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 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 .
1. Description of the KZ monodromy of
The KZ monodromy of is the subgroup of generated by the matrices
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
so that the KZ monodromy is the subgroup .
Remark 1 This change of basis is purely cosmetical: it makes that the symplectic form preserved by these matrices in is
Denote by the subgroup of unipotent upper triangular matrices in .
2. Arithmeticity of the KZ monodromy of
A result of Tits says that a Zariski-dense subgroup such that has finite-index in must be arithmetic (i.e., has finite-index in ).
Since we already know that is Zariski-dense (cf. Proposition 3 in this post here), it suffices to check that has finite-index in .
For this sake, we follow the strategy in Section 2 of this paper of Singh and Venkataramana, namely, we study matrices in fixing the first basis vector and, a fortiori, stabilizing the flag .
After asking Sage to compute a few elements of (conjugates under of words on , , and of size ) fixing the basis vector , we found the following interesting matrices:
In order to check that has finite index in , we observe that
are elements in generating the positive root groups of . In particular, has finite-index in , 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 with tiled by squares (as well as for the origami 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…