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Tropical dynamics of area-preserving maps
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 |
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 $.
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. |
[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. 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. |
[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. |
show all references
Dedicated to the memory of Bill Veech
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. |
[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. 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. |
[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. |







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 | $1$ | $-1$ | $\pm i$ | $\pm j$ | $\pm k$ | |
$\chi_1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
$\chi_i$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ |
$\chi_j$ | $1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ |
$\chi_k$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ |
$\mathop{\mathrm{tr}} \chi_2$ | $2$ | $2$ | $-2$ | $0$ | $0$ | $0$ |
$d$ | 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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