WINDING AND UNWINDING AND ESSENTIAL INTERSECTIONS IN H 3

Transcription

1 WINDING AND UNWINDING AND ESSENTIAL INTERSECTIONS IN H 3 JANE GILMAN AND LINDA KEEN Abstract. Let G = A, B be a non-elementary two generator subgroup of Isom(H 2 ). If G is discrete and free and geometrically finite, its quotient is a pair of pants and in [5] we produced a formula for the number of essential self intersections (ESIs) of any primitive geodesic on the quotient. An ESI is a point where the geodesic has a self-intersection on a seam. Self-intersections of geodesics on arbitrary hyperbolic surfaces have recently been studied by Basmajian [1] and Chas [2]. Here we extend our results to groups two generator groups G Isom(H 3 ) which are discrete, free and geometrically finite. We generalize our definition of ESIs and give a geometric interpretation of them in the quotient manifold. We show that they satisfy the same formulas. 1. Introduction In [5] we considered some geometric implications of the Gilman- Maskit discreteness algorithm, [9], for subgroups of Isom(H 2 ). We showed that when the algorithm decides that the group is discrete and free and represents a three holed sphere, it does so by finding a special set of stopping generators that correspond to the only three simple geodesics on the surface, those around the waist and the cuffs. Every other geodesic that winds around the waist and cuffs and has selfintersections. Some of these intersections necessarily lie on the seams, that is, on the geodesic arcs joining the simple geodesics. We call these essential self-intersections to differentiate them from the others. We produce a formula to count the essential self intersections of primitive geodesics. We showed that reversing the steps of the algorithm by starting with the simple geodesics of the stopping generators and, going step by step to the original generators, corresponds to winding the simple geodesics around themselves creating the essential intersections on the seams. Date: August 14, Some of this work was carried out while the author was a visitor at ICERM. This work was supported by a CUNY Collaborative grant and grants from PSC- CUNY. 1

2 2 JANE GILMAN AND LINDA KEEN In this paper we generalize our results to two generator subgroups of Isom(H 3 ). We use the approach to the discreteness problem we presented in [8] using the right angled hexagons defined by Fenchel, [3] along with a technique described in [11, 12]. In [8] we produced a semialgorithm which, in the case that G is discrete, free and geometrically finite is an algorithm. That is, it stops after finitely many steps when it finds a special hexagon that we call a canonical hexagon as well as a triple of group elements corresponding to three alternating sides of the hexagon; these are stopping generators for this situation. These elements project to simple geodesics in the quotient manifold that are special in the sense that they are the shortest among all geodesics coming from elements of the group. The other three alternating sides project to the set of geodesic arcs in the quotient manifold invariant under the hyperelliptic involution, see [4] These are the generalization of the seams. The other group elements produced by the semialgorithm now project to simple curves in the 3-manifold they have no self intersections. We show however, that there are special pairs of points on these curves that we call ESI points which have properties analogous to the essential self-intersections in the two dimensional case. Our main result is that the ESIs can be counted by using the same formula as in the two dimensional case. We also show that, reversing the semi-algorithm corresponds to a generalization of winding the special geodesics around the seams to get back to the geodesics of the original generators. The paper is organized as follows. In Section 2 we give some background and define the terms we use. In Section 3 we the Generalized Essential Self intersection points and in Section 4 we prove the main theorem that gives an inductive formula to count the number of these points. In Section 5 we apply this counting formula and give geometric interpretations of it r in the quotient three-manifold. 2. Background and Terminology We use the following terminology. If M is a geodesic in H 3 we let H M denote the half-turn about M. If X is an isometry we let Ax X denote the invariant geodesic joining its fixed points; this is called its axis. If M is any geodesic orthogonal to Ax X, there is another geodesic M orthogonal to Ax X such that X is the product of the two half-turns, X = L M L M. Note that any two geodesics in H 3 have a unique common perpendicular. If G = X, Y H 3 there is a standard way to associate a right angled hexagon H = H(X, Y ) to G. Namely, we consider the axes

