# Creating pseudo-Anosov Maps from Permutations and Matrices

## TL;DR

This paper introduces a combinatorial method to characterize pseudo-Anosov maps on closed surfaces using permutations called ordered block permutations, enabling a systematic construction of such maps from permutation data.

## Contribution

It provides a novel integral combinatorial framework to encode and construct pseudo-Anosov maps via admissible permutations, linking surface topology with permutation theory.

## Key findings

- Characterization of pseudo-Anosov maps using ordered block permutations
- Construction of pseudo-Anosov maps from admissible permutations
- Every orientable invariant foliation can be generated from permutation data

## Abstract

We provide an integral combinatorial characterization of pseudo-Anosov maps on closed oriented surfaces of genus g > 1. We show that an orientation-preserving pseudo-Anosov homeomorphism with orientable foliations and fixing all critical trajectories can be encoded as a permutation of 2g+v-1 positive integers, where v is the number of singular points of the foliations (disregarding multiplicity). We call such a permutations an ordered block permutation (OBP), and it satisfies an admissiblity condition. Conversely, we show that a surface along with measured foliations (up to scaling) and the pseudo-Anosov map can be uniquely constructed out of the data of an admissible permutation of 2g+v-1 positive integers. In particular, for closed surfaces, we construct every orientable foliation invariant under a pseudo-Anosov homeomorphism.