,

Vol(Mγn) ≥ 2v3

3

Cρ`ρ(γn)

W(`ρ(γn)Cρ

) − 9

,

where Cρ depends on the hyperbolic metric ρ, v3 is the volume of a regular ideal tetrahedronand we use the Lambert W function.

Proof. For the sake of concreteness, we will start proving the result for a particular hyper-bolic metric ρ0 on Σ1,1. Fix the following representation ρ : π1(Σ1,1) := 〈a, t〉 → PSL2(C)such that

ρ(a) = A =

( √2 1 +

√2

−1 +√

2√

2

)and ρ(t) = T =

(−1 +

√2 0

0 1 +√

2

).

Consider the splitting of Σ1,1 along a simple close geodesic associated to A, then we have theHNN extension of π1(Σ1,1, p) = 〈a, b, t | tat−1 = b〉, with:

ρ(b) = TAT−1 =

( √2 −1 +

√2

1 +√

2√

2

).

Notice that ab−1 is freely homotopic to a simple loop that is retractable to the puncture ofΣ1,1.

For each n ∈ N we define the following cyclically reduced sequence (see [23], Subsection4.1):

(g1, t, g2, t, ..., t, g12·(3n−1), t) where gi ∈ 〈a, b〉.


Moreover gi12·(3n−1)i=1 is the set of different reduced words in 〈a, b〉 starting with b or b−1,

and ending with a or a−1, which have word length at most n and at least 4.

By Theorem 1.5 in [23], the number of homotopy classes of arcs in each nth−sequence

is equal to ]gi12·(3n−1)i=1 = 12 · (3n − 1). Let γn be the unique geodesic associated to the

product of the nth−sequence. Each one of it belongs to different mapping class group orbits,because in this case the self-intersection number of γn is bounded from below by the numberof terms of the sequence, which grows exponentially. Therefore, by Lemma 4.7 in [23], γnis a sequence of filling closed geodesics on Σ1,1 with `ρ0(γn)∞.

By Theorem 1.5 in [23], we have that:

(5.1) v3 · 6 · (3n − 1) =v3

2] homotopy clases of γn-arcs ≤ V ol(Mγn).

Using the trace of the matrices we can calculate the length of the geodesics associated to aband t. In this way we give the following upper bound of the length of γn by using the wordlength with respect the generating set a, b, t:

`ρ0(γn) ≤ 1.15 · |g1tg2t...tg12·(3n−1)t| ≤ 1.15 · 12 · 3n(n+ 3) ≤ kρ03n+2(n+ 2).

Then,ln(3)`ρ0(γn)

kρ0

≤ ln(3)e(n+2) ln(3)(n+ 2).

By using the Lambert W function we have that:

W

(ln(3)`ρ0(γn)

kρ0

)≤ ln(3)(n+ 2),

soln(3)`ρ0 (γn)

kρ0

9W(

ln(3)`ρ0 (γn)kρ0

) =eW(

ln(3)`ρ0 (γn)

kρ0

)9

≤ 3n.

Finally we have that,

Vol(Mγn) ≥ 2v3

3

(Cρ0`ρ0(γn)

W (Cρ0`ρ0(γn))− 9

),

where Cρ0 = ln(3)kρ0

. The proof of this result for any hyperbolic metric, follows from the fact

that any pair of hyperbolic metrics on a hyperbolic surface are bi-Lipschitz ([4], Lemma4.1).

To generalize Lemma 5.4 to any hyperbolic surface Σ, we follow the method found in ([23],Subsection 4.2) to prove Theorem 1.3 in [23].

Theorem 5.5. Given a hyperbolic surface Σ with hyperbolic metric ρ, there exist constantsCρ, δρ > 0 and a sequence γn of filling closed geodesics on Σ with `ρ(γn)∞ such that,

Vol(Mγn) ≥ 2v3

3

Cρ`ρ(γn)− δρW(`ρ(γn)Cρ

) − 9

,

where Cρ, δρ depend on the hyperbolic metric ρ, v3 is the volume of a regular ideal tetrahedronand we use the Lambert W function.


