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On stretch factors of pseudo-Anosov maps

  • Author(s): Pankau, Joshua Charles
  • Advisor(s): Long, Darren
  • et al.
Abstract

In 1974, Thurston proved that, up to isotopy, every automorphism of closed orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has lead to a rich theory with applications ranging from dynamical systems to low dimensional topology. Associated with every pseudo-Anosov map is a real number $\lambda > 1$ known as the \textit{stretch factor}. Thurston showed that every stretch factor is an algebraic unit but it is unknown exactly which units can appear as stretch factors. Though this question remains open, we provide a partial answer by showing a large class of units are obtainable as stretch factors using a construction due to Thurston. We will show that every Salem number has a power that is the stretch factor of a pseudo-Anosov map arising from Thurston's construction, and then we will use the techniques to generalize to a much larger class of units. We also show that every totally real number field $K$ is of the form $K = \mathbb{Q}(\lambda + \lambda^{-1})$, where $\lambda$ is the stretch factor of a pseudo-Anosov map arising from Thurston's construction. Finally, we develop a new method of constructing closed orientable surfaces from positive integer matrices that will be crucial to proving the above results.

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