Periods of continued fractions and volumes of modular knots complements
PPERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OFMODULAR 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 com-plete hyperbolic metric. We show that there exist sequences of closed geodesics for whichthis volume is bounded linearly in terms of the period of the geodesic’s continued fractionexpansion. Also, for sequences of figure-eight type closed geodesics, as defined in this paper,on a thrice-punctured sphere we give volume bounds in terms of the geodesic length of thegeodesic. 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 (cid:98) γ in its unit tangent bundle T Σ , namely thecorresponding periodic orbit of the geodesic flow. Let M (cid:98) γ denote the complement of a regularneighborhood of the canonical lift in T Σ . As a consequence of the Hyperbolization Theorem, M (cid:98) γ admits a finite volume complete hyperbolic metric if and only if γ fills Σ [14]. Suchmetric is unique up to isometry, by Mostow’s Rigidity Theorem, meaning that any geometricinvariant is a topological invaraint. We will be interested in estimating the volume of M (cid:98) γ interms of properties of the close geodesic γ. Bergeron, Pinsky and Silberman have already studied in [4] the problem of finding anupper bound for the volume of M (cid:98) γ , by giving one which is linear in the length of the geodesic.Nevertheless, it is easy to construct sequences of closed geodesics with length approching toinfinity but whose associated canonical lift complements are homoemorphic. For example,the iterations under an infinite-order diffeomorphism of the surface, of a given filling closedgeodesic. In ([17], Theorem 1.1) we constructed more interesting sequences of closed geodesicswhose associated canonical lift complements are not homeomorphic with each other and thesequence of the corresponding volumes is bounded. Also in ([17], Theorem 1.5) we gave alower bound of the volume of M (cid:98) γ in terms of the number homotopy classes of arcs of γ ineach pair of pants of a given a pants decomposition of Σ . It is interesting to point out thatin a collaboration with Tommaso Cremaschi and Andrew Yarmola in [9] we study the sameproblem for large families of filling collections of simple closed geodesics, such as pair of fillingclosed geodesics, and found bounds for the volume of the corresponding link complement interms of expressions involving distances in the pants graph.We focus here mainly in the case of the modular surface Σ mod = H / PSL ( Z ) . Thishyperbolic 2-orbifold is particularly interesting since its unit tangent bundle is homeomorphicto the complement of the trefoil knot in S . Therefore, in the modular surface case, M (cid:98) γ canbe considered as a two component link complement in S . Moreover, in [15] Ghys observedthat the periodic orbits of the geodesic flow over the modular surface are the Lorenz knots.For facts about such knots see [12]. a r X i v : . [ m a t h . G T ] A ug J. A. RODRIGUEZ-MIGUELES
In the modular surface case, ([4], Section 3) Bergeron, Pinsky and Silberman also gavean upper bound for the volume of M (cid:98) γ which is proportional to the period of the geodesic’scontinued fraction expansion plus the sum of the logarithms of its corresponding coefficients.Nevertheless, in ([17], Corollary 1.2) we constructed sequences of closed geodesics with theperiod approaching to infinity, but whose sequence of the corresponding volumes is uniformlybounded. In this paper, we prove that there exist a sequence of closed geodesics for which thevolume of the canonical lift complement has an upper bound linearly in terms of the period: Theorem 1.1.
For the modular surface Σ mod , there exist a sequence { γ k } of closed geodesicson Σ mod with n γ k (cid:37) ∞ such that, Vol( M (cid:99) γ k ) < v (5 n γ k + 2) , where n γ k is half the period of the continued fraction expansion of γ k and v is the volume ofa regular ideal tetrahedron. As a consequence of Theorem 1.4 in [17], we have that up to a constant, Theorem 1.1 issharp.
Corollary 1.2.
For the modular surface Σ mod , there exist a sequence { γ k } of closed geodesicson Σ mod with n γ k (cid:37) ∞ such that, v n γ k ≤ Vol( M (cid:99) γ k ) < v (5 n γ k + 2) , where n γ k is half the period of the continued fraction expansion of γ k and v is the volume ofa regular ideal tetrahedron. We obtain the sequences of canonical lifts in Theorem 1.1 by performing annular Dehnfilling surgery on the canonical lift complement of some closed geodesics on the modularsurface. Recall that every closed geodesic on the modular surface is represented by a primitiveelement in the semigroup generated by the parabolic elements X = (cid:18) (cid:19) and Y = (cid:18) (cid:19) in PSL ( Z ) . So far we have discussed the particular case of the modular surface, but it is also remark-able that for a thrice-punctured sphere one can consider the closed geodesics associated toprimitive elements in the semigroup generated by { X, Y } the free homotopy class of twodifferent boundary components. The closed geodesics obtained in this way, will be denotedas figure-eight type closed geodesics. For some sequences of figure-eight type closed geodesicswe estimate the corresponding canonical lift complement volume in terms of the geodesiclength: Theorem 1.3.
Given a hyperbolic metric ρ on a thrice-punctured sphere Σ , , then thereexist a constant C ρ > and a sequence { γ k } of closed geodesics on Σ with (cid:96) ρ ( γ k ) (cid:37) ∞ suchthat v (cid:18) C ρ (cid:96) ρ ( γ k ) − W ( C ρ (cid:96) ρ ( γ k )) (cid:19) ≤ Vol( M (cid:98) γ k ) ≤ v (cid:18) C ρ (cid:96) ρ ( γ k ) + 4 W ( C ρ (cid:96) ρ ( γ k )) (cid:19) , where v is the volume of a regular ideal tetrahedron and we use the Lambert W function. The main ingredients to prove Theorem 1.3, is our next result which estimates a lowerbound for the volume of the canonical lift complement of figure-eight type closed geodesicson the thrice-punctured sphere Σ , , in terms of combinatorial data of the reduced wordrepresenting the conjugacy class of the geodesic in π (Σ , ) . ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 3
Theorem 1.4.
