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The limit set of non-orientable mapping class groups

The author was partially supported by NSF Grant DMS 1856155

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  • We provide evidence both for and against a conjectural analogy between geometrically finite infinite covolume Fuchsian groups and the mapping class group of compact non-orientable surfaces. In the positive direction, we show the complement of the limit set is open and dense. Moreover, we show that the limit set of the mapping class group contains the set of uniquely ergodic foliations and is contained in the set of all projective measured foliations not containing any one-sided leaves, establishing large parts of a conjecture of Gendulphe. In the negative direction, we show that a conjectured convex core is not even quasi-convex, in contrast with the geometrically finite setting.

    Mathematics Subject Classification: Primary: 57K20; Secondary: 37E35.

    Citation:

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  • Figure 1.  A quadratic differential $ q $ on $ \mathscr{S}_2 $ given by the slit torus construction

    Figure 2.  A quadratic differential on $ \mathscr{N}_3 $

    Figure 3.  Two possibilities for first return to $ \eta_i $: on the left, the arc returns without the local orientation flipping, and on the right, the arc returns with the local orientation flipped

    Figure 4.  The curve $ c_i $ is colored blue. Since the leaf from $ p_0 $ returns with the local orientation flipped to both $ p_{i-1} $ and $ p_{i} $, the curve $ c_i $ is two-sided

    Figure 5.  Construction of the blue curve $ c_i $ when the leaf always returns with orientation flipped from the "up" or "down" direction

    Figure 6.  The arcs $ \widetilde{q_1} $ and $ \widetilde{q_2} $

    Figure 7.  Homotopy taking $ q_2 $ to $ q_1 $

    Figure 8.  The curves restricted to a pair of pants

    Figure 9.  The right angled hexagon obtained by cutting the pants along the seams

    Figure 10.  A DQD on $ \mathscr{N}_4 $

    Figure 11.  A DQD on $ \mathscr{N}_9 $. To display the gluing maps on the small slits, we have a zoomed in picture in the ellipses

  • [1] C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces, Ann. Sci. École Norm. Sup., 23 (1990), 469-494.  doi: 10.24033/asens.1608.
    [2] V. Delecroix, J. Rüth, S. Lelièvre, F. Chapoton, C. Fougeron and D. Davis, flatsurf/sage-flatsurf: 0.4.7, 2022., doi: 10.5281/zenodo.6810926.
    [3] V. Erlandsson, M. Gendulphe, I. Pasquinelli and J. Souto, Mapping class group orbit closures for non-orientable surfaces, arXiv: 2110.02644, 2021.,
    [4] A. FathiF. Laudenbach and  V. PoénaruThurston's Work on Surfaces, Mathematical Notes, vol. 48, Princeton University Press, 2012. 
    [5] A. GamburdM. Magee and R. Ronan, An asymptotic formula for integer points on Markoff-Hurwitz varieties, Ann. of Math., 190 (2019), 751-809.  doi: 10.4007/annals.2019.190.3.2.
    [6] M. Gendulphe, What's wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces, arXiv: 1706.08798, 2017.,
    [7] A. B. Katok, Invariant measures of flows on oriented surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778. 
    [8] R. P. Kent and C. J. Leininger, Subgroups of the mapping class group from the geometrical viewpoint, in In the Tradition of Ahlfors–Bers, IV, 119-141, Contemp. Math., vol. 432, Amer. Math. Soc., Providence, RI, 2007. doi: 10.1090/conm/432/08306.
    [9] A. Lenzhen and K. Rafi, Length of a curve is quasi-convex along a Teichmüller geodesic, J. Differential Geom., 88 (2011), 267-295.  doi: 10.4310/jdg/1320067648.
    [10] M. Magee, Counting one-sided simple closed geodesics on Fuchsian thrice punctured projective planes, Int. Math. Res. Not. IMRN, 2020 (2020), 3886-3901.  doi: 10.1093/imrn/rny112.
    [11] H. Masur, Measured foliations and handlebodies, Ergodic Theory Dynam. Systems, 6 (1986), 99-116.  doi: 10.1017/S014338570000331X.
    [12] J. McCarthy and A. Papadopoulos, Dynamics on Thurston's sphere of projective measured foliations, Comment. Math. Helv., 64 (1989), 133-166.  doi: 10.1007/BF02564666.
    [13] Y. N. Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom., 35 (1992), 151-217.  doi: 10.4310/jdg/1214447809.
    [14] M. Mirzakhani, Growth of the number of simple closed geodesies on hyperbolic surfaces, Ann. of Math., 168 (2008), 97-125.  doi: 10.4007/annals.2008.168.97.
    [15] A. Nogueira, Almost all interval exchange transformations with flips are nonergodic, Ergodic Theory Dynam. Systems, 9 (1989), 515-525.  doi: 10.1017/S0143385700005150.
    [16] P. Norbury, Lengths of geodesics on non-orientable hyperbolic surfaces, Geom. Dedicata, 134 (2008), 153-176.  doi: 10.1007/s10711-008-9251-3.
    [17] A. Skripchenko and S. Troubetzkoy, On the Hausdorf dimension of minimal interval exchange transformations with flips, J. Lond. Math. Soc., 97 (2018), 149-169.  doi: 10.1112/jlms.12099.
    [18] W. P. Thurston, The Geometry and Topology of Three-Manifolds, lecture notes, Princeton University, 1979.
    [19] W. A. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222-272.  doi: 10.1007/BF02790174.
    [20] S. Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math., 109 (1979), 323-351.  doi: 10.2307/1971114.
    [21] A. Wright, From rational billiards to dynamics on moduli spaces, Bull. Amer. Math. Soc., 53 (2016), 41-56.  doi: 10.1090/bull/1513.
    [22] A. Wright, A tour through Mirzakhani's work on moduli spaces of Riemann surfaces, Bull. Amer. Math. Soc., 57 (2020), 359-408.  doi: 10.1090/bull/1687.
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