Algebraic & Geometric Topology

Least dilatation of pure surface braids

Marissa Loving

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We study the minimal dilatation of pseudo-Anosov pure surface braids and provide upper and lower bounds as a function of genus and the number of punctures. For a fixed number of punctures, these bounds tend to infinity as the genus does. We also bound the dilatation of pseudo-Anosov pure surface braids away from zero and give a constant upper bound in the case of a sufficient number of punctures.

Article information

Algebr. Geom. Topol., Volume 19, Number 2 (2019), 941-964.

Received: 17 February 2018
Revised: 31 July 2018
Accepted: 13 August 2018
First available in Project Euclid: 19 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Secondary: 20F36: Braid groups; Artin groups 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 30F60: Teichmüller theory [See also 32G15] 37B40: Topological entropy 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 57M07: Topological methods in group theory 57M99: None of the above, but in this section

mapping class group pseudo Anosov dilatation stretch factor pure surface braids


Loving, Marissa. Least dilatation of pure surface braids. Algebr. Geom. Topol. 19 (2019), no. 2, 941--964. doi:10.2140/agt.2019.19.941.

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  • J W Aaber, N Dunfield, Closed surface bundles of least volume, Algebr. Geom. Topol. 10 (2010) 2315–2342
  • I Agol, C J Leininger, D Margalit, Pseudo-Anosov stretch factors and homology of mapping tori, J. Lond. Math. Soc. 93 (2016) 664–682
  • L V Ahlfors, Lectures on quasiconformal mappings, 2nd edition, University Lecture Series 38, Amer. Math. Soc., Providence, RI (2006)
  • T Aougab, S Huang, Minimally intersecting filling pairs on surfaces, Algebr. Geom. Topol. 15 (2015) 903–932
  • T Aougab, S J Taylor, Pseudo-Anosovs optimizing the ratio of Teichmüller to curve graph translation length, from “In the tradition of Ahlfors–Bers, VII” (A S Basmajian, Y N Minsky, A W Reid, editors), Contemp. Math. 696, Amer. Math. Soc., Providence, RI (2017) 17–28
  • H Baik, A Rafiqi, C Wu, Constructing pseudo-Anosov maps with given dilatations, Geom. Dedicata 180 (2016) 39–48
  • M Bauer, An upper bound for the least dilatation, Trans. Amer. Math. Soc. 330 (1992) 361–370
  • J S Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–238
  • J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton Univ. Press (1974)
  • P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkhäuser, Boston (1992)
  • S Dowdall, Dilatation versus self-intersection number for point-pushing pseudo-Anosov homeomorphisms, J. Topol. 4 (2011) 942–984
  • B Farb, C J Leininger, D Margalit, The lower central series and pseudo-Anosov dilatations, Amer. J. Math. 130 (2008) 799–827
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
  • A Fathi, F Laudenbach, V Poénaru, Thurston's work on surfaces, Mathematical Notes 48, Princeton Univ. Press (2012)
  • F R Gantmacher, The theory of matrices, I, Chelsea, New York (1959)
  • F W Gehring, Quasiconformal mappings which hold the real axis pointwise fixed, from “Mathematical essays dedicated to A J Macintyre” (H Shankar, editor), Ohio Univ. Press, Athens, Ohio (1970) 145–148
  • J-Y Ham, W T Song, The minimum dilatation of pseudo-Anosov $5$–braids, Experiment. Math. 16 (2007) 167–179
  • E Hironaka, Small dilatation mapping classes coming from the simplest hyperbolic braid, Algebr. Geom. Topol. 10 (2010) 2041–2060
  • E Hironaka, E Kin, A family of pseudo-Anosov braids with small dilatation, Algebr. Geom. Topol. 6 (2006) 699–738
  • S Hirose, E Kin, The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups, Q. J. Math. 68 (2017) 1035–1069
  • Y Imayoshi, M Ito, H Yamamoto, On the Nielsen–Thurston–Bers type of some self-maps of Riemann surfaces with two specified points, Osaka J. Math. 40 (2003) 659–685
  • N V Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs 115, Amer. Math. Soc., Providence, RI (1992)
  • E Kin, M Takasawa, Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior, J. Math. Soc. Japan 65 (2013) 411–446
  • I Kra, On the Nielsen–Thurston–Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981) 231–270
  • E Lanneau, J-L Thiffeault, On the minimum dilatation of braids on punctured discs, Geom. Dedicata 152 (2011) 165–182
  • J Malestein, A Putman, Pseudo-Anosov dilatations and the Johnson filtration, Groups Geom. Dyn. 10 (2016) 771–793
  • B Martelli, M Novaga, A Pluda, S Riolo, Spines of minimal length, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017) 1067–1090
  • H Minakawa, Examples of pseudo-Anosov homeomorphisms with small dilatations, J. Math. Sci. Univ. Tokyo 13 (2006) 95–111
  • J Pankau, Salem number stretch factors and totally real fields arising from Thurston's construction, preprint (2017)
  • R C Penner, Bounds on least dilatations, Proc. Amer. Math. Soc. 113 (1991) 443–450
  • H Shin, B Strenner, Pseudo-Anosov mapping classes not arising from Penner's construction, Geom. Topol. 19 (2015) 3645–3656
  • W T Song, Upper and lower bounds for the minimal positive entropy of pure braids, Bull. London Math. Soc. 37 (2005) 224–229
  • W T Song, K H Ko, J E Los, Entropies of braids, J. Knot Theory Ramifications 11 (2002) 647–666
  • B Strenner, L Liechti, Minimal pseudo-Anosov stretch factors on nonorientable surfaces, preprint (2018)
  • O Teichmüller, Ein Verschiebungssatz der quasikonformen Abbildung, Deutsche Math. 7 (1944) 336–343
  • W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417–431
  • C-Y Tsai, The asymptotic behavior of least pseudo-Anosov dilatations, Geom. Topol. 13 (2009) 2253–2278
  • M Yazdi, Lower bound for dilatations, J. Topol. 11 (2018) 602–614