Trace field degrees in the Torelli group

TL;DR

This paper demonstrates that for surfaces of genus g ≥ 2, all trace field degrees from 1 up to 3g-3 can be realized by pseudo-Anosov elements in the Torelli group, confirming a longstanding conjecture.

Contribution

It shows that all integers from 1 to 3g-3 are achievable as trace field degrees in the Torelli group, using Thurston-Veech constructions, and provides examples with algebraic degrees up to 6g-6.

Findings

All integers 1 to 3g-3 are realized as trace field degrees.

Examples with algebraic degree up to 6g-6 are constructed.

Validates Thurston's 1980s claim about trace field degrees.

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 two and $6g-6$. This validates a claim by Thurston from the 1980s.