Given a thrice-punctured sphere Σ , , and γ a figure-eight type closed geodesic,we have that: v (cid:93) { exponents of X in ω γ } + (cid:93) { exponents of Y in ω γ } − ≤ Vol( M (cid:98) γ ) , where v is the volume of the regular ideal tetrahedron, ω γ is the cyclically reduced wordrepresenting the conjugacy class of γ in π (Σ , ) . Theorem 1.4 uses a result due to Agol, Storm and Thurston [3] giving a lower bound for thevolume of M (cid:98) γ in terms of the simplicial volume of the double of the manifold constructed bycutting M (cid:98) γ along an incompressible surface. Here we apply it to the incompressible surfacecoming from the pre-image under the map T (Σ , ) → Σ , of a simple geodesic arc whosevertices points belong to the same boundary component. Moreover, Theorem 1.4 is an analogof a general lower bound for the volume of canonical lift complements of geodesics on surfacesadmiting a non-trivial pants decomposition ([17], Theorem 1.5). 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 acombinatorial description of the canonical lift of figure-eight type closed geodesics. And insection 4 we prove Theorem 1.1, Theorem 1.4, Corollary 1.2 and Theorem 1.3. Acknowledgments:
I would like to thank Pierre Dehornoy for some interesting conversa-tions on this and other topics. I gratefully acknowledge the support of Pekka Pankka andfrom the Academy of Finland project 297258 “Topological Geometric Function Theory”.2.
Coding of closed geodesics
Modular surface case.
We know that the conjugation classes in PSL ( 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 PSL ( Z ) in a uniqueway: Lemma 2.1.
Let A be an element of SL ( Z ) which has two distinct eigenvalues in R ∗ + . Theconjugation class of A in SL ( Z ) contains a representative of the form: n A (cid:89) i =1 X k i Y m i where X = (cid:18) (cid:19) and Y = (cid:18) (cid:19) , n A > and k i , m i are strictly positive integers.In addition, the representation is unique up to cyclic permutation of the factors X k i Y m i . Conversely, any product not empty of such factors is an element of
PSL ( Z ) with two distincteigenvalues in R ∗ + . Consider the model of the upper half-plane for the hyperbolic space H , provided with aFarey’s triangulation F (the ideal triangle with the vertices 0 , ∞ , and all its images bysuccessive reflections with respect to its sides). We know that the group of oriented isometriesof H , which preserves F , is identified with PSL ( Z ) . Let ˜ α be the hyperbolic line going from the repulsive fixed point of A to the attractive fixedpoint. Then ˜ α crosses an infinity of ideal triangles ( ..., t − , t , t , ... ) of F . You can formallywrite a bi-infinite word: ω A := ...LRRRLLRR... J. A. RODRIGUEZ-MIGUELES where the k th letter is R (resp. L ) if and only if the line ˜ α comes out from t k by the rightside (resp. on the left) with respect the side where it enters, in this case we will say that ˜ α turns right (resp. turns left) to t k . The word ω A contains at least one R and one L , becausethe ends of ˜ α are distinct. The image of t by A is a certain t m ( m >
1) and ω A is periodical.We associate the matrix X to R and Y to L, which are parabolic transformations of H which fix the points 0 and ∞ , respectively. Let B be any subword of ω A of length m, welook at B as a product of transvection matrices and therefore as an element of SL ( Z ) . Bystudying the action of X and Y on F we can easily see that A and B are conjugated inPSL ( Z ) since both are of strictly positive trace.We check the uniqueness in the following way: on one hand, if A and B are conjugate, thereis an element of PSL ( Z ) (preserving F ) which sends the axis of A on the axis of B, therefore A and B define the same word ω A up to translation. On the other hand, by consideringthe action of X and Y on H , we see that a product of standard transvection matrices asin the statement of Lemma 2.1 always defines the word ω A = (cid:81) n A i =1 X k i Y m i , which repeatsinfinitely. We denote n A as the period of ω A , which is the same as the number of (cyclic)subwords of the form xy in ω A . The continued fraction expansion of a geodesic in the modular surface.
In this subsec-tion, we will specify how the sequence ( k , m , ..., k n A , m n A ) of Lemma 2.1 is related to thecontinued fraction expansion of the attractive fixed point of A. For more details on the proofsof the results in this subsection see [11].The continuous 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. This arc crosses a succession 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 L n R n L n ... with n i ∈ N ,is called the cut sequence of x. If x > L , while if 0 < x < R. Note that the cutting sequence is independent of the initial pointon the imaginary axis. The key observation is:
Lemma 2.2.
Let x > with a cutting sequence L n R n L n ... with n i ∈ N . Then x = [ n ; n , n , ... ] . By the same reason < x < has a cutting sequence R n L n R n ... with n i ∈ N , then x = [0; n , n , ... ] . To manage negative numbers, simply replace the negative number x with − /x = [ b ; b , b , b , ... ]with b ≥ , by using an element of the modular group. Example 2.3.
Let A = (cid:18) − (cid:19) the matrix associated with the semi-geodesic ˜ α whichconnects the points i and (1 + i ) / . Then ˜ α crosses the imaginary axis and ends with ( √ − / . Following ˜ α, we notice that ˜ α has the cutting sequence LRLRLR... . Then ( √ − / , , , , ... ] . We say that two numbers x = [ a ; a , ... ] , y = [ b ; b , ... ] have the same tails if there is k, l ∈ N such that a k + r = b l + r for all r ≥ . We say that they have the same tails mod 2 if k + l is even. To understand this definition, we need the following lemma: ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 5
Lemma 2.4.