3 WINDING AND UNWINDING AND ESSENTIAL INTERSECTIONS IN H 3 3 Ax X, Ax Y, Ax X 1 Y and let L be the common perpendicular of Ax X and Ax Y, L X the common perpendicular of Ax X and Ax X 1 Y and L Y the common perpendicular of Ax Y and Ax X 1 Y. Further, X factors as H L H LX and Y as H L H LY. The six geodesics form a six-sided polygonal curve in H 3 that we denote by H(X, Y ). In [8] L is termed the core geodesic. We term the sides of the hexagon respectively half-turn sides and axes sides. This hexagon can be defined whatever the geometric type of X and Y, but for simplicity here we assume that X and Y are both loxodromic and that their axes are disjoint axes Skew acute, skew obtuse hexagons. The hexagon H is defined so that its sides meet at right angles at the vertices. However, there are other important angles related to the hexagon. Denote the sides S i, i = 1,..., 6, where i is always taken mod 6. Now let p i be the vertex where the sides S i and S i+1 meet. Let N i be the normal to S i, S i+1 at p i and let V i be the tangent to the side S i at p i. Denote the forward and backwards parallel transports of V i to p i+1 and p i 1 respectively, by V i and V i. Then we can measure the transport angle θ i between V i 1 and V i+1 at p i and take it in the interval ( π, π]. If θ i π we call it obtuse and if θ i < π/2 we call it acute. We call a hexagon skew acute if all the transport angles are acute. This notation comes from the fact that when the hyperbolic lengths of the half-turn sides are all zero, the hexagon reduces to a triangle and the transport angles for these half-turn sides are the external angles of the triangle Palindromic Enumeration. In [6] we define an enumeration scheme to find all pairs of primitive elements 1 of a rank two free group from a given pair. Each conjugacy class of a primitive element contains words of minimal length. These are unique up to cyclic permutation. In the enumeration scheme, each pair of primitives consists of either two palindromic words or one palindrome and a product of the two palindromes that appeared at the previous step in the scheme. At a given point in the scheme there are always two ways to choose the next pair, what we call a left choice and a right choice. The sequence of integers [n 1,..., n k ], where n i is the number of left (right) choices before the next right (left) choice, is identified with the continued fraction 1 There are many uses of the term primitive; here we use it as in [10], an element in a free group of rank n is primitive if it can be extended to be part of a minimal generating set. When a group is a discrete free group, by a primitive curve on the quotient we mean the image of a primitive element in the group.

4 4 JANE GILMAN AND LINDA KEEN of a rational number. We use this identification to label the primitives with rational numbers The Procedure. In [8] we define a semi-algorithm we call The Procedure for two generator non-elementary subgroups of G Isom(H 3 ). In the case that G Isom(H 2 ), the Procedure is equivalent to the Gilman-Maskit algorithm. Given G = A, B, the Procedure begins by setting A 0 = A, B 0 = B. At step k, it looks at the hexagon for A k, B k, H(A k, B k ). If it is skew acute, it stops. If not, it decides whether to make a left or right choice to determine the next pair of elements A k+1, B k+1. One of the main results in that paper is that if G is discrete, free and geometrically finite, it does stop. In that case, we denote the last pair A k, B k by C, D and we prove the skew-acute hexagon H(C, D) is essentially unique and call it the Canonical hexagon for G. The Procedure records the successive numbers of left and right choices in a sequence [n 1,..., n k ]. We note that in this case we can apply the Procedure to the stopping generators to arrive at our starting pair by using steps determined by the backward sequence of steps [n k,..., n 1 ]. 3. Generalized Essential Self Intersections The groups G and 3G = L A, L, L B are simultaneously discrete, free, geometrically finite groups. Therefore, the images of the half-turn lines under G tessellate H 3. We call this collection of lines the half-turn tessellation and denote it by T H. In the introduction, we gave an informal description of the essential self-intersection points of a geodesic on a hyperbolic pair of pants. Here we give a formal definition which we will then generalize to H 3. Definition 1. Suppose that G Isom(H 2 ) is free and discrete and that X is an element in G such that there are half-turn geodesics L 1, L 2 T H such that X = H L1 H L2. Let I X be the segment of Ax X between its intersection with the half-turn line L 1 and the half-turn L 2. Let E X be the set of points on I X where Ax X meets lines in T H. These are the Essential Self-Intersection points of X. This definition agrees with our informal definition. To see this, note first that the elements of T H project to the seams of the quotient surface. Then note that a fundamental segment for X is I X H L1 (I X ) and since T H is invariant under H L1 the essential self-intersection points on I X are paired with their images on H L1 (I X ). Note that if X is primitive, any intersection of its axis with a geodesic of T H in I X cannot be orthogonal. It follows that if γ X is the projection

