Efficient geodesics and an effective algorithm for distance in the complex of curves

Abstract

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.

Appendix: Webb’s algorithm

In this appendix we give an exposition of Webb’s algorithm for computing distance in $\mathscr {C}(S)$. As with the efficient geodesic algorithm we will make the inductive hypothesis that for some $n \ge 3$ we have an algorithm to determine if the distance between two vertices is $0, \dots ,n-1$ and we would like to give an algorithm for determining if the distance between two vertices is n. First we introduce an auxiliary tool, the arc complex for a surface with boundary.

Arc complex Let F be a compact surface with nonempty boundary. The arc complex $\mathscr {A}(F)$ is the simplicial complex with k-simplices corresponding to $(k+1)$-tuples of homotopy classes of essential arcs in F with pairwise disjoint representatives. Here, homotopies are allowed to move the endpoints of an arc along $\partial F$, and an arc is essential if it is not homotopic into $\partial F$.

The algorithm In what follows, we assume that $\chi (F) < 0$. A maximal simplex of $\mathscr {A}(F)$ can be regarded as a triangulation of the surface obtained from F by collapsing each component of the boundary to a point. If F is a compact, orientable surface of genus g with m boundary components, then the number of edges in any such triangulation is $6g+3m-6$.

Let v and w be two vertices of $\mathscr {C}(S)$ with $d(v,w) \ge 3$. As in the efficient geodesic algorithm, it suffices by the induction hypothesis to list all candidates for vertices $v_1$ on a tight geodesic $v=v_0,\dots ,v_{n}=w$. Since there are finitely many vertices in each simplex of $\mathscr {C}(S)$ it further suffices to list all candidates for simplices $\sigma _1$ on a tight multigeodesic $v=\sigma _0,\dots ,\sigma _{n}=w$.

Suppose we have such a tight multigeodesic $v=\sigma _0,\dots ,\sigma _{n}=w$. We can choose representatives $\alpha _i$ of the $\sigma _i$ so that $\alpha _i \cap \alpha _{i+1} = \emptyset$ for all i and so that each $\alpha _i$ lies in minimal position with $\alpha _0$. If we cut S along $\alpha _0$, we obtain a compact surface $S'$, some of whose boundary components correspond to $\alpha _0$.

For each $i > 1$, the representative $\alpha _i$ gives a collection of disjoint arcs in $S'$ and hence a simplex $\tau _i$ of $\mathscr {A}(S')$ (some arcs of $\alpha _i$ might be parallel and these get identified in $\mathscr {A}(S')$). For $i \ge 3$, the collection of arcs is filling, which means that when we cut $S'$ along these arcs we obtain a collection of disks and boundary-parallel annuli, and we say that the corresponding simplex of $\mathscr {A}(S')$ is filling.

Since there is a unique configuration for $\alpha _{n}$ and $\alpha _0$ in minimal position, there is a unique possibility for $\tau _{n}$. As $\tau _{n} \cup \tau _{n-1}$ is contained in a simplex of the arc complex of $S'$ and since $\tau _{n}$ is filling, there are finitely many possibilities for $\tau _{n-1}$ (and we can explicitly list them). This is the key point: there are infinitely many vertices of $\mathscr {C}(S)$ that correspond to any given simplex in the arc complex, but there are finitely many choices for the simplex itself.

Because $\tau _i$ is filling whenever $i \ge 3$, we can continue this process inductively, and explicitly list all possibilities for $\tau _2$. Now, by the definition of a tight multigeodesic in $\mathscr {C}(S)$, the simplex $\sigma _1$ is represented by the union of the essential components of the boundary of a regular neighborhood of $\alpha _0 \cup \alpha _2$. Equivalently, any such $\sigma _1$ is given by a regular neighborhood of the union of $\partial S'$ with a representative of $\tau _2$. Hence there are finitely many (explicitly listable) possibilities for $\sigma _1$, as desired.

A bound on the number of candidates In the introduction we stated that the number of candidate simplices $\sigma _1$ produced by Webb’s algorithm when $d(v,w)=n$ is bounded above by

$$\begin{aligned} 2^{(72g+12)\min \{n-2,21\}} (2^{6g-6}-1). \end{aligned}$$

We will now explain this bound; we are grateful to Richard Webb for supplying us with the details.

We can think of the sequence $\tau _{n},\dots ,\tau _3$ as a path in the filling multi-arc complex, that is, the simplicial complex whose vertices are simplices of $\mathscr {A}(S')$ whose geometric realizations fill $S'$ and whose edges correspond to simplices with geometric intersection number zero. Then we obtain $\tau _2$ by extending this path by one more edge and taking some nonempty subset of the simplex of $\mathscr {A}(S')$ represented by the endpoint $\hat{\tau }_2$ of this extended path.

Webb proved that the degree of an arbitrary vertex of this filling multi-arc complex is bounded above by $2^{72g+12}$ (this is for the case where we start with a closed surface of genus g and cut along a single simple closed curve, as above); see his paper [19]. Our extended path from $\tau _n$ to $\hat{\tau }_{n-2}$ has length $n-2$ and so this a priori gives a bound of $2^{(72g+12)(n-2)}$ for the number of possibilities for $\hat{\tau }_2$. However, there is a version of the bounded geodesic image theorem which tells us that, because the $\tau _i$ arise from a geodesic in $\mathscr {C}(S)$, the actual distance in the filling multi-arc complex between $\tau _n$ and $\hat{\tau }_2$ is bounded above by 21. This gives the first multiplicand in the desired bound. The second multiplicand comes from the number of ways of choosing a nonempty sub-simplex $\tau _2$ of $\hat{\tau }_2$. The number of vertices of $\tau _2$ is bounded above by $6g-6$, and so there are $2^{6g-6}-1$ ways to choose $\tau _2$ from $\hat{\tau }_2$.