Let ˜ α and ˜ α (cid:48) be oriented geodesics lines of H with the same positive endpoint x, then the cutting sequences of ˜ α and ˜ α (cid:48) coincide from a certain rank. In addition, x = [ a ; a , ... ] and y = [ b ; b , ... ] have the same tails mod 2 if and only if there is a matrix g ∈ SL ( Z ) such that g ( ˜ α ) = ˜ α (cid:48) . It is not difficult to see algebraically that any number whose continuous fraction is almostperiodic is quadratic, meaning a solution of an equation of the form: ax + bx + c = 0 avec a ∈ N ∗ et b, c ∈ Z . Conversely, any quadratic number has an almost periodic continuous fraction expantion. Thefollowing result makes possible to establish a relation between the fixed points of a hyperbolicisometry of SL ( Z ) and the quadratic numbers: Lemma 2.5.
Let x be irrational. The following properties are equivalent: ( i ) x is fixed by a hyperbolic isometry of PSL ( Z ) ; ( ii ) x is quadratic. In conclusion, an element A of PSL ( Z ) is hyperbolic if and only if it is conjugated inPSL ( Z ) to an isometry of the form (cid:81) ki =1 X n i − Y n i , with n j ∈ N and its fixed point hasthe same tail mod 2 as the quadratic number x = [0; n , n , ..., n k ] . Notice that k is exactlyhalf the period of the continued fraction of x, also denoted as the period n A (see subsection2.1) of its word representation.2.2. Coding figure-eight type closed geodesics on hyperbolic surfaces.Definition 2.6.
Given a hyperbolic surface Σ , we say that a closed geodesic γ in Σ is a figure-eight type closed geodesic if there exist X, Y ∈ π (Σ) representing the free homotopyclass of two disjoint oriented simple closed curves α and β, such that γ represents the element: ω γ := n γ (cid:89) i =1 X k i Y m i , where k i , m i ∈ N , n γ is the period of γ and XY represents the free homotopy class of a nonsimple closed curve.Notice that all figure-eight type closed geodesic relative to X, Y are contained in a subsur-face of Σ which is a pair of pants.3.
Parametrization of the canonical lift
Modular knots inisde the Lorenz template.
In order to study the geometry of M (cid:98) γ , we will start by constructing an isotopy class representing (cid:98) γ, from the word ω γ . For moredetails on the algorithm see [5].A template [6] is a branched surface with boundary and an expansive semi flow. Thefollowing is a theorem of Ghys [15], relying on a theorem of Birman and Williams [6].
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 -sphere, excluding the boundary curves of T . On any finite subset, thecorrespondence is by an ambient isotopy.
J. A. RODRIGUEZ-MIGUELES
Figure 1.
The template T inside the unit tnagent bundle of the modularsurface (trefoil knot complement in S ) with the canonical lift correspondingto the word X Y . The template comes with a symbolic dynamics given by the symbols X and Y that cor-respond to passing through the left or through the right ear (or equivalently through theleft or the right half of the branch line). There is not starting point for the orbit, then thewords X, Y used to describe them are primitive up to cyclic permutation. Ghys proves in[15] 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 PSL ( Z ) . X Y XX YY
Figure 2.
The figure on the left is the Lorenz template. The figure on theright shows the splitting of the template to obtain the braid.Williams [22] constructed an algorithm to find the periodic orbits inside the template T , just from the representing word ω. In the right of Figure 2, the template has been cut opento give a related template for braids, which inherit an orientation from the template, top tobottom.We will ilustrate his algorithm, in the figure 3, by showing how to recover the orbit fromthe word ω := X Y XY . We start by writing the 10 cyclic permutations ω = ω , ω , ..., ω , in the natural order. We reorder lexicographically these 10 words using the rule X < Y.
Thenew position µ i is given after each ω i : X Y XY Y XY X Y X Y XY X Y XY X Y X Y XY X XY X Y XY XY X Y X Y X Y XY X Y X Y XY , , , , , , , , , , and indices a permutation braid,where the strand i begins with µ i and ends ends with µ i +1 . The i th strand of the braid is ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 7 an overcrossing strand if and only if µ i < µ i +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 X Y XY (see figure 3). The braid obtainedis called the Lorenz braid associated with X Y XY . X YX Y Figure 3.
The Lorenz braid associatied with X Y XY 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 > i th strand begins at i and ends at i + d i . Since two undercrossing strands of never cross, we have the following series of positiveintegers: d ≤ d ≤ ... ≤ d p − ≤ d p . We collect this data in the following vector:¯ v = (cid:104) d , ..., d p (cid:105) X , ≤ d and d i ≤ d i +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, thearrows separate the left and right strands. Each d i with i = 1 , ..., p is the difference betweenthe initial and final positions of the i th overcrossing strand. The integer d i is also the numberof strands that pass under the i th braid strand. The vector ¯ v determines the closed braidwith n = ( p + d p ) strands, which we call the representation of the Lorenz braid of ω . Allperiodic orbits on the template T appear in this way.The overcrossing strands travel in groups of parallel strands, which are strands of the sameslope, or equivalent strands whose associated d i coincide. If d µ j = d µ j +1 = ... = d µ j + s j +1 , where s j is the number of strands in the j th group then let r j = d µ j . Thus, we can write ¯ v on the form: ¯ v = (cid:104) d s µ , ..., d s k µ k (cid:105) X = (cid:104) r s , ..., r s k k (cid:105) X , ≤ s i and r i < r i +1 . J. A. RODRIGUEZ-MIGUELES
Note that p = s + ... + s k , d = r , d p = r k . The period of the word ω is found in terms of the braid representation by the followingnumber: t = (cid:93) { i | i + d i > p where 1 ≤ i ≤ p } . In the example of figure 3, (cid:104) , , , , (cid:105) X = (cid:104) , , , (cid:105) X . Also, p = 5 , k = 4 , r k = 5 ,n = p + r k = 10 and the period is 2 . Lemma 3.2.
