Efficient geodesics and an effective algorithm for distance in the complex of curves (original) (raw)

2014, arXiv (Cornell University)

We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for all distances accessible by computer. Our method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky. FIGURE 1. Vertices of C (S 2) with distance 4 and intersection number 12; this is the smallest possible intersection for vertices with distance 4 2 JOAN BIRMAN, DAN MARGALIT, AND WILLIAM MENASCO One goal of this paper is to give an algorithm for distance-the efficient geodesic algorithm-that actually can be implemented, at least for small distances. The third author and Glenn, Morrell, and Morse [9] have in fact already developed an implementation of our algorithm, called Metric in the Curve Complex [8]. Their program is assembling a data bank of examples as we write. Known examples. Let S g denote a closed, connected, orientable surface of genus g and let i min (g, d) denote the minimal intersection number for vertices of C (S g) with distance d. The Metric in the Curve Complex program has been used to show that: (1) i min (2, 4) = 12 and (2) i min (3, 4) ≤ 21. The highly symmetric example in Figure 1-which realizes i min (2, 4)-was discovered using the program. See Section 2 for a discussion of this example and a proof using the methods of this paper that the distance is actually 4. We are only aware of one other explicit picture in the literature of a pair of vertices of C (S 2) that have distance four, namely, the example of Hempel that appears in the notes of Saul Schleimer [16, Figure 2] (see [9, Example 1.6] for a proof that the distance is 4). This example has geometric intersection number 25. Using the bounded geodesic image theorem [12, Theorem 3.1] of Masur and Minsky (as quantified by Webb [21]) it is possible to explicitly construct examples of vertices with any given distance; see [17, Section 6]. We do not know how to keep the intersection numbers close to the minimum with this method, but Aougab and Taylor did in fact use this method to give examples of vertices of arbitrary distance whose intersection numbers are close to the minimum in an asymptotic sense; see their paper [3] for the precise statement. Local infinitude. One reason why computations with the complex of curves are so difficult is that it is locally infinite and moreover there are infinitely many geodesics (i.e. shortest paths) between most pairs of vertices. Masur and Minsky [12] addressed this issue by finding a preferred set of geodesics, called tight geodesics, and proving that between any two vertices there are finitely many tight geodesics; see Section 2.2 for the definition. Our first goal is to give a new class of geodesics that still has finitely many elements connecting any two vertices but is more amenable to certain computations. Efficient geodesics. Our approach to geodesics in C (S) is defined in terms of intersections with arcs. First, suppose that γ is an arc in S and α is a simple closed curve in S. We say that γ and α are in minimal position if α is disjoint from the endpoints of γ and the number of points of intersection of α with γ is smallest over all simple closed curves that are homotopic to α through homotopies that do not pass through the endpoints of γ. Let v 0 ,. .. , v n be a geodesic of length at least three in C (S), and let α 0 , α 1 , and α n be representatives of v 0 , v 1 , and v n that are pairwise in minimal position (this configuration is unique up to isotopy of S). A reference arc for the triple α 0 , α 1 , α n EFFICIENT GEODESICS AND AN EFFECTIVE ALGORITHM FOR DISTANCE 3 is an arc γ that is in minimal position with α 1 and whose interior is disjoint from α 0 ∪ α n ; such arcs were considered by Leasure [11, Definition 3.2.1]. We say that the oriented geodesic v 0 ,. .. , v n is initially efficient if |α 1 ∩ γ| ≤ n − 1 for all choices of reference arcs γ (this is independent of the choices of α 0 , α 1 , and α n by the uniqueness statement above). Finally, we say that v = v 0 ,. .. , v n = w is efficient if the oriented geodesic v k ,. .. , v n is initially efficient for each 0 ≤ k ≤ n− 3 and the oriented geodesic v n , v n−1 , v n−2 , v n−3 is also initially efficient. We emphasize that to test the initial efficiency of v k ,. .. , v n we should look at reference arcs for the triple v k , v k+1 , and v n and we allow n − k − 1 points of intersection of (a representative of) v k+1 with any such reference arc. Existence of efficient geodesics. Our main result is that efficient geodesics always exist, and that there are finitely many between any two vertices.