A user's quide to the mapping class group: Once punctured surfaces

An automatic structure for the mapping class group of a surface of finite type was described in [M]. This document is intended as a practical guide to computations using a variant of this automatic structure, in the special case of a once-punctured, oriented surface S. As such, we shall try to be more descriptive and less theoretical than in [M], leaving the reader to consult [M] for detailed proofs. Our primary goal is that the reader may learn, as quickly as possible, how to compute in the mapping class group of a once-punctured surface: we describe a quadratic time algorithm for the word problem, henceforth called the algorithm, which can be implemented with pencil and paper. A Mathematica version of the algorithm is (or will soon be) available; check the software library at the Mathematical Sciences Research Institute (e-mail address: msri.org), or the Geometry Center (geom.umn.edu). As with any computational method, it is necessary to learn some of the theory in order to learn the algorithm. We spend some time developing various combinatorial tools, with enough justification supplied to aid understanding and lessen the steepness of the learning curve. There is a trade-off involved here: time invested understanding theory may be time wasted gaining proficiency; I do not know if I have found the right balance. Also, despite my stated purpose, in a few places I have put in perhaps too much detail about items of combinatorica that interest me, but which are not really to the point, so the reader is forewarned. The algorithm described herein can be adapted to arbitrary punctured surfaces, with or without boundary and orientation. The data structures needed do not lend themselves quite so nicely to pencil and paper calculation, so we do not pursue the issue here; details can be found in [M]. And while an automatic structure for the mapping class group of a closed surface is described in [M], in this case the results are not suited for practical calculation, because of the non-constructive nature of the proof; hopefully a practical automatic structure will emerge from a deeper understanding of closed surfaces. For the rest of the paper, let S be an oriented, once-punctured surface which is not the 2-sphere. We regard S as a closed surface with a distinguished point p, the puncture. The mapping class group is MCG(S) = Homeo(S)/ Homeo 0 (S), where Homeo(S) is the group of …