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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.


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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

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