Minimal pseudo-Anosov stretch factors on nonoriented surfaces

Liechti, Livio; Strenner, Balázs

doi:10.2140/agt.2020.20.451

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Minimal pseudo-Anosov stretch factors on nonoriented surfaces

Livio Liechti and Balázs Strenner

Algebraic & Geometric Topology 20 (2020) 451–485

DOI: 10.2140/agt.2020.20.451

Abstract

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus $4$ , $5$ , $6$ , $7$ , $8$ , $10$ , $12$ , $14$ , $16$ , $18$ and $20$ . We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliations on the closed orientable surfaces of genus $1$ , $3$ , $5$ , $7$ , $9$ and $11$ . As a byproduct, we obtain that the stretch factor of a pseudo-Anosov map on a nonorientable surface or an orientation-reversing pseudo-Anosov map on an orientable surface does not have Galois conjugates on the unit circle. This shows that the techniques that were used to disprove Penner’s conjecture on orientable surfaces are ineffective in the nonorientable cases.

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Keywords

small stretch factor, minimal dilatation, pseudo-Anosov map, dilatation, Penner's construction, nonorientable surface

Mathematical Subject Classification 2010

Primary: 57M20

Secondary: 11C08, 37E30, 57M99

References

Publication

Received: 13 December 2018

Revised: 12 March 2019

Accepted: 1 April 2019

Published: 23 February 2020

Authors

Livio Liechti
Department of Mathematics University of Fribourg Fribourg Switzerland

Balázs Strenner
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

Volume 20, issue 1 (2020)