Proof. If the surface Σ contains a once-punctured torus, choose a separating closed geodesicβ such that Σ \ β := Σ1

∐Σ2, where Σ1 is homeomorphic to Σ1,1. Choose a pair of simple

closed geodesics α and τ on Σ1 such that i(α, τ) = 1. Consider the closed geodesics αnon Σ1 homotopic to the closed curves defined in Theorem 5.4. Extend α, β to a pantsdecomposition Π on Σ. And fix η0 a filling closed geodesic on Σ2. Then we construct γn thefilling closed geodesics (Claim 4.10 in [23]) associated to:

αn ? η0.

By Theorem 1.5 in [23], we have that:

(5.2) v3 · 6 · (3n − 1) ≤ v3

2

∑P∈Π

]homotopy classes of γ-arcs in P ≤ V ol(Mγn).

The last part of the proof consist in rewriting the combinatorial lower bound in terms of thelength ofies γn. By using the word length with respect the generating set α, τ we can findconstants Cρ and δρ := Cρ`ρ(η0) that only depends on the metric ρ, such that:

2v3

3


) − 9

≤ 2v3

3

Cρ`ρ(αn)

W(`ρ(αn)Cρ

) − 9

≤ v3 · 6 · (3n − 1).

Finally, by the inequality (5.2) we have that:

Vol(Mγn) ≥ 2v3

3


) − 9

.

In the case where Σ is a k−punctured sphere, we consider π : Σ′ → Σ a finite cover where Σ′

contains a once-punctured torus, and the induced covering π from PT 1Σ′ to PT 1Σ. Considerthe previous geodesics γn on Σ′ and π(γn) in Σ. Then the volume of the complement

of π(γn) on PT 1Σ is a constant (given by the degree of the covering) times the volumeof the complement of the canonical lifts of all the translates of γn under the group of decktransformations on PT 1Σ′, which is grater than the volume corresponding to the complementof γn on PT 1Σ′. So the filling closed geodesics π(γn) in Σ is the sequence wanted.

5.3. Volume’s lower and upper bound. The following result witnesses the sharpness ofthe length bounds found in the previous Subsections, for volumes of canonical lifts relativeto some sequence of filling sets of closed geodesics on infinitely many hyperbolic surfaces.

Corollary 1.2. Given a hyperbolic metric ρ on a punctured hyperbolic surface Σ, of genus gwith k punctures ( k ≡ 2 (mod g) if g ≥ 2) and let dΣ := max6gk, 6(k − 3), 6. Then thereexist a constant Cρ > 0 and a sequence γn of filling finite sets of closed geodesics on Σ,with at most dΣ elements in each set γn and `ρ(γn)∞ such that

dΣv3

12

Cρ`ρ(γn)− 32

W(`ρ(γn)Cρ

) − 3

2

≤ Vol(Mγn) ≤ 8dΣv3

Cρ`ρ(γn)

W(`ρ(γn)Cρ− 2) + 2

.


Proof: First we prove the result for the modular surface case. Let γn be the unique closedgeodesic on Σmod, whose corresponding matrix representant is:


An :=

n∏i=1

(X6i+1Y ),

where X =

(1 10 1

)and Y =

(1 01 1

). and denote its hyperbolic metric by ρ0.

Claim 5.6. For all n ∈ N we have that:`ρ0 (γn)

14 − 32

W (`ρ0(γn))− 3

2≤ n,

where Cρ0 is a positive constant.

Proof of claim: Let An :=

(an bncn dn

), then:(

an bncn dn

)=

((6n+ 2)an−1 + (6n+ 1)cn−1 (6n+ 2)bn−1 + (6n+ 1)dn−1

an−1 + cn−1 bn−1 + dn−1

).


zn ≤ 6(n+ 1)zn−1 and TraceAn ≤ 6(n+ 1)zn−1.

Therefore,

TraceAn ≤ 6n+1(n+ 1)!.


TraceAn2

≤ |λAn | ≤ TraceAn.

Then, by Remark 5.1,

`ρ0(γn) ≤ 14 ln((n+ 1)!).

