Minimal pseudo-Anosov stretch factors on nonoriented surfaces

Liechti, Livio¹ ; Strenner, Balázs²

¹Department of Mathematics, University of Fribourg, Fribourg, Switzerland
²School of Mathematics, Georgia Institute of Technology, Atlanta, GA, United States

Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 451-485

Résumé

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.

Détail

DOI : 10.2140/agt.2020.20.451

Classification : 57M20, 11C08, 37E30, 57M99
Keywords: small stretch factor, minimal dilatation, pseudo-Anosov map, dilatation, Penner's construction, nonorientable surface

Bibliographie

[1] J W Aaber, N Dunfield, Closed surface bundles of least volume, Algebr. Geom. Topol. 10 (2010) 2315 | DOI

[2] C C Adams, The noncompact hyperbolic 3–manifold of minimal volume, Proc. Amer. Math. Soc. 100 (1987) 601 | DOI

[3] I Agol, Volume change under drilling, Geom. Topol. 6 (2002) 905 | DOI

[4] I Agol, P A Storm, W P Thurston, Lower bounds on volumes of hyperbolic Haken 3–manifolds, J. Amer. Math. Soc. 20 (2007) 1053 | DOI

[5] P Arnoux, J C Yoccoz, Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 75

[6] M Bauer, An upper bound for the least dilatation, Trans. Amer. Math. Soc. 330 (1992) 361 | DOI

[7] R Breusch, On the distribution of the roots of a polynomial with integral coefficients, Proc. Amer. Math. Soc. 2 (1951) 939 | DOI

[8] J H Cho, J Y Ham, The minimal dilatation of a genus-two surface, Experiment. Math. 17 (2008) 257 | DOI

[9] D Gabai, R Meyerhoff, P Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009) 1157 | DOI

[10] E Hironaka, Small dilatation mapping classes coming from the simplest hyperbolic braid, Algebr. Geom. Topol. 10 (2010) 2041 | DOI

[11] E Hironaka, E Kin, A family of pseudo-Anosov braids with small dilatation, Algebr. Geom. Topol. 6 (2006) 699 | DOI

[12] N V Ivanov, Coefficients of expansion of pseudo-Anosov homeomorphisms, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 167 (1988) 111 | DOI

[13] E Kin, S Kojima, M Takasawa, Entropy versus volume for pseudo-Anosovs, Experiment. Math. 18 (2009) 397 | DOI

[14] E Kin, M Takasawa, Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior, J. Math. Soc. Japan 65 (2013) 411 | DOI

[15] S Kojima, G Mcshane, Normalized entropy versus volume for pseudo-Anosovs, Geom. Topol. 22 (2018) 2403 | DOI

[16] E Lanneau, J L Thiffeault, On the minimum dilatation of braids on punctured discs, Geom. Dedicata 152 (2011) 165 | DOI

[17] E Lanneau, J L Thiffeault, On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus, Ann. Inst. Fourier Grenoble 61 (2011) 105 | DOI

[18] C J Leininger, On groups generated by two positive multi-twists : Teichmüller curves and Lehmer’s number, Geom. Topol. 8 (2004) 1301 | DOI

[19] L Liechti, On the arithmetic and the geometry of skew-reciprocal polynomials, Proc. Amer. Math. Soc. 147 (2019) 5131 | DOI

[20] L Liechti, B Strenner, The Arnoux–Yoccoz mapping classes via Penner’s construction, preprint (2018)

[21] L Liechti, B Strenner, Minimal Penner dilatations on nonorientable surfaces, J. Topol. Anal. (2019) | DOI

[22] C T Mcmullen, Polynomial invariants for fibered 3–manifolds and Teichmüller geodesics for foliations, Ann. Sci. École Norm. Sup. 33 (2000) 519 | DOI

[23] C T Mcmullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003) 857 | DOI

[24] P Milley, Minimum volume hyperbolic 3–manifolds, J. Topol. 2 (2009) 181 | DOI

[25] H Minakawa, Examples of pseudo-Anosov homeomorphisms with small dilatations, J. Math. Sci. Univ. Tokyo 13 (2006) 95

[26] R C Penner, A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988) 179 | DOI

[27] R C Penner, Bounds on least dilatations, Proc. Amer. Math. Soc. 113 (1991) 443 | DOI

[28] A Schinzel, H Zassenhaus, A refinement of two theorems of Kronecker, Michigan Math. J. 12 (1965) 81 | DOI

[29] H Shin, B Strenner, Pseudo-Anosov mapping classes not arising from Penner’s construction, Geom. Topol. 19 (2015) 3645 | DOI

[30] B Strenner, Algebraic degrees of pseudo-Anosov stretch factors, Geom. Funct. Anal. 27 (2017) 1497 | DOI

[31] B Strenner, Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant, Math. Res. Lett. 25 (2018) 677 | DOI

[32] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417 | DOI

[33] C Y Tsai, The asymptotic behavior of least pseudo-Anosov dilatations, Geom. Topol. 13 (2009) 2253 | DOI

[34] A D Valdivia, Sequences of pseudo-Anosov mapping classes and their asymptotic behavior, New York J. Math. 18 (2012) 609

[35] M Yazdi, Pseudo-Anosov maps with small stretch factors on punctured surfaces, preprint (2018)

[36] A Y Zhirov, On the minimum dilation of pseudo-Anosov diffeomorphisms of a double torus, Uspekhi Mat. Nauk 50 (1995) 197