Class number for pseudo-Anosovs

Dahmani, François; Mj, Mahan

doi:10.1090/tran/9310

Class number for pseudo-Anosovs
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by François Dahmani and Mahan Mj;

Trans. Amer. Math. Soc. 378 (2025), 1465-1482

DOI: https://doi.org/10.1090/tran/9310

Published electronically: December 4, 2024

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Abstract:

Given two automorphisms of a group $G$, one is interested in knowing whether they are conjugate in the automorphism group of $G$, or in the abstract commensurator of $G$, and how these two properties may differ. When $G$ is the fundamental group of a closed orientable surface, we present a uniform finiteness theorem for the class of pseudo-Anosov automorphisms. We present an explicit example of a commensurably conjugate pair of pseudo-Anosov automorphisms of a genus $3$ surface, that are not conjugate in the mapping class group, and we also show that infinitely many pairwise non-commuting pseud-Anosov automorphisms have class number equal to one. In the appendix, we briefly survey the Latimer-MacDuffee theorem that addresses the case of automorphisms of $\mathbb {Z}^n$, with a point of view that is suited to an analogy with surface group automorphisms.

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Class number for pseudo-Anosovs
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