Class number for pseudo-Anosovs

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 independent automorphisms of hyperbolic orbifolds 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.