Is a typical bi-Perron algebraic unit a pseudo-Anosov dilatation?

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 units 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 >= 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.