Let ( k i ) ni =1 ∈ N n such that k + 1 < k , k i < k i +1 for ≤ i ≤ n − , and γ theclosed geodesics on Σ mod associated to n (cid:81) i =1 ( X k i Y ) . Then the associated Lorenz braid to γ is: (cid:104) s , s , ..., ( n − s n − , n s n (cid:105) X , where s i = i ( k n +1 − i − k n − i ) for ≤ i ≤ n − , s n − = ( n − k − k − , and s n = n ( k + 1) − . Proof:
By induction over n. For the base of induction consider n = 3 , it determines thefollowing permutation, obtained by reordering lexicographically the k + k + k + 3 wordsinduced by ω γ under cyclic permutation: (1 ,...,k − k , ( k − k j, ( k − k k − k l, ( k − k j, ( k − k k − k l, ( k − k k − k l ) , where 0 ≤ j ≤ k − k − , and 0 ≤ l ≤ k . (1) The number of overcorssing strands shifted one place are the first k − k then s = k − k . (2) The number of overcorssing strands shifted two places are the ones whose ends havea j > s = 2( k − k − . (3) The number of overcorssing strands shifted three places are the ones whose ends hasa l index, with the exception of ( k − k ) + 2( k − k ) + 1 which correspond to anundercorssing strand, then s = 3( k + 1) − . If our statement is true for n = m, then after multiplying m (cid:81) i =1 ( X k i Y ) with X k m +1 Y we willmodify the Lorenz braid by adding k m +1 overcorssing strands:(1) k m +1 − k m at the begining of the braid, then s m +1 = k m +1 − k m . (2) k m +1 − i − k m − i are added in the i th collection of parallel strand for 1 ≤ i ≤ m − , because one new overcrossing strand enters each collection of parallel strands of thepreviouse Lorenz braid. This implies that: s i +1 = i ( k m +1 − i − k m − i ) + k m +1 − i − k m − i = ( i + 1)( k m +1 − i − k m − i ) . (3) For the penultimate and last collection of parallel overcorssing strands we will add k − k − k + 1 respectively because of the new strand entering into eachcollection of parallel overcrossing strands. This implies that: s m = ( m − k − k −
1) + k − k − s m +1 = ( m + 1)( k + 1) − . (cid:3) ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 9
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 Σ , where γ isfilling. We are going to construct an explicit representant of the isotopy class of (cid:98) γ in T (Σ , ) . Lemma 3.3.
The set of canonical lifts relative to figure-eight type closed geodesics on Σ , with respect two fixed boundary components, is in bijective correspondence with the set ofperiodic orbits on the template T . Proof.
Let γ be a figure-eight type closed geodesic on Σ , with respect two fixed boundarycomponents.First we will fix a vector field on Σ , , by considering the asocciate periodic orbit in T , induced by the word ω γ . Notice that there is a natural projection of T to a pair of pants Σ , which is obtained by the overlaping the ears of the template T (see figure 4). Figure 4.
The projection map from T to Σ , Moreover, the periodic orbit is mapped under this projection to a figure-eight type closedcurve in minimal position and representing the same homotopy class of γ in Σ , . The vector field ξ is obtained by extending the the foliation given by the oriented arcs of γ in Σ , outside the overlapping triangle of the projection map from the template T to Σ , . The foliation is obtain first by enclosing each arc to a familly of disjoint concenric circles tothe corresponding boundary component, and then doing parallel copies of the closed simpleloops which converges from each side to the boundary of the piece of Σ , obtained afterspliting along a mid edge connecting the third boundary component of Σ , (see figure 5).The vector field ξ induces a global section inside T (Σ , ) and the canonical lift (cid:98) γ is isotopicto the embedding of the corresponding periodic orbit in T . (cid:3) Figure 5.
The foliation induced by the γ -arcs outside the non-injective pieceof the projection map.4. Sequences of geodesics whose volume complement is bounded by the period
In this section we prove, in Theorem 1.1, an upper bound for the volume of the complementof canonical lifts of some sequences of geodesics in the modular surface linearly in terms theperiod of the geodesic’s continued fraction expantion . Also for all figure-eight type geodesicson any hyperbolic surface we give a combinatorial lower bound, in Theorem 1.4, for the volumeof the corresponding canonical lift complement in terms of its reduced word representationin the fundamental group of the surface.4.1.
On the modular surface.
To get to imagine the canonical lift in our sequences we willexemplify for the canonical lift associated to the closed geodesic X Y X Y X Y X Y XY onthe modular surface. As we saw in the previous section we can also associate it with theLorenz braid (cid:104) , , , , (cid:105) X . Equivalently if we change the setting of the knot using therule
Y < X, we also have the vector (cid:104) , , , , (cid:105) Y . In order to simplify the figures inthis we will remove the treifol component from the link associated to de boundary componentof T Σ mod (see figure 1). X Y
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Figure 6. X Y X Y X Y X Y XY.
In general consider the Lorenz braid (cid:104) r s , ..., r s k k (cid:105) X associated with γ. Equivalently if wechange the setting of the knot using the rule
Y < X, we also have the vector of the same size (cid:104) l q , ..., l q k k (cid:105) Y . ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 11
Vertical rings.
