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Minimal pseudo-Anosov stretch factors on nonoriented surfaces

Livio Liechti and Balázs Strenner

Algebraic & Geometric Topology 20 (2020) 451–485

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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small stretch factor, minimal dilatation, pseudo-Anosov map, dilatation, Penner's construction, nonorientable surface
Mathematical Subject Classification 2010
Primary: 57M20
Secondary: 11C08, 37E30, 57M99
Received: 13 December 2018
Revised: 12 March 2019
Accepted: 1 April 2019
Published: 23 February 2020
Livio Liechti
Department of Mathematics
University of Fribourg
Balázs Strenner
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States