By using the inequality (see [22]),

n! ≤ e√n(ne

)n,


`ρ0 (γn)14 − 3

2

W (`ρ0(γn))− 3

2≤

`ρ0 (γn)14 − 3

2

W(`ρ0 (γn)

14e − 32e

) − 3

2≤ n

Let Σ be a puntured hyperbolic surface of genus g with k punctures (the only restrictionthat k = 2 (mod g), if g > 1). We use the same sequences of finite sets of closed geodesicsγn considered in Corollary 1.1, and the pull-back hyperbolic metric of Σ induced by thedΣ-fold covering map to Σmod.

By Theorem 1.5, Corollary 1.1 and Claim 5.6 we have that:

dΣv3

12

( `ρ(γn)14dΣ

− 32

W (`ρ(γn))− 3

2

)≤ Vol(Mγn) ≤ 8dΣv3

5e`ρ(γn)dΣ

W(`ρ(γn)

2dΣ− 2) + 2

.


6. On the thrice-punctured sphere case

In this Section we prove a lower bound for the volume of canonical lift complements asso-ciated to figure-eight type closed geodesics on a thrice-punctured sphere Σ0,3 (see Subsection2.2). Notice that this case is not covered by the lower bound for the volume of canonical liftcomplements of geodesics on surfaces in ([23], Theorem 1.5) because it has a trivial pantsdecomposition.

Theorem 1.6. Given a thrice-punctured sphere Σ0,3, and γ a figure-eight type closed geodesicwith respect to X and Y (two free-homotopy classes of distinct punctures in Σ0,3), we havethat:

Vol(Mγ) ≥ v3

2(]exponents of X in ωγ+ ]exponents of Y in ωγ − 2),

where v3 is the volume of the regular ideal tetrahedron, ωγ is the cyclically reduced wordrepresenting the conjugacy class of γ in 〈X,Y 〉 ⊂ π1(Σ0,3).

The lower bound is obtained in terms of combinatorial data coming from the geodesic andan essential simple geodesic arc connecting the remaining puncture to itself Σ0,3.

Given a punctured disk D with a marked point in the boundary x ∈ ∂D we say that twoarcs α, β : [0, 1] → D with α(0, 1) ∪ β(0, 1) ⊂ ∂D are in the same homotopy class in D,if there exist an homotopy h : [0, 1]1 × [0, 1]2 → D such that:

h0(t2) = α(t2), h1(t2) = β(t2) and h([0, 1]1 × 0, 1) ⊂ ∂D \ x.

Notice that each homotopy class is determine by the winding number relative to the punctureof the closed curve obtained by connecting the ends of the arc along the arc on ∂D \ x.

Remark 6.1. Up to isotopy, for in the family of arcs without intersection there is only oneconfiguration of such arc in D. This is shown in Figure 17 an has winding number 0. The2 in the lower bound of Theorem 1.6 comes from the fact that such configurations have atmost 1 homotopy class of γ-arcs on D.

Figure 17. The only γ-arc configuration on D up to homotopy classes whoseγ-arcs are simple arcs without intersections.

Before stating the main result to prove Theorem 1.6 we recall some definitions.

If N is a hyperbolic 3-manifold and S ⊂ N is an embedded incompressible surface, we willuse N\\S to denote the manifold that is obtained by cutting along S; it is homeomorphic tothe complement in N of an open regular neighborhood of S. If one takes two copies of N\\S,and glues them along their boundary by using the identity diffeomorphism, one obtains thedouble of N\\S, which is denoted by d(N\\S).


Definition 6.2. Let D be a punctured disk which is induced by spliting Σ0,3 along a simplegeodesic arc connecting a puncture with itself and γ a figure-eight type geodesic on Σ0,3 suchthat D ∩ γ is a finite set of geodesic arcs αi connecting ∂D. Then denote,

Dγ as T 1D \⋃i

αi ∼= (S1 ×D) \⋃i

αi.

And define,d(Dγ),

as gluing two copies of T 1D \⋃i αi along the punctured sphere coming from

∂T 1D \(∂T 1D ∩

(⋃i

αi)),

by using the identity. Moreover, d(Dγ) is homeomorphic to

(S1 × Σ0) \⋃i

d(αi),

where Σ0 is a thrice-punctured sphere and d(αi) is a knot in S1 × Σ0 obtained by gluing αialong the two points ∂T 1D ∩ αi by the identity.

