Trace field degrees in the Torelli group

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.

💡 Research Summary

This paper provides a complete classification of the possible trace field degrees and stretch factor degrees for pseudo-Anosov mapping classes residing in the Torelli group of a closed orientable surface of genus g ≥ 2. The Torelli group, denoted I(S_g), is the kernel of the symplectic representation of the full mapping class group, consisting of elements that act trivially on the first homology of the surface.

The main results are threefold. First, Theorem 1 establishes that for any genus g ≥ 2, every integer d between 1 and 3g-3 can be realized as the trace field degree (the degree of the field extension Q(λ+λ^{-1}) over Q) of some pseudo-Anosov element in I(S_g). Second, Theorem 2 states that every even integer 2d between 2 and 6g-6 can be realized as the stretch factor degree (the degree of Q(λ) over Q) of a pseudo-Anosov element in I(S_g). Theorem 3 synthesizes these results, showing that for g ≥ 3, such examples can be explicitly constructed using the classical Thurston-Veech construction, thereby also substantiating a claim made by Thurston in the 1980s that the theoretical upper bound of 6g-6 for the stretch factor degree is sharp.

The proof strategy is innovative and hinges on two key components. The authors introduce the concept of the “multicurve intersection degree” d of a pair of filling multicurves (α, β), defined as the algebraic degree of the spectral radius of the matrix XX^⊤, where X encodes their geometric intersection numbers. Their pivotal Theorem 4 (a nonsplitting criterion) demonstrates that for such a pair, there exists a positive integer n such that the mapping class T_α^n ∘ T_β^(nε) (for any nonzero integer ε) is pseudo-Anosov with a stretch factor λ of degree 2d over Q. The choice ε = -1 is crucial for landing in the Torelli group. The second, more intricate, part of the proof involves constructing, for every degree 1 ≤ d ≤ 3g-3, a specific pair of filling multicurves (α, β) with intersection degree d, where all curves in α and β are either separating or form bounding pairs. This geometric condition ensures that T_α ∘ T_β^{-1} acts trivially on homology, i.e., belongs to I(S_g). Applying Theorem 4 with ε = -1 to these constructed pairs then yields the desired pseudo-Anosov elements in the Torelli group with the claimed degrees.

The genus g=2 case is handled separately via ad-hoc examples constructed using conjugates of a separating curve Dehn twist, verified with the software “Flipper,” and then transferred to the closed surface by filling a puncture.

Beyond the main theorems, the paper offers several insights. It discusses the challenge of constructing examples with odd stretch factor degrees within the Torelli group, which remains an open problem. It also revisits Thurston’s upper bound in the context of prescribed singularity strata. Furthermore, Theorem 5 reveals a parallel between the degree-2 field extension Q(λ) over Q(λ+λ^{-1}) and the extension Q(√μ) over Q(μ), where √μ is the spectral radius of the bipartite graph defined by the multicurve pair, showing that all even “multicurve bipartite degrees” up to 6g-6 are also attainable.

In summary, this work decisively answers a question posed by Margalit regarding dynamical invariants in the Torelli group, validates a long-standing conjecture of Thurston, and provides a powerful method based on combinatorial geometry and number theory to control the algebraic properties of pseudo-Anosov stretch factors.