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

Pankau, Joshua

doi:10.2140/gt.2020.24.1695

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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

DOI: 10.2140/gt.2020.24.1695

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 $λ > 1$ , 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 $K = ℚ (λ + λ^{- 1})$ , where $λ$ 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

References

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

Volume 24, issue 4 (2020)