Posted by: matheuscmss | November 7, 2016

## Arithmeticity of the Kontsevich-Zorich monodromy of a certain origami of genus three

Gabriela Weitze-Schmithüsen is currently visiting me in Paris and I took the opportunity to revisit some of my favorite questions about square-tiled surfaces / origamis.

Last week, we spent a couple of days revisiting the content of my blog post on Sarnak’s question about thin KZ monodromies and we realized that the origami ${\mathcal{O}_1}$ of genus 3 discussed in this post turns out to exhibit arithmetic KZ monodromy! (In particular, this answers my Mathoverflow question here.)

In this very short post, we show the arithmeticity of the KZ monodromy of ${\mathcal{O}_1}$.

1. Description of the KZ monodromy of ${\mathcal{O}_1}$

The KZ monodromy ${\Gamma_{\mathcal{O}_1}}$ of ${\mathcal{O}_1}$ is the subgroup of ${\mathrm{Sp}(4,\mathbb{Z})}$ generated by the matrices

$\displaystyle A=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right) \quad \textrm{and} \quad B=\left(\begin{array}{cccc} -1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$

of order three: see Remark 9 in this post here.

For the sake of exposition, we are going to permute the second and fourth vectors of the canonical basis using the permutation matrix

$\displaystyle P=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right),$

so that the KZ monodromy ${\Gamma_{\mathcal{O}_1}}$ is the subgroup ${P\cdot\langle A, B\rangle\cdot P}$.

Remark 1 This change of basis is purely cosmetical: it makes that the symplectic form preserved by these matrices in ${P\cdot\langle A, B\rangle\cdot P}$ is

$\displaystyle \Upsilon=\left(\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$

Denote by ${U(\mathbb{Z})}$ the subgroup of unipotent upper triangular matrices in ${\mathrm{Sp}(4,\mathbb{Z})}$.

2. Arithmeticity of the KZ monodromy of ${\mathcal{O}_1}$

A result of Tits says that a Zariski-dense subgroup ${\Gamma\subset \mathrm{Sp}(4,\mathbb{Z})}$ such that ${\Gamma\cap U(\mathbb{Z})}$ has finite-index in ${U(\mathbb{Z})}$ must be arithmetic (i.e., ${\Gamma}$ has finite-index in ${\mathrm{Sp}(4,\mathbb{Z})}$).

Since we already know that ${\Gamma_{\mathcal{O}_1}}$ is Zariski-dense (cf. Proposition 3 in this post here), it suffices to check that ${\Gamma_{\mathcal{O}_1}\cap U(\mathbb{Z})}$ has finite-index in ${U(\mathbb{Z})}$.

For this sake, we follow the strategy in Section 2 of this paper of Singh and Venkataramana, namely, we study matrices in ${\Gamma_{\mathcal{O}_1}}$ fixing the first basis vector ${e_1}$ and, a fortiori, stabilizing the flag ${\mathbb{Q} e_1\subset e_1^{\perp}:=\{v\in\mathbb{Q}^4:\Upsilon(v,e_1)=0\}\subset \mathbb{Q}^4}$.

After asking Sage to compute a few elements of ${\Gamma_{\mathcal{O}_1}}$ (conjugates under ${P}$ of words on ${A}$, ${B}$, ${A^2}$ and ${B^2}$ of size ${\leq 10}$) fixing the basis vector ${e_1}$, we found the following interesting matrices:

$\displaystyle x:=P\cdot (A^2B)^2 (AB^2)^2 \cdot P = \left(\begin{array}{cccc} 1 & 0 & 3 & -3 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right),$

$\displaystyle y:=P\cdot A B A^2 B A (AB^2)^2\cdot P = \left(\begin{array}{cccc} 1 & 3 & 0 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \end{array}\right),$

and

$\displaystyle z:=P\cdot A^2 B A^2 (B^2 A)^2 B\cdot P = \left(\begin{array}{cccc} 1 & 0 & 3 & 0 \\ 0 & 1 & -3 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right).$

In order to check that ${\Gamma_{\mathcal{O}_1}\cap U(\mathbb{Z})}$ has finite index in ${U(\mathbb{Z})}$, we observe that

$\displaystyle \alpha = [y,x] = yxy^{-1}x^{-1} = \left(\begin{array}{cccc} 1 & 0 & 0 & 18 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right),$

$\displaystyle \beta = x^6[y,x] = \left(\begin{array}{cccc} 1 & 0 & 18 & 0 \\ 0 & 1 & 0 & 18 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

$\displaystyle \gamma = y^6[y,x]^{-1} = \left(\begin{array}{cccc} 1 & 18 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -18 \\ 0 & 0 & 0 & 1 \end{array}\right)$

$\displaystyle \delta = z^6 \beta^{-1} = z^6 (x^6 [y,x])^{-1} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -18 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

are elements in ${\Gamma_{\mathcal{O}_1}\cap U(\mathbb{Z})}$ generating the positive root groups of ${\textrm{Sp}(4,\mathbb{R})}$. In particular, ${\Gamma_{\mathcal{O}_1}\cap U(\mathbb{Z})}$ has finite-index in ${U(\mathbb{Z})}$, so that the argument is complete.

Remark 2 It is worth noticing that all non-arithmetic Veech surfaces in genus two provide examples of thin KZ monodromy, but this is not the case for origamis (arithmetic Veech surfaces) of genus 2 in the stratum ${\mathcal{H}(2)}$ with tiled by ${\leq 6}$ squares (as well as for the origami ${\mathcal{O}_1}$ of genus three mentioned above). In particular, this indicates that Sarnak’s question about existence and/or abundance of thin KZ monodromies among origamis might have an interesting answer…

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