5 WINDING AND UNWINDING AND ESSENTIAL INTERSECTIONS IN H 3 5 of Ax X to the quotient surface, then E X together with the endpoints of I X contains one lift of each of the points where γ X intersects a seam. Moreover, γ X cannot intersect the seam orthogonally except at the projections of the endpoints of I X, so the points are true intersection points. If X is not primitive, I X may contain more than one lift of each intersection point with a seam and it may contain points where γ X does not intersect itself but crosses the seam orthogonally. We generalize the Essential Self intersection points to the current setting, H 3. Here, we no longer have points but line segments. Definition 2. Suppose that G Isom(H 3 ) is free, discrete, and nonelelementary and that X is an element in G such that there are half-turn geodesics L 1, L 2 T H such that X = H L1 H L2. Let I X be the segment of Ax X between its intersection with L 1 and L 2. Let L U be any line in T H and let O U be the common orthogonal to the pair of lines (L U, A X ). Let T O be the collection of these orthogonal lines. Now let E X be the intersection points of the subset of O X of T O consisting of those orthogonals that intersect Ax X within I X. These intersection points on I X are the Generalized Essential Self Intersection points of X, or for brevity, the ESI points of X. By abuse of notation, denote the segments of the lines in O X between L U and Ax X by O X again. If we consider the interval ĪX = H L (I X ), we can define the sets ŌX and ĒX as above. Since T H is invariant under H L there are the same number of points in ĒX as in E X and similarly the same number of segments in ŌX as in O X. We denote this number by ESI(X). As in the two dimensional case, let γ X be the projection of Ax X to the quotient manifold. The projections of the segments in ŌX and O X form connectors between points in the projections of the points in E X and E X on γ X. These are orthogonal to the invariant lines of the hyperelliptic involution. If X is primitive, the lifts of the connectors are in one to one correspondence with the points in E X (or ĒX). 4. Counting Generalized Essential Self Intersections Theorem 4.1 (Products). Suppose X and Y are not necessarily primitive elements of a group G, a two-generator, free discrete, non-elementary, geometrically finite group. Assume further that the common perpendicular to the axes of X and Y belongs to T H and that X and Y have ESI numbers ESI(X) and ESI(Y ). Then the ESI number of their

6 6 JANE GILMAN AND LINDA KEEN product is given by the formula ESI(XY ) = ESI(X) + ESI(Y ) + 1. Proof. This is modification of the proof of [11, 12]. Suppose the common orthogonal to Ax X and Ax Y is a half-turn line. To keep the notation simple, denote it here by L and suppose that L X, L Y T H are such that X = H LX H L, Y = H LY H L. It follows that XY = H LX H L H LY H L. Set L Ȳ = H L (L Y ). It is invariant under H L H LY H L. Let Γ XY be the polygonal curve in H 3 formed by following the segment Ax X from its intersection with L X to its intersection with L, then following L to its intersection with Ax Y and finally following Ax Y to its intersection with L Ȳ. Then Γ XY H L (Γ XY ) projects to a curve γ XY in the quotient manifold that is freely homotopic to γ XY, the projection of the axis of XY. For readability set m = ESI(X), n = ESI(Y ). Let L X i, i = 1,... m be the half turn lines in T H whose common orthogonal O X i with Ax X meets it in I X. Define L Y i, O Y i, i = 1 + m,..., m + n, similarly. Let O XY i, i = 1,... m + n be the orthogonals from L X i and L Y i to Ax XY. Let O XY n+m+1 be the orthogonal between the L and Ax XY. Consider the projections of the points where all these orthogonals meet Ax XY to γ XY. Then because of the homotopy, we can find exactly one lift of each of these points on γ XY to the interval I XY of Ax XY between L X and L Ȳ. Therefore there are exactly m + n + 1 points on I XY that are intersection points of orthogonals between Ax XY and lines in T H as claimed. 5. Canonical hexagons and Winding We first prove that the Canonical hexagon has the property that primitive elements whose axes are its axis sides have zero ESI number. Theorem 5.1 (No Essential Intersections). If G = A, B is discrete, free and geometrically finite, then the primitives that are the axes sides of its Canonical hexagon have no ESI points. Proof. Let (C, D, C 1 D) be the triple whose axes are sides of the Canonical Hexagon H = H(C, D). In [8] we proved that the Canonical Hexagon has the property that its axis sides have shortest hyperbolic length among all hexagons formed from primitive elements. Suppose there is a half-turn side, ˆL T H such that the orthogonal between ˆL and (for definiteness) Ax C intersects Ax C between its vertices in H(C, D). Then Ax C and ˆL are sides of a skew-obtuse hexagon Ĥ in which the transport angle along Ax C is obtuse. This means that if we

