Trace field degrees in the Torelli group

Abstract

We show that for $g\ge 2$, all integers $1\le d\le 3g-3$ arise as trace field degrees of pseudo-Anosov mapping classes in the Torelli group of the closed orientable surface of genus $g$. Our method uses the Thurston–Veech construction of pseudo-Anosov maps, and we provide examples where the stretch factor has algebraic degree any even number between 2 and $6g-6$. This validates a claim by Thurston from the 1980s.