Let m X be the larger index such that there is a strand coming out ofthe interval (cid:32) m X − (cid:88) k =0 s k , m X (cid:88) k =0 s k (cid:33) which remains in the X -band. So let m Y be the analogous index but for the Y -band.We begin by adding the following m X +1 vertical rings A X i of delimited by unknots parallelto the X -band, which enclose the intervals (cid:32) i − (cid:88) k =0 s k + 3 / , i (cid:88) k =0 s k + 1 / (cid:33) where s = 0 and i < m X , and for i = m X we have two cases: (cid:32) m X − (cid:88) k =0 s k + 3 / , m X (cid:88) k =0 s k + 1 / (cid:33) , if s m X ≤ r m X . Or: (cid:32) m X − (cid:88) k =0 s k + 3 / , m X − (cid:88) k =0 s k + (cid:22) s m X r m X (cid:23) r m X + 1 / (cid:33) , if s m X > r m X . Finally, for i = m X + 1 we add a last vertical ring which encloses the interval: (cid:32) m X (cid:88) k =0 s k + 3 / , p + 1 / (cid:33) , if s m X ≤ r m X . Or: (cid:32) m X − (cid:88) k =0 s k + (cid:22) s m X r m X (cid:23) r m X + 3 / , p + 1 / (cid:33) , if s m X > r m X . In case there is no such m X we consider only the interval (0 , p + 1 / . Note that the strands coming out of the same ring A X i , for i ≤ m X , are parallel.We will do the same construction for the Y -band this time by considering the vector (cid:104) l q , ..., l q k k (cid:105) Y . So we will have the family vertical rings { A Y j } m Y +1 j =1 in the Y -band. Lemma 4.1.
The total number of vertical rings is at most two times the period of the closedgeodesic γ plus two. Proof:
As each s i is associated to a family of parallel arcs, then we know that there are atleast m X free spaces for arcs coming from the Y -band which fall into the X -band. As theperiod of the X -vector is the same as the Y -vector associated to γ we know that there areexactly the period number of arcs comming from Y -band to the X -band, so m X ≤ t. So thenumber of vertical ring is m X + 1 + m Y + 1 , then we obtain the wanted estimate. (cid:3) In the following section we explain the Dehn surgery that we will apply along the boundarycomponents of the vertical rings.
X Y
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Figure 7.
The vertical rings of X Y X Y X Y X Y XY.
Annular Dehn surgery.
Given A an embedded ring into an oriented 3-manifold N, let ∂A = L +1 ∪ L − . Let us orient L +1 and L − with a compatible orientation of N . Let { m i , l i } ,i = ± , be a basis where m i is the meridian over ∂N ( L i ) and l i a longitud over ∂N ( L i )induced by A for i = ± l i = ∂N ( L i ) ∩ A ) . With these basis, the Dehn filling along the slope 1 /n in L +1 and − /n in L − givesan homeomorphism N ∼ = N ( L + ∪ L − ) (cid:0) n , − n (cid:1) . This homeomorphism is obtained by cutting N ∼ = N ( L + ∪ L − ) along A then wind n -times, and trivialy filling the boundary components of A . See the figure 8 for the case n = − A. Notice that this homeomorphism is the identity outside a normal neighborhood of A. A A
Figure 8.
The annular Dehn surgery surgery makes the curve to twist along 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 − ) (cid:18) n , − n (cid:19) restricted to S is a composition of a n -Dehn twist. If n > S on the sameside as L + , then the Dehn twist is on the opposite sense.4.1.3. Construction of the link L γ in S . In this section we construct the family of links on S such that after Dehn filling will obtain the sequences of canonical lifts on the unit tangentbundle of the modular surface. ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 13
Theorem 1.1.
For the modular surface Σ mod , there exist a sequence { γ k } of closed geodesicson Σ mod with n γ k (cid:37) ∞ such that, Vol( M (cid:99) γ k ) < v (5 n γ k + 2) , where n γ k is half the period of the continued fraction expansion of γ k and v is the volume ofa regular ideal tetrahedron. Proof:
Let ( k i ) ni =1 ∈ N n such that k + 1 < k , k i < k i +1 for 2 ≤ i ≤ n − , and γ the closedgeodesic on Σ mod associated to n (cid:81) i =1 ( X k i Y ) . We will use the fact [20] 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 L γ associatedwith γ in S such that by Dehn filling along some components of L γ , we get M (cid:98) γ , and alsoby notcing that S \ L γ 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 bounded from above by the self-intersection number of the closed curves.By Lemma 3.2 the associated Lorenz braid to γ is: (cid:104) s , s , ..., ( n − s n − , n s n (cid:105) X , where s i = i ( k n +1 − i − k n − i ) for 1 ≤ i ≤ n − , s n − = ( n − k − k − , and s n = n ( k + 1) − . B Figure 9.
The link ¯ τ ∪ B ∪ { ∂ L A X i , ∂ R A X i } n +1 i =1 ∪ { ∂ L A Y , ∂ R A Y } The link L γ consist of 2 n + 7 components, where 2 n + 6 are :(1) The trefoil knot corresponding to the boundary of T Σ mod , denoted by ¯ τ , (2) The boundary components of each vertical ring A X i and A Y in T Σ mod , denoted by { ∂ L A X i , ∂ R A X i } n +1 i =1 and { ∂ L A Y , ∂ R A Y } , (3) An unknot B enclosing only the links in (2) (see figure 9).Before constructing the last component of L γ notice that the link complement formed by(1) , (2) , and (3) is homeomorphic to the complement of the link ¯ τ on T Σ , n +4 and ¯ τ projectsinjectively to a closed curve τ on Σ , n +4 (see figure 10). Figure 10.
The closed curve τ on Σ , n +4 . We will construct the last knot component of L γ , denoted by σ γ 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 σ γ , on Σ , n +4 and specify the crossing information with itself.We start by marking intervals I X i and I Y on Σ , n +4 whose preimage under the projectionmaps are the corresponding vertical rings A X i and A Y . x x A x x A x x A x x A x x A x A y Y A y y y y y Figure 11.
