New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

Hieu  Trung Do; Thomas A. Schmidt

doi:10.3934/jmd.2016.10.541

Article Contents

2016, Volume 10: 541-561. Doi: 10.3934/jmd.2016.10.541

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New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

Hieu Trung Do^1, and
Thomas A. Schmidt^1,

1.
Department of Mathematics, Oregon State University, Corvallis, OR 97331

Received: February 29, 2016

Revised: August 31, 2016

Published: October 2016

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Abstract

We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant. In particular, we give a new infinite family of such maps in genus 3.

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Mathematics Subject Classification: Primary: 37E30, 57M50; Secondary: 11R06.

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