## Algebraic & Geometric Topology

### Least dilatation of pure surface braids

Marissa Loving

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1552960831

Digital Object Identifier
doi:10.2140/agt.2019.19.941

Mathematical Reviews number (MathSciNet)
MR3924180

Zentralblatt MATH identifier
07075117

#### Citation

Loving, Marissa. Least dilatation of pure surface braids. Algebr. Geom. Topol. 19 (2019), no. 2, 941--964. doi:10.2140/agt.2019.19.941. https://projecteuclid.org/euclid.agt/1552960831

#### References

• 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