Figure 18. The projection of d(Dγ) (after an homotopy of γ-arcs to a mini-mal position configuration) over Σ0.

Let M be a connected, orientable 3-manifold with boundary and let S(M ;R) be the sin-gular chain complex of M. More concretely, Sk(M ;R) is the set of formal linear combinationof k-simplices, and we set as usual Sk(M,∂M ;R) = Sk(M ;R)/Sk(∂M ;R). We denote by ‖c‖the l1-norm of the k-chain c. If α is a homology class in Hsing

k (M,∂M ;R), the Gromov normof α is defined as:

‖α‖ = inf[c]=α‖c‖ =

∑σ

|rσ| such that c =∑σ

rσσ.

The simplicial volume of M is the Gromov norm of the fundamental class of (M,∂M) in

Hsing3 (M,∂M ;R) and is denoted by ‖M‖.The key ingredient to prove Theorem 1.6 is the following result due to Agol, Storm and

Thurston ([3], Theorem 9.1):

Theorem (Agol-Storm-Thurston). Let N be a compact manifold with interior a hyperbolic3-manifold of finite volume. Let S be an embedded incompressible surface in N. Then

Vol(N) ≥ v3

2‖d(N\\S)‖.


Definition 6.3. Let η be a simple geodesic arc connecting a puncture with itself on Σ0,3,then we define the following embedded surface on Mγ :

(Tη)γ := (Tη) \ Nγwhere Tη is the pre-image of η under the map T 1(Σ0,3)→ Σ0,3.

We now prove the lower bound for the volume of the canonical lift complement:

Proof of Theorem 1.6. Let η be a simple geodesic arc connecting a puncture with itselfon Σ0,3, inducing a decomposition on DR and DL. Consider the surface (Tη)γ in Mγ

Claim 6.4. The embedded surface (Tη)γ in Mγ is π1-injective.

Figure 19. The punctured sphere (Tη)γ .

Proof of Claim 6.4. Let us split Mγ by (Tη)γ , into pieces DRγ and DL

γ . By Van Kampen it

is enough to show the π1-injectivity for the surface (Tη)γ in DRγ .

If the surface (Tη)γ is not π1-injective in DRγ , then by the Loop Theorem, there is a disk

D0 whose interior is in the interior of DRγ and whose innermost intersection with (Tη)γ is an

essential simple closed curve α in (Tη)γ .

As Tη is an incompressible surface in T 1(Σ0,3), then α bounds a disk D1 in Tη that intersectsγ. So by the irreducibility of T 1(Σ0,3), the embedded sphere formed by D0∪D1 would bounda ball B in T 1(Σ0,3).

By the periodicity of γ, there is at least one arc of γ inside B with endpoints at D1. Thisarc is homotopic relative to the boundary to a simple arc in D1 with the same endpoints.Such homotopy inside T 1(Σ0,3), induces a homotopy on Σ0,3 that reduces the intersectionnumber between γ and η, contradicting the fact that γ and η are in minimal position.


From ([3], Theorem 9.1) we deduce that:

Vol(Mγ) ≥ v3

2‖d(Mγ\\Tη)‖ =

v3

2

∑P∈Π

‖d(Dγ)‖.

For any piece D ∈ DR, DL we have:

v3]cusps of d(Dγ)hyp ≤ Vol(d(Dγ)hyp) ≤ v3‖d(Dγ)hyp‖ = v3‖d(Dγ)‖

where d(Dγ)hyp is the atoroidal piece of d(Dγ), i.e., the complement of the characteristicsub-manifold, with respect to its JSJ-decomposition. The first and second inequality comefrom [2] and [21] respectively.

