# Galois conjugates of pseudo-Anosov stretch factors are dense in the complex plane

## TL;DR

This paper demonstrates that Galois conjugates of pseudo-Anosov stretch factors are dense in the complex plane, revealing extensive distribution properties and no restrictions on their locations, except in low-complexity cases.

## Contribution

It proves the density of Galois conjugates of stretch factors in the complex plane for pseudo-Anosov elements, expanding understanding of their algebraic and geometric properties.

## Key findings

- Galois conjugates are dense in the complex plane for most cases.
- No restrictions on the location of Galois conjugates from Penner's construction.
- Exception in low-complexity cases where density does not hold.

## Abstract

In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, we obtain that in a sense there is no restriction on the location of Galois conjugates of stretch factors arising from Penner's construction. This complements an earlier result of Shin and the author stating that Galois conjugates of stretch factors arising from Penner's construction may never lie on the unit circle.