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A family of quaternionic monodromy groups of the Kontsevich–Zorich cocycle

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  • For all $ d $ belonging to a density-$ 1/8 $ subset of the natural numbers, we give an example of a square-tiled surface conjecturally realizing the group $ \mathrm{SO}^*(2d) $ in its standard representation as the Zariski-closure of a factor of its monodromy. We prove that this conjecture holds for the first elements of this subset, showing that the group $ \mathrm{SO}^*(2d) $ is realizable for every $ 11 \leq d \leq 299 $ such that $ d = 3 \bmod 8 $, except possibly for $ d = 35 $ and $ d = 203 $.

    Mathematics Subject Classification: Primary: 37D40; Secondary: 32G15.


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  • Figure 1.  An illustration of $X_g^{(d)}$

    Figure 2.  An illustration of $\widetilde{X}^{(3)}$ showing its four singularities. Horizontally, the each copy of $X^{(3)}$ is cyclically glued to the copy on its right or left, but this does not hold for the vertical gluings (as the top sides of $X_k^{(3)}$, for example, are glued to the bottom sides of $X_{-i}^{(3)}$)

    Figure 3.  The $\mathrm{SL}(2, \mathbb{Z})$-orbit of $X^{(d)}$ using $T$ and $S$ as generators. It consists of three distinct square-tiled surfaces, which we call $Z^{(d)}$, $X^{(d)}$ and $Y^{(d)}$ from left to right. The labels in the $Y^{(d)}$ and $Z^{(d)}$ show the identification of the sides. Unlabelled horizontal sides are identified with the only horizontal having the same horizontal coordinates, and similarly for unlabelled vertical sides

    Figure 4.  The "canonical" fundamental domain of the action of the theta subgroup on the upper half-plane. The resulting Teichmüller curve has genus zero and two cusps

    Figure 5.  An illustration of $Y_g^{(d)}$ and of the cut-and-paste operations used to obtain this description

    Figure 6.  Direction (-1, 2) on $Y_g^{(d)}$

    Figure 7.  Direction $(-1, 2)$ on $T^2 \cdot Y_g^{(d)}$. The gluings are cyclically shifted and the signs of elements of $Q$ on the labels $\eta_\bullet^1$ are changed

    Table 1.  Character table of Q

    Dimension$1$$-1$$\pm i$$\pm j$$\pm k$
    $\mathop{\mathrm{tr}} \chi_2$$2$$2$$-2$$0$$0$$0$
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    Table 2.  The index of ${\rm SL}(\widetilde{X}^{(d)})$ and the genus and number of cusps of the resulting Teichmüller curve for small values of $d$

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  • [1] A. AvilaC. Matheus and J.-C. Yoccoz, The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces, J. Mod. Dyn., 14 (2019), 21-54.  doi: 10.3934/jmd.2019002.
    [2] P. Deligne, La conjecture de Weil. Ⅱ, Publ. Math. Inst. Hautes Études Sci., 52 (1980), 137-252. 
    [3] A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich–Zorich cocycle, Ann. of Math. (2), 188 (2018), 281–313. doi: 10.4007/annals.2018.188.1.5.
    [4] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $ \mathfrak sl(2, \mathbb{R})$ action on moduli space, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95-324.  doi: 10.1007/s10240-018-0099-2.
    [5] A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $ \mathfrak sl(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673–721. doi: 10.4007/annals.2015.182.2.7.
    [6] S. FilipG. Forni and C. Matheus, Quaternionic covers and monodromy of the Kontsevich–Zorich cocycle in orthogonal groups, J. Eur. Math. Soc. (JEMS), 20 (2018), 165-198.  doi: 10.4171/JEMS/763.
    [7] S. Filip, Semisimplicity and rigidity of the Kontsevich–Zorich cocycle, Invent. Math., 205 (2016), 617-670.  doi: 10.1007/s00222-015-0643-3.
    [8] S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle, Duke Math. J., 166 (2017), 657-706.  doi: 10.1215/00127094-3715806.
    [9] G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn., 8 (2014), 271-436.  doi: 10.3934/jmd.2014.8.271.
    [10] C. Matheus, J.-C. Yoccoz and D. Zmiaikou, Homology of origamis with symmetries, Ann. Inst. Fourier (Grenoble), 64 (2014), 1131–1176., doi: 10.5802/aif.2876.
    [11] C. Matheus, J.-C. Yoccoz and D. Zmiaikou, Corrigendum to "Homology of origamis with symmetries", Ann. Inst. Fourier (Grenoble), 66 (2016), 1279–1284., doi: 10.5802/aif.3038.
    [12] C. A. M. Peters and J. H. M. Steenbrink, Monodromy of variations of Hodge structure, Acta Appl. Math., 75 (2003), 183-194.  doi: 10.1023/A:1022344213544.
    [13] A. Wright, Translation surfaces and their orbit closures: An introduction for a broad audience, EMS Surv. Math. Sci., 2 (2015), 63-108.  doi: 10.4171/EMSS/9.
    [14] A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006,437–583. doi: 10.1007/978-3-540-31347-2_13.
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