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

Rodolfo Gutiérrez-Romo

doi:10.3934/jmd.2019008

Article Contents

2019, Volume 14: 227-242. Doi: 10.3934/jmd.2019008

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

Rodolfo Gutiérrez-Romo

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

Received: November 16, 2018

Revised: January 26, 2019

Published online: March 29, 2019

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Abstract

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 $.

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

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

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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 $X_{-i}^{(3)}$)

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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

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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

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

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

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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

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Table 1. Character table of Q

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

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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 $d$

$d$	Index	Genus	Cusps
$3$	$12$	$0$	$3$
$11$	$16896$	$225$	$960$
$19$	$1867776$	$30721$	$94208$

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