The marking of the intervals associated to X Y X Y X Y X Y XY. (1) From each I X i with 1 ≤ i ≤ n − α i starting at x i ∈ I X i to y i ∈ I Y passing one time through each interval I X j for i ≤ j ≤ n + 1 (see figure 12). Draw allarcs in a way that are disjoint from each other.(2) Construct an arc α n from a point x n ∈ I X n to a point y n ∈ I Y disjoint from theprevious arcs and intervals. x x A x x A x x A x x A x x A x A y Y Ay y y y y Figure 12. α i arcs with respect to X k Y X k Y X k Y X k Y X k Y. (3) Connect the point y i with x i +1 with 1 ≤ i ≤ n − β i disjoint from theintervals and parallel between them (see figure 13).(4) Draw an arc β n from y n to x intersecting once each of the other β i arcs.Once we joint all arcs we will have the closed curve σ γ . The crossing information of thelink σ γ is given by the fact that the strand β n is under the other β i strands (see figure 14). Remark 4.2.
The self-intersection number of the closed curve σ γ is n − . Claim 4.3. M (cid:98) γ is obtained by making annular Dehn filling along the boundary of the verticalrings components of L γ and trivial Dehn filling on B. ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 15 x x A x x A x x A x x A x x A x A y Y Ay y y y y Figure 13.
The closed curve σ γ associated to X k Y X k Y X k Y X k Y X k Y. Figure 14.
The knot σ γ associated to X k Y X k Y X k Y X k Y X k Y. X Y
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 401 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. L γ and the vertical rings. Proof of claim:
First we isotope σ γ 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 A X i with i < n − k n +1 − i − k n − i . (2) For A X n − we will do an annular Dehn filling of type k − k − . (3) For A X n we will do an annular Dehn filling of type k . (4) For A X n +1 and A Y we will do an annular Dehn filling of type 1 . X Y
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 401 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 X Y X Y X Y X Y XY.
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 β i that intersects avertical ring. Notice that for each A X i is intersected by the arc β i . (1) For the vertical rings A X i with i < n − i ( k n +1 − i − k n − i )overcrossing strands with the same slope.(2) For A X n − we will obtain exactly ( n − k − k −
1) overcrossing strands with thesame slope.(3) For A X n and A X n +1 we will obtain exactly n ( k + 1) − X Y X Y X Y X Y XY. (cid:3)
Finally, let Γ = τ ∪ σ γ , by [20]Vol( M (cid:98) γ ) < (cid:107) T (Σ , n +4 ) Γ (cid:107) , and by ([10], Thm 1.5) (cid:107) T (Σ , n +4 ) Γ (cid:107) ≤ v i (Γ , Γ) ≤ v (5 n + 2) , giving us an upper bound of the volume depending linearly only on n. (cid:3) Corollary 1.2.
For the modular surface Σ mod , there exist a sequence { γ k } of closed geodesicson Σ mod with n γ k (cid:37) ∞ such that, v n γ k ≤ Vol( M (cid:99) γ k ) < v (5 n γ k + 2) , ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 17 where n γ k is half the period of the continued fraction expansion of γ k and v is the volume ofa regular ideal tetrahedron. Proof:
The sequence can be obtained for any infinite subsequence of the following sequenceof closed geodesics: n (cid:89) i =1 ( X k i +1 Y ) where { k i } ∈ N N , and k i < k i +1 . The upper bound for the volume of the corresponding canonical lift complements is aconsequence of Theorem 1.2 and the lower bound is proven in ([17], Theorem 1.4). (cid:3)
On the thrice-punctured sphere.
In this section we prove a lower bound for thesimplicial volume of canonical lift complements associated to figure-eight type closed geodesicson a thrice-punctured sphere Σ , . Theorem 1.4.
Given a thrice-punctured sphere Σ , , and γ a figure-eight type closed geodesic,we have that: v (cid:93) { exponents of X in ω γ } + (cid:93) { exponents of Y in ω γ } − ≤ Vol( M (cid:98) γ ) , where v is the volume of the regular ideal tetrahedron, ω γ is the cyclically reduced wordrepresenting the conjugacy class of γ in π (Σ , ) . The lower bound is obtained in terms of combinatorial data coming from the geodesic andan essential simple geodesic arc connecting the same boundary component on Σ , . Given a punctured disk D with a marked point in the boundary x ∈ ∂D we say that twoarcs α, β : [0 , → D with α ( { , } ) ∪ β ( { , } ) ⊂ ∂D are in the same homotopy class in D, if there exist an homotopy h : [0 , × [0 , → D such that: h ( t ) = α ( t ) , h ( t ) = β ( t ) and h ([0 , × { , } ) ⊂ ∂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 boundary arc notcontaining x. Remark 4.4.
Up to isotopy, for a family of simple arcs without intersection there is onlyone configuration of arc in D . This is shown in Figure 17 an was winding number 0 . The 2 inthe lower bound of Theorem 1.4 comes from the fact that such configurations have at most1 homotopy class of γ -arcs on D. Figure 17.
The only γ -arcs configuration on D up to homotopy classeswhose γ -arcs are simple arcs without intersections. Before stating the main result to prove Theorem 1.4 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 4.5.
Let D be a punctured disk which is induced by spliting Σ , along a simplearc connecting the same cusp component and γ a figure-eight type geodesic on Σ , such that D ∩ γ is a finite set of geodesic arcs { α i } connecting ∂D. Then denote, D (cid:98) γ as T D \ (cid:91) i (cid:98) α i ∼ = ( S × D ) \ (cid:91) i (cid:98) α i . And define, d ( D (cid:98) γ ) , as gluing two copies of T D \ (cid:83) i (cid:98) α i along the punctured sphere coming from ∂T D \ (cid:0) ∂T D ∩ (cid:0) (cid:91) i (cid:98) α i (cid:1)(cid:1) , by using the identity. Moreover, d ( D (cid:98) γ ) is homeomorphic to( S × Σ ) \ (cid:91) i d ( (cid:98) α i ) , where Σ is a thrice-punctured sphere and d ( (cid:98) α i ) is a knot in S × Σ obtained by gluing (cid:98) α i along the two points ∂T D ∩ (cid:98) α i by the identity. Figure 18.
