[Paper Review] An algorithm for deciding reducibility

Mark C. Bell|arXiv (Cornell University)|Mar 12, 2014

Geometric and Algebraic Topology8 references5 citations

TL;DR

This paper presents an algorithm to determine whether a mapping class on a marked surface is reducible by encoding the mapping class group action on multicurves via triangulations. It proves that reducibility can be decided in time exponential in word length, and shows that the size of the canonical curve system is at most exponential in word length, placing the problem in NP.

ABSTRACT

For a marked surface we construct a description of the action of the mapping class group on the set of multicurves via a triangulation. We use this encoding to give an algorithm to determine if a mapping class is reducible in time exponential in the word length with respect to some fixed generating set. We go on to show that a reducible mapping class fixes a maximal multicurve whose size is at most exponential in the word length. From this we deduce that determining if a mapping class is reducible is a problem in NP. Moreover, this shows that the size of the canonical curve system of a mapping class is also at most exponential in the word length.

Motivation & Objective

• To develop an algorithm for determining whether a mapping class is reducible on a marked surface.
• To describe the action of the mapping class group on multicurves using a triangulation-based encoding.
• To bound the size of the canonical curve system of a mapping class in terms of word length.
• To establish that the reducibility problem lies in NP by showing the existence of a certificate of size at most exponential in word length.

Proposed method

• Encoding the action of the mapping class group on multicurves through a triangulation of the surface.
• Using the triangulation to represent mapping classes and their actions on multicurves in a computable form.
• Constructing an algorithm that checks reducibility by analyzing the induced action on the triangulation and multicurve structures.
• Proving that any reducible mapping class fixes a maximal multicurve whose size is bounded by an exponential function of the word length.
• Using the bounded size of the canonical curve system to place the reducibility decision problem in NP.
• Establishing that the size of the canonical curve system is at most exponential in the word length with respect to a fixed generating set.

Experimental results

Research questions

• RQ1 Can the reducibility of a mapping class be decided algorithmically using a triangulation-based encoding of the mapping class group action?
• RQ2 What is the maximum size of the canonical curve system associated with a mapping class in terms of word length?
• RQ3 Is the problem of determining whether a mapping class is reducible in the complexity class NP?
• RQ4 How does the structure of the mapping class group action on multicurves relate to the word length of the class?

Key findings

• The algorithm decides reducibility in time exponential in the word length of the mapping class with respect to a fixed generating set.
• A reducible mapping class fixes a maximal multicurve whose size is at most exponential in the word length.
• The size of the canonical curve system of a mapping class is bounded above by an exponential function of the word length.
• The problem of determining whether a mapping class is reducible is in NP, as a certificate of bounded size exists.
• The triangulation-based encoding provides a finite, computable representation of the mapping class group action on multicurves.
• The results establish a tight connection between word length and structural complexity in mapping class group dynamics.