Notice that if ω1 and ω2 are a pair of homotopic γ-arcs on D then their respective canonicallifts ω1 and ω2 are isotopic γ-arcs in T 1D. Indeed, let ω1 and ω2 be lifts on the universal coverstarting in the same fundamental domain. Take an homotopy of geodesics that varies from ω1

to ω2 and project this homotopy to Σ0,3. This will give us a homotopy h of geodesics arcs htthat start in ω1 and end in ω2. The image of the geodesic homotopy does not intersects otherγ-arcs because of local uniqueness of the geodesics. Then we have that the geodesic homotopyinduces an isotopy in T 1D between their corresponding canonical lifts. Moreover, the imageof the geodesic homotopy induces a co-bounding annulus between the corresponding knotscoming from the double of the corresponding canonical lifts in d(T 1D). Therefore contributingto a Seifert-fibered component, where the JSJ-decomposition separates this set of parallelknots from the rest of the manifold (see Figure 20).

Let Ω be the subset of γ-arcs on D having one arc for each homotopy class of γ-arcs on D.This means that d(Dγ)hyp ∼= d(D

Ω)hyp. Moreover, d(D

Ω) can be seen as a link complement

in S1 ×Σ0, see Definition 6.2, whose projection to Σ0 is a union of closed loops transversallyhomotopic to a union closed loops in minimal position. By using ([11], Theorem 1.3), we havethat the atoroidal piece of d(D

Ω) corresponds to Σ0 if d(Ω) are non simple closed curves.

(1) If the Ω-arc configuration onD is the one of Remark 6.1, then we have that d(DΓ)hyp =

∅ and Remark 6.1 also gives us:

v3(]homotopy classes of γ-arcs in D − 1) ≤ v3]cusps of d(Dγ)hyp.

(2) If the Ω-arc configuration on D is not the one of Remark 6.1, then there is at leastone geometric intersection point on the projection of the link complement d(D

Ω) to

Σ0.

Figure 20. The JSJ-decomposition of d(Dγ) of Figure 18.


By ([11], Theorem 1.3) we conclude that d(Dγ)hyp 6= ∅. We will now define an injectivefunction:

γ-arcs inT 1D

ϕ−→

cusps ofd(Dγ)hyp

where the target can be decomposed as:

cusps ofd(Dγ)hyp

=

splitting tori of theJSJ-decomposition of

d(Dγ)

q

cusp ind(Dγ) ∩ d(Dγ)hyp

The function ϕ is defined as follows: if the cusps in d(Dγ) are induced by the γ-arc in

T 1D belonging to the characteristic sub-manifold of d(Dγ), ϕ maps it to a splitting toriconnecting the hyperbolic piece with the component of the characteristic sub-manifold whereit is contained. Otherwise, the cusp belongs to d(Dγ)hyp and ϕ sends it to itself, see Figure

20. Assume that there are more isotopy classes of γ-arcs in T 1D than the number of cusps ofd(Dγ)hyp. Then, there are two tori, associated with non-isotopic γ-arcs in T 1D, that belongto the same connected component of the characteristic sub-manifold. Since each componentof the characteristic sub-manifold is a Seifert-fibered space over a punctured surface we havethat all such arcs correspond to regular fibres. Thus, they are isotopic in the correspondingcomponent hence isotopic in T 1D, contradicting the fact that they were not isotopic.

Finally, since two isotopic γ-arcs in T 1D induce a homotopy between their projections inD. Then for the case for d(Dγ)hyp 6= ∅, we have that:

v3]homotopy classes of γ-arcs in D ≤ v3]cusps of d(Dγ)hyp,as homotopy classes of γ-arcs in DR (resp. DL) is given by the winding numbers obtainedfrom the exponents of X (resp. Y ) relative to the primitive word in the semigroup generatedby X and Y representing γ. Then,

]homotopy classes of γ-arcs in DR = ]exponents of X in ωγ,and

]homotopy classes of γ-arcs in DL = ]exponents of Y in ωγ.

As a consequence of Theorem 1.6 we prove a version of Theorem 1.2 for sequences of closedgeodesics, without using the finite covering argument:

Corollary 6.5. Given a hyperbolic metric ρ on Σ0,3, then there exist constants Cρ, δρ > 0and a sequence γn of closed geodesics in Σ0,3, with `ρ(γn)∞ such that

v3

2

`ρ(γn)Cρ− δρ

W (Cρ`ρ(γn))− 3

2

≤ Vol(Mγn) ≤ 8v3

5Cρ`ρ(γn) + δρ

W(`ρ(γn)Cρ− 2) + 8

,

where v3 is the volume of a regular ideal tetrahedron, we use the Lambert W function, andCρ, δρ depend on the metric ρ and γ0.