The projection of d ( D (cid:98) γ ) (after an homotopy of γ -arcs to a mini-mal position configuration) over Σ . 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, S k ( M ; R ) is the set of formal linear combinationof k -simplices, and we set as usual S k ( M, ∂M ; R ) = S k ( M ; R ) /S k ( ∂M ; R ). We denote by (cid:107) c (cid:107) the l -norm of the k -chain c . If α is a homology class in H singk ( M, ∂M ; R ), the Gromov normof α is defined as: (cid:107) α (cid:107) = inf [ c ]= α {(cid:107) c (cid:107) = (cid:88) σ | r σ | such that c = (cid:88) σ r σ σ } . The simplicial volume of M is the Gromov norm of the fundamental class of ( M, ∂M ) in H sing ( M, ∂M ; R ) and is denoted by (cid:107) M (cid:107) . ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 19
The key ingredient to prove Theorem 1.4 is the following result due to Agol, Storm andThurston ([3], Theorem 9.1):
Theorem (Agol-Storm-Thurston) . Let N be a compact manifold with interior a hyperbolic -manifold of finite volume. Let S be an embedded incompressible surface in N. Then
Vol( N ) ≥ v (cid:107) d ( N \\ S ) (cid:107) . Definition 4.6.
Let η be a simple geodesic arc connecting the same cusp component on Σ , , then we define the following embedded surface on M (cid:98) γ :( T η ) (cid:98) γ := ( T η ) \ N (cid:98) γ where T η is the pre-image of η under the map T (Σ , ) → Σ , . We now prove the lower bound for the volume of the canonical lift complement:
Proof of Theorem 1.4.
Let η be a simple geodesic arc connecting the same cusp componenton Σ , , inducing a decomposition on D R and D L . Consider the surface ( T η ) (cid:98) γ in M (cid:98) γ Claim 4.7.
The embedded surface ( T η ) (cid:98) γ in M (cid:98) γ is π -injective. Figure 19.
The punctured sphere ( T η ) (cid:98) γ . Proof of Claim 4.7.
Let us split M (cid:98) γ by ( T η ) (cid:98) γ , into pieces D R (cid:98) γ and D L (cid:98) γ . By Van Kampen itis enough to show the π -injectivity for the surface ( T η ) (cid:98) γ in D R (cid:98) γ . If the surface ( T η ) (cid:98) γ is not π -injective in D R (cid:98) γ , then by the Loop Theorem, there is a disk D whose interior is in the interior of D R (cid:98) γ and whose innermost intersection with ( T η ) (cid:98) γ is anessential simple closed curve α in ( T η ) (cid:98) γ . As T η is an incompressible surface in T (Σ , ) , then α bounds a disk D in T η that intersects (cid:98) γ. So by the irreducibility of T (Σ , ) , the embedded sphere formed by D ∪ D would bounda ball B in T (Σ , ) . By the periodicity of (cid:98) γ, there is at least one arc of (cid:98) γ inside B with endpoints at D . Thisarc is homotopic relative to the boundary to a simple arc in D with the same endpoints.Such homotopy inside T (Σ , ) , induces a homotopy on Σ , that reduces the intersectionnumber between γ and η, contradicting the fact that γ and η are in minimal position. (cid:3) From ([3], Theorem 9.1) we deduce that:Vol( M (cid:98) γ ) ≥ v (cid:107) d ( M (cid:98) γ \\ T η ) (cid:107) = v (cid:88) P ∈ Π (cid:107) d ( D (cid:98) γ ) (cid:107) . For any piece D ∈ { D R , D L } we have: v (cid:93) { cusps of d ( D (cid:98) γ ) hyp } ≤ Vol( d ( D (cid:98) γ ) hyp ) ≤ v (cid:107) d ( D (cid:98) γ ) hyp (cid:107) = v (cid:107) d ( D γ ) (cid:107) where d ( D (cid:98) γ ) hyp is the atoroidal piece of d ( D (cid:98) γ ) , i.e., the complement of the characteristicsub-manifold, with respect to its JSJ-decomposition. The first and second inequality comefrom [2] and [16] respectively.Notice that if ω and ω are a pair of homotopic γ -arcs on D then their respective canonicallifts (cid:99) ω and (cid:99) ω are isotopic (cid:98) γ -arcs in T D. Indeed, let (cid:102) ω and (cid:102) ω be lifts on the universal coverstarting in the same fundamental domain. Take an homotopy of geodesics that varies from (cid:102) ω to (cid:102) ω and project this homotopy to Σ , . This will give us a homotopy h of geodesics arcs h t that start in ω and end in ω . The image of the geodesic homotopy does not intersects other (cid:98) γ -arcs because of local uniqueness of the geodesics. Then we have that the geodesic homotopyinduces an isotopy in T D 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 D ) . 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 ( D (cid:98) γ ) hyp ∼ = d ( D (cid:98) Ω ) hyp . Moreover, d ( D (cid:98) Ω ) can be seen as a link complementin S × Σ , see Definition 4.5, whose projection to Σ is a union of closed loops transversallyhomotopic to a union closed loops in minimal position. By using ([10], Theorem 1.3), we havethat the atoroidal piece of d ( D (cid:98) Ω ) corresponds to Σ if d (Ω) are non simple closed curves.(1) If the Ω-arc configuration on D is the one of Remark 4.4, then we have that d ( D (cid:98) Γ ) hyp = ∅ and Remark 4.4 also gives us: v ( (cid:93) { homotopy classes of γ -arcs in D } − ≤ v (cid:93) { cusps of d ( D (cid:98) γ ) hyp } . (2) If the Ω-arc configuration on D is not the one of Remark 4.4, then there is at leastone geometric intersection point on the projection of the link complement d ( D (cid:98) Ω ) toΣ . Figure 20.
