Is action of MCG on the curve complex computable for closed surfaces? [Yes: Birman Exact Sequence]

Question

$\DeclareMathOperator\MCG{MCG}$Let $\Gamma=\MCG(S_{g, 0,0 })$ be mapping class group of closed hyperbolic surfaces. Let $V=C^0(S)$ be the set of vertices of the simplicial curve complex. We are studying the well known group action $\Gamma \times V \to V$, yet we find ourselves utterly incapable of computing and translating elements $([\phi], [\alpha]) \mapsto [\phi(\alpha)]$ where $\alpha$ represents a "simple closed curve" in $V$. In otherwords it appears that the action of the MCG on the curve complex is incomputable for closed surfaces.

I am aware of Mark C. Bell's curver program. It's documentation indicates that it uses results of Saul Schleimer relating paths in a "flip graph" with words in the pointed mapping class groups. I am not aware of any flip graph analogy for the closed mapping class groups.

My question: What is best computational approach to being able to solve the following problem for the group action $\Gamma \times V \to V$:

``Given $[\phi] \in \Gamma, [\alpha] \in V, [\beta] \in V$, determine whether $[\phi(\alpha)]=[\beta]$ is True or False in $V$."

Evidently the fact that we have no linear representations of $\Gamma$ is one computational obstruction, for otherwise all the quantities could essentially be represented as matrices.

In my research I have interest in constructing finite subsets $I$ for which a chain sum $\sum_{[\phi] \in I} \sum_{[\alpha] \in B}[\phi(\alpha)]=0$ mod 2. Here $B$ is a finite subset of $V$ based on Nathan Broaddus' homology spheres. (e.g. $B$ is the vertex set if the two dimensional spheres in genus $g=2$ discovered by Broaddus).

(**) I'll accept Sam Nead's answer since it suggests that Birman's exact sequence tells us that forgetful map from pointed mapping classes to unpointed mapping classes is onto, with kernel equal to the image of $\pi_1(S, pt)$ under the Push map. So it's obvious from Birman's sequence that Bell-Webb's curver program can be used to compute pure mapping class actions on the curve complex. Although this raises the question of whether the membership problem for image of $\pi_1(S, pt)$ by the push map is computable in the pointed groups. Assuming it is, then to solve the subset-sum problem I'll start with the pointed groups in curver.

Answer 1

You write "it appears that the action of the MCG on the curve complex is incomputable for closed surfaces."

This is not correct. Geva Yashfe points out one approach in the comments. Here is another, with references given below.

Fix $S$ a connected, closed, oriented surface of genus $g$. Fix a one-vertex triangulation of $S$. We represent simple closed curves via "edge coordinates" (or we could use normal coordinates). Note that (as $S$ is closed) two curves can have different edge coordinates yet be isotopic.

Thus your problem reduces to determining isotopy equivalence among such representatives. (The presence of a mapping class is dealt with by understanding (say) how Dehn twists act on edge coordinates.) It is a theorem that two curves are isotopic if and only if they cobound an annulus, perhaps after performing a sequence of "bigon" moves. Detecting such annuli or bigons, and performing bigon moves, can be done computationally.

A naive implementation will be polynomial time in the edge weights. Thinking more deeply gives an algorithm that is polynomial time in the logarithm of the edge weights. This brings us to the beginnings of modern research in this area, and so answers your question (in italics).

You can find versions and discussions of this in papers such as

• Schaefer, Sedgwick, and Štefankovič [2002]
• Agol, Hass, and Thurston [2005]
• Erickson and Nayyeri [2012]

as well as work of Bell (partly with Webb).