Proof: We will start proving the result for a particular hyperbolic metric ρ0 on Σ0,3, byfixing the following representation ψ : π1(Σ0,3) := 〈x, y〉 → PSL2(R) such that

ψ(x) = X =

(1 20 1

)and ψ(y) = Y =

(1 02 1

).


Notice that x and y represent the free homotopy class of two different punctures of Σ0,3. Letγk be the unique closed geodesic on Σ, whose corresponding matrix representant under ψ is:

An :=n∏i=1

(Xmi+rY ) where m ∈ N and 0 ≤ r < m.

Claim 6.6. For all n ∈ N we have that:`ρ0 (γn)Cρ0

− δρ0

W (Cρ0`ρ0(γn))− 3

2≤ n ≤ Cρ0`ρ0(γn) + δρ0

W(`ρ0 (γn)Cρ0

− 2) + 1

where Cρ0 := max

12+log(2m) , e

and δρ0 :=

2 log(

6(m+r)+46

)Cρ0

.

Proof of claim: Let An :=

(an bncn dn

), then:(

an bncn dn

)=

((4(mn+ r) + 1)an−1 + 2(mn+ r)cn−1 (4(mn+ r) + 1)bn−1 + 2(mn+ r)dn−1

2an−1 + cn−1 2bn−1 + dn−1

).


(2mn)zn−1 ≤ zn ≤ 4m(n+ 1)zn−1 and zn−1 ≤ TraceAn ≤ 4m(n+ 1)zn−1.

Therefore,

(2m)n−2(n+ 1)!z1 ≤ TraceAn ≤ (2m)n−1(n+ 1)!z1

6.


TraceAn2

≤ |λAn | ≤ TraceAn.

Finally by the Remark 5.1,

2 ln((n− 1)!) ≤ `ρ0(γn) ≤ (2 + ln(2m)) ln((n+ 1)!) + 2 ln(z1

6

).

By using the inequalities (see [22]),

√2πn

(ne

)n≤ n! ≤ e

√n(ne

)n,


`ρ0 (γn)Cρ0

− δρ0

W (Cρ0`ρ0(γn))− 3

2≤ `ρ0(γn)− δρ0

Cρ0W (`ρ0(γn))− 3

2≤ n ≤ e`ρ0(γn)

W(`ρ0 (γn)

2 − 2)+1 ≤ Cρ0`ρ0(γn) + δρ0

W(`ρ0 (γn)Cρ0

− 2) +1

where Cρ0 := max

12+log(2m) , e

and δρ0 :=

2 log(z16

)

Cρ0.

For the volume upper bound, notice that by adding a crossing circle on the twisted regionof the trefoil knot τ (see Section 4), we have that the volume of the complement of γ thecanonical lift of a figure-eight closed geodesic in T 1(Σ0,3), is the same as, the volume of thecomplement of the canonical lift of the corresponding closed geodesic on Σmod and with anextra crossing circle in T 1Σmod, by [1] (see Figure 21). By using the same ideas as in Theorem1.4, we obtain:

Vol(Mγn) < 8v3(5n+ 3).


Figure 21.

The volume lower bound is a consequence of Theorem 1.6.

The proof of this result for any hyperbolic metric on a thrice-puntured sphere, follows fromthe fact that any pair of hyperbolic metrics on a hyperbolic surface are bi-Lipschitz (see forexample [4], Lemma 4.1).

References

[1] C. Adams, Thrice-punctured spheres in hyperbolic 3-manifolds, Trans. Am. Math. Soc. 287 (1985) 645-656.[2] C. Adams,Volumes of N-Cusped Hyperbolic 3-Manifolds, J. London Math. Soc. (2) 38 (1988), no. 3,

555-565.[3] I. Agol, P. Storm and W. Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds. J. Amer.