7 WINDING AND UNWINDING AND ESSENTIAL INTERSECTIONS IN H 3 7 apply the Procedure to Ĥ we obtain a new hexagon H that has an axis side shorter than any of the sides of Ĥ, and hence a side shorter than the axis side Ax C. The same is true for with C replaced by D or C 1 D. Therefore H cannot be canonical. We next count the ESIs of a primitive element using the palindromic enumeration scheme: Lemma 5.2. Let (U, V ) be a primitive pair of elements occurring in the palindromic enumeration scheme and let its ESI numbers be ESI(U) and ESI(V ). A left choice for the next pair is (U, UV ) and a right choice is (V U, V ). If we make m left choices, we have the pair (U m V, V ) and we have ESI(U m V ) = 1 + mesi(u) + ESI(V ). Proof. Using Theorem 4.1 we see that ESI(UV ) = ESI(V U) = ESI(U) + ESI(V ) + 1. If we apply it inductively, the lemma follows. We note that if we start with the primitive pair (C, D) whose hexagon is the canonical hexagon, we have a formula for the generalized Essential Self-Intersections for all primitive elements of G. We can use the sequence determined by the Procedure to get back to the original primitive pair (A, B). Note that if the sequence that determines how to get from (A, B) to (C, D) is denoted by [q 1,..., q t ] and the sequence that determines how to get from (C, D) to (A, B) is denoted by [n 1,..., n t ] then [n 1,..., n t ] = [q t,..., q 1 ]; that is n k = q t k. Theorem 5.3. Main Theorem Suppose H(C, D) is canonical hexagon for the given group G. Suppose U is a primitive element obtained using the enumeration sequence that corresponds to the rational p/q whose continued fraction is [n 1,..., n t ]. Then if U k corresponds to the k th approximant of p/q, [n 1,..., n k ] = r k /s k, ESI(U k ) can be written in terms of ESI(U k 1 ) and ESI(U k 2 ) as follows: ESI(U k ) = n k ESI(U k 1 ) + ESI(U k 2 ) + 1. Proof. This follows directly from the above lemmas and corollaries because U 0 = C, U 1 = D, ESI(C) = ESI(D) = ESI(C 1 D) = 0. Remark 5.1. Since the number ESI(X) is a conjugacy invariant, Theorem 5.3 allows on to find the ESI for any primitive curve in the group.

8 8 JANE GILMAN AND LINDA KEEN Remark 5.2. Given G = A, B such that G is discrete free and geometrically finite, the Procedure of [8] determines a sequence [q 1,..., q t ] as it finds the Canonical hexagon. By the theorem above, the ESI numbers of the primitive elements it encounters go down to zero. This is analogous to the unwinding of a primitive element on a pair of pants described in [5]. To see this, we can look at the quotient manifold. If we move along γ X, the projection of the axis of a primitive X, we encounter its intersections with the orthogonals to the half-turn lines. Let p i be one such intersection and let σ i be the projection of the connector, that is the orthogonal from a half-turn line L i to Ax X that hits Ax X at p i. Let σ i be the image of σ i under the hyperelliptic involution. Now let γ i be the segment of Ax X between p i and p i+1. We have a loop made by following in order, σ i, γ i, σ i+1, σ i+1, γ i 1, σ i. The loop goes through exactly two of the invariant lines of the hyperelliptic involution and is invariant under the involution. It is homotopic to the projection of the axis of one of primitives of the Canonical hexagon. The curve γ X γ X is a closed curve homotopic to the union of all the loops. The number of loops is ESI(X). Thus we can think of it as wound around the simplest loops and the Procedure as determining an unwinding. Going from the Canonical hexagon generators to any other primitive by a series of left right choices in the enumeration scheme corresponds to winding. References 1. Basmajian, A. Universal length bounds for non-simple closed geodesics on hyperbolic surfaces J. Topol. 6 (2013), no. 2, Chas, M. Self-intersection numbers of length-equivalent curves on surfaces Exp. Math. 23 (2014), no. 3, Fenchel, Werner Elementary Geometry in Hyperbolic Space degruyter, Gilman, Jane and Keen, Linda, Two generator Discrete Groups: Hyperelliptic Handlebodies Geom. Dedicata, Volume 110(no. 1), (2205), Gilman, Jane and Keen, Linda, Word Sequence and Intersection Numbers, Cont. Math 311 (2002), Gilman, Jane and Keen, Linda, Discreteness Criteria and the Hyperbolic Geometry of Palindromes, Journal of Conformal Geometry and Dynamics 13 (2009), 7. Gilman, Jane and Keen, Linda Enumerating Palindromes in Rank Two Free Groups, Journal of Algebra, (2011) Gilman, Jane and Keen, Linda, Canonical Hexagons and the P SL(2, C) Discreteness Problem preprint, arxiv. 9. Gilman, Jane and Maskit, Bernard, An Algorithm for Two-generator Discrete Groups Michigan Math J 38 (1991), Magnus, Wilhelm; Karass, Abraham; and Solitar, Donald Combinatorial Group Theory (1966) John Wiley & Sons, NYC. 11. Mailk, Vidur, Thesis, Rutgers University Newark, 2007.

9 WINDING AND UNWINDING AND ESSENTIAL INTERSECTIONS IN H Mailik, Vidur, Curves Generated on Surfaces by the Gilman-Maskit Algorithm, AMS-CONM, vol. 510 (2010), Department of Mathematics and Computer Science, Rutgers University, Newark, NJ address: gilman@rutgers.edu Department of Mathematics, The Graduate Center and Lehman College, CUNY, New York, NY address: LKeen@gc.cuny.edu