# American Institute of Mathematical Sciences

February  2019, 14: 227-242. doi: 10.3934/jmd.2019008

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

 Institut de Mathématiques de Jussieu – Paris Rive Gauche, UMR 7586, Bátiment Sophie Germain, 75205 PARIS Cedex 13, France

Dedicated to the memory of Bill Veech

Received  November 16, 2018 Revised  January 26, 2019 Published  March 2019

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 $d = 203$.

Citation: Rodolfo Gutiérrez-Romo. A family of quaternionic monodromy groups of the Kontsevich–Zorich cocycle. Journal of Modern Dynamics, 2019, 14: 227-242. doi: 10.3934/jmd.2019008
##### References:
 [1] A. Avila, C. 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. Google Scholar [2] P. Deligne, La conjecture de Weil. Ⅱ, Publ. Math. Inst. Hautes Études Sci., 52 (1980), 137-252. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] S. Filip, G. 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. Google Scholar [7] S. Filip, Semisimplicity and rigidity of the Kontsevich–Zorich cocycle, Invent. Math., 205 (2016), 617-670. doi: 10.1007/s00222-015-0643-3. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### References:
 [1] A. Avila, C. 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. Google Scholar [2] P. Deligne, La conjecture de Weil. Ⅱ, Publ. Math. Inst. Hautes Études Sci., 52 (1980), 137-252. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] S. Filip, G. 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. Google Scholar [7] S. Filip, Semisimplicity and rigidity of the Kontsevich–Zorich cocycle, Invent. Math., 205 (2016), 617-670. doi: 10.1007/s00222-015-0643-3. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
An illustration of $X_g^{(d)}$
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 $X_{-i}^{(3)}$)
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
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
An illustration of $Y_g^{(d)}$ and of the cut-and-paste operations used to obtain this description
Direction (-1, 2) on $Y_g^{(d)}$
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
Character table of Q
 Dimension $1$ $-1$ $\pm i$ $\pm j$ $\pm k$ $\chi_1$ $1$ $1$ $1$ $1$ $1$ $1$ $\chi_i$ $1$ $1$ $1$ $1$ $-1$ $-1$ $\chi_j$ $1$ $1$ $1$ $-1$ $1$ $-1$ $\chi_k$ $1$ $1$ $1$ $-1$ $-1$ $1$ $\mathop{\mathrm{tr}} \chi_2$ $2$ $2$ $-2$ $0$ $0$ $0$
 Dimension $1$ $-1$ $\pm i$ $\pm j$ $\pm k$ $\chi_1$ $1$ $1$ $1$ $1$ $1$ $1$ $\chi_i$ $1$ $1$ $1$ $1$ $-1$ $-1$ $\chi_j$ $1$ $1$ $1$ $-1$ $1$ $-1$ $\chi_k$ $1$ $1$ $1$ $-1$ $-1$ $1$ $\mathop{\mathrm{tr}} \chi_2$ $2$ $2$ $-2$ $0$ $0$ $0$
The index of ${\rm SL}(\widetilde{X}^{(d)})$ and the genus and number of cusps of the resulting Teichmüller curve for small values of $d$
 $d$ Index Genus Cusps $3$ $12$ $0$ $3$ $11$ $16896$ $225$ $960$ $19$ $1867776$ $30721$ $94208$
 $d$ Index Genus Cusps $3$ $12$ $0$ $3$ $11$ $16896$ $225$ $960$ $19$ $1867776$ $30721$ $94208$
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