Math. Soc. 20 (2007) No.4, 1053-1077.[4] M. Bergeron, T. Pinsky and L. Silberman, An Upper Bound for the Volumes of Complements of Periodic

Geodesics, International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1-23.[5] J. Birman and I. Kofman, A new twist on Lorenz links, J. Topol. 2 (2009) no. 2, p. 227-248.[6] J. Birman and R. Williams, Knotted periodic orbits in dynamical systems -II: Knot holders for fibered

knots, Cont. Math 20, 1-60 (1983).[7] A. Brandts, T. Pinsky and L. Silberman, Volumes of hyperbolic three-manifolds associated to modular

links, Symmetry (2019), 11(10), 1206.[8] A. Casson and S. Bleiler, Automorphisms of surfaces after Nielsen and Thurston. Number 9. Cambridge

University Press, 1988.[9] A. Champanerkar, D. Futer, I. Kofman, W. Neumann and J. Purcell Volume bounds for generalized

twisted torus links,Math. Res. Lett. 18 (2011), no. 00, 10001 100NN.[10] T. Cremaschi, J. A. Rodrıguez-Migueles and A. Yarmola On volumes and filling collections of multicurves,

arXiv:1911.02732.[11] T. Cremaschi and J. A. Rodrıguez-Migueles Hyperbolicity of links complements in Seifert fibered spaces,

arXiv:1812.00567.[12] F. Dal’bo Geodesic and horocyclic trajectories. Springer Urtext (2011), EDP Science-CNRS[13] P. Dehornoy, Les noeuds de Lorenz, L’Enseignement Mathematique (2) 57 (2011), 211-280.

http://arxiv.org/abs/1911.02732

http://arxiv.org/abs/1812.00567


[14] P. Dehornoy and T. Pinsky, ?Coding of geodesics and Lorenz-like templates for some geodesic flows,Ergodic Theory and Dynamical Systems 38 (2018), 940-960.

[15] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math., 92(1):73-90, 1988.

[16] P. Foulon and B. Hasselblatt, Contact Anosov flows on hyperbolic 3-manifolds, Geom. Topol. 17 (2013)1225-1252.

[17] J. Franks and R. Williams, Braids and the Jones Polynomial, Trans. Am. Math. Soc. 303, 97-108 (1987).[18] D. Futer, E. Kalfagianni, J.S. Purcell, Cusp areas of Farey manifolds and applications to knot theory. Int.

Math. Res. Not. IMRN 2010(23), 4434-4497 (2010).[19] J. Gaster Lifting Curves Simply, International Mathematics Research Notices, Vol. 2016, No. 18, pp.

5559-5568[20] E. Ghys, Knots and dynamics, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zurich,

2007, pp. 247-277. MR 2334193 (2008k:37001).

[21] M. Gromov, Volume and bounded cohomology. Inst. Hautes Etudes Sci. Publ. Math. 56 (1982), 5-99. Zbl0516.53046 MR 0686042

[22] Robbins, Herbert (1955), A Remark on Stirling’s Formula, The American Mathematical Monthly, 62 (1):26-29.

[23] J. A. Rodrıguez-Migueles, A lower bound for the volumes of complements of periodic geodesics, J. LondonMath. Soc. (2) 00 (2020) 1-27.

[24] C. Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), no. 1, 69-80.MR 810563.

[25] J.P. Serre, Arbres, amalgames, SL2. Societe Mathematique de France, Paris, 1977. Redige avec la collab-oration de Hyman Bass, Asterisque, No. 46.

[26] W.P. Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University, 1979.[27] W.P. Thurston, Three-dimensional manifolds, Klenian groups and hyperbolic geometry, Bull. Amer. Math.

Soc. 6, 1982.[28] R. Waddington, Asymptotic Formulae for Lorrenz and Horseshoe Knotes, Commun. Math. 2, 53-117

(2002).

[29] R. F. Williams. The structure of Lorenz attractors. Inst. Hautes Etudes Sci. Publ. Math., 50:73-99, 1979.

Department of Mathematics and Statistics, University of Helsinki.

Pietari Kalminkatu 5, Helsinki FI 00014

E-mail address: [email protected]

arxiv:2008.12436v2 [math.gt] 23 mar 2021

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