#### Volume 24, issue 4 (2020)

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Salem number stretch factors and totally real fields arising from Thurston's construction

### Joshua Pankau

Geometry & Topology 24 (2020) 1695–1716
##### Abstract

In 1974, Thurston proved that, up to isotopy, every automorphism of a closed orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has led 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 stretch factor. Thurston showed that every stretch factor is an algebraic unit but it is unknown exactly which units can appear as stretch factors. We show that every Salem number has a power that is the stretch factor of a pseudo-Anosov map arising from a construction due to Thurston. We also show that every totally real number field $K$ is of the form $K=ℚ\left(\lambda +{\lambda }^{\phantom{\rule{-0.17em}{0ex}}-1}\right)$, where $\lambda$ is the stretch factor of a pseudo-Anosov map arising from Thurston’s construction.

##### Keywords
topology, pseudo-Anosov, Salem number, stretch factor, Thurston's construction, mapping class group
##### Mathematical Subject Classification 2010
Primary: 11R80, 37E30, 57M99
##### Publication
Received: 16 November 2017
Revised: 21 September 2019
Accepted: 12 December 2019
Published: 10 November 2020
Proposed: Étienne Ghys
Seconded: Bruce Kleiner, Jean-Pierre Otal
##### Authors
 Joshua Pankau Department of Mathematics The University of Iowa Iowa City, IA United States