The JSJ-decomposition of d ( D (cid:98) γ ) of Figure 18. ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 21
By ([10], Theorem 1.3) we conclude that d ( D (cid:98) γ ) hyp (cid:54) = ∅ . We will now define an injectivefunction: (cid:26) (cid:98) γ -arcs in T D (cid:27) ϕ −→ (cid:26) cusps of d ( D (cid:98) γ ) hyp (cid:27) where the target can be decomposed as: (cid:26) cusps of d ( D (cid:98) γ ) hyp (cid:27) = splitting tori of theJSJ-decomposition of d ( D (cid:98) γ ) (cid:113) (cid:26) cusp in d ( D (cid:98) γ ) ∩ d ( D (cid:98) γ ) hyp (cid:27) The function ϕ is defined as follows: if the cusps in d ( D (cid:98) γ ) are induced by the (cid:98) γ -arc in T D belonging to the characteristic sub-manifold of d ( D (cid:98) γ ) , 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 ( D (cid:98) γ ) hyp and ϕ sends it to itself, see Figure20. Assume that there are more isotopy classes of (cid:98) γ -arcs in T D than the number of cusps of d ( D (cid:98) γ ) hyp . Then, there are two tori, associated with non-isotopic (cid:98) γ -arcs in T D, 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 D, contradicting the fact that they were not isotopic.Finally, since two isotopic (cid:98) γ -arcs in T D induce a homotopy between their projections in D. Then for the case for d ( D (cid:98) γ ) hyp (cid:54) = ∅ , we have that: v (cid:93) { homotopy classes of γ -arcs in D } ≤ v (cid:93) { cusps of d ( D (cid:98) γ ) hyp } , as homotopy classes of γ -arcs in D R (resp. D L ) 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, (cid:93) { homotopy classes of γ -arcs in D R } = (cid:93) { exponents of X in ω γ } , and (cid:93) { homotopy classes of γ -arcs in D L } = (cid:93) { exponents of Y in ω γ } . (cid:3) Theorem 1.3.
Given a hyperbolic metric ρ on a thrice-punctured sphere Σ , , then thereexist a constant C ρ > and a sequence { γ k } of closed geodesics on Σ with (cid:96) ρ ( γ k ) (cid:37) ∞ suchthat v (cid:18) C ρ (cid:96) ρ ( γ k ) − W ( C ρ (cid:96) ρ ( γ k )) (cid:19) ≤ Vol( M (cid:98) γ k ) ≤ v (cid:18) C ρ (cid:96) ρ ( γ k ) + 4 W ( C ρ (cid:96) ρ ( γ k )) (cid:19) , where v is the volume of a regular ideal tetrahedron and we use the Lambert W function. Proof:
We will start proving the result for a particular hyperbolic metric ρ on Σ , , byfixing the following representation ψ : π (Σ , ) := (cid:104) x, y (cid:105) → PSL ( R ) such that ψ ( x ) = X = (cid:18) (cid:19) and ψ ( y ) = Y = (cid:18) (cid:19) . Notice that x and y represent the free homotopy class of two different boundary componentsof Σ , . Let γ k be the unique closed geodesic on Σ , whose corresponding matrix representantunder ψ is: A k := k +1 (cid:89) i =1 i (cid:54) =2 ( X i Y ) if k > , and A = XY if k = 1 . Claim 4.8.
For all k ∈ N we have that:2 ln( k !) − ≤ (cid:96) ρ ( γ k ) ≤ k !) . Proof of claim:
Let A k := (cid:18) a k b k c k d k (cid:19) , then: (cid:18) a k b k c k d k (cid:19) = (cid:18) (4 k + 5) a k − + (2 k + 2) c k − (4 k + 5) b k − + (2 k + 2) d k − a k − + c k − b k − + d k − (cid:19) and A = (cid:18) (cid:19) . We briefly recall that in the case A ∈ PSL ( R ) is a hyperbolic element of trace t, theeigenvalues of A are − t ±√ t − . Let λ A be the eigenvalue satisfying | λ A | > . Then, the lengthof the closed geodesic determine by A is 2 ln | λ A | . Let us denote z k := a k + b k + c k + d k then:5 kz k − ≤ z k ≤ kz k − and z k − ≤ Trace A k ≤ kz k − . Therefore, 5 k k ! ≤ Trace A k ≤ k k ! . Notice that the eigenvalue of A k whose absolute value is bigger than one, denoted as λ A k , isbounded as follows: Trace A k ≤ | λ A k | ≤ Trace A k . Finally, 2 ln( k !) − ≤ k ln(5) + ln( k !)) − k k !) − ≤ (cid:96) ρ ( γ k )and, (cid:96) ρ ( γ k ) ≤ k k !) = 2( k ln(11) + ln( k !)) ≤ k !) . (cid:3) Figure 21.
By using the inverse of Stirling’s aproximation we have: k (cid:16) (cid:96) ρ ( γ k )ln( (cid:96) ρ ( γ k )) . For the volume upper bound, notice that by adding a crossing circle on the twisted regionof the τ , we have that the volume of the complement of (cid:98) γ the canonical lift of a figure-eightclosed geodesic in T (Σ , ) , is the same as, the volume of the complement of the canonical ERIODS OF CONTINUOUS FRACTIONS AND VOLUMES OF MODULAR KNOTS COMPLEMENTS 23 lift of the corresponding closed geodesic on Σ mod and the crossing circle in T Σ mod , by [1](see figure 21). By using the same idea as in Theorem 1.1, we obtain:Vol( M (cid:99) γ k ) < v (5 k − . The volume lower bound is a consequence of Theorem 1.4.The proof of this result for any hyperbolic metric, follows from the fact that any pairof hyperbolic metrics on a hyperbolic surface are bi-Lipschitz (see for example [4], Lemma4.1). (cid:3)
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Department of Mathematics and Statistics, University of Helsinki.Pietari Kalminkatu 5, Helsinki FI 00014