Is a typical bi-Perron number a pseudo-Anosov dilatation?

TL;DR

This paper investigates whether typical bi-Perron algebraic units are pseudo-Anosov dilatations, showing that most do not correspond to such dilatations for surfaces of genus at least 10, with implications for moduli space geodesics.

Contribution

It provides a partial answer to the question of whether typical bi-Perron numbers are pseudo-Anosov dilatations, especially for high genus surfaces, and relates this to geodesic counts in moduli space.

Findings

Most bi-Perron algebraic units of degree up to 2n do not correspond to pseudo-Anosov dilatations for genus n≥10.

Asymptotic analysis shows the rarity of such dilatations among bi-Perron units.

Results imply bounds on the number of closed geodesics of the same length in certain moduli spaces.

Abstract

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic unit whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$. As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area one abelian differentials for low genus cases.