The pseudo-Anosov and conjugacy problems are in $\textbf{NP} \cap \textbf{co-NP}$

TL;DR

This paper proves that the problems of identifying pseudo-Anosov mapping classes and conjugacy of mapping classes are in NP and co-NP, providing polynomial bounds and exponential time algorithms for these problems.

Contribution

It establishes the NP and co-NP complexity classifications for the pseudo-Anosov and conjugacy problems, with polynomial bounds on splitting sequences.

Findings

Deciding pseudo-Anosov mapping classes is in NP.

Deciding conjugacy of mapping classes is in co-NP.

Provides exponential time algorithms based on polynomial bounds.

Abstract

For a fixed marked surface $S$, we construct polynomial bounds on the periodic and preperiodic lengths of the maximal splitting sequences of a projectively invariant measured train track. We give two consequences of these bounds. Firstly, that the problem of deciding whether a mapping class is pseudo-Anosov lies in $\textbf{NP}$. This is dual to the previously known result that the pseudo-Anosov problem is in $\textbf{co-NP}$. Secondly, that the problem of deciding whether two mapping classes are conjugate lies in $\textbf{co-NP}$. Similarly, this is the dual to the previously known result that the conjugacy problem is in $\textbf{NP}$. As usual, in both cases we immediately obtain exponential time solutions to these problems. A version of these algorithms have been implemented as part of flipper.