Consider the following two symplectic matrices $$ A \ = \ \left(\begin{array}{rrrr}% 1&0&0&0\\% 0&1&0&0\\% 0&0&-1&1\\% 0&0&-1&0\\% \end{array}\right), \ \ \ B \ = \ \left(\begin{array}{rrrr}% -1&0&0&-1\\% 0&0&-1&0\\% 0&1&-1&0\\% 1&0&0&0\\% \end{array}\right). $$ Is it true that the (Zariski-dense) group $\langle A,B \rangle$ generated by $A$ and $B$ has infinite index in ${\rm Sp}(4,\mathbb{Z})$ (i.e., $\langle A, B \rangle$ is thin in ${\rm Sp}(4,\mathbb{Z}))$?
This question emerged from some calculations performed by Vincent Delecroix and myself with the monodromy of certain square-tiled surfaces (and, in their turn, these calculations were motivated by a question posed by Peter Sarnak to Alex Eskin and Alex Wright).
More precisely, our considerations led to a representation $p: G \to {\rm Sp}(4,\mathbb{Z})$ of the level $4$ congruence group $G = \langle a,b \rangle$ generated by the order three matrices $$ a \ = \ \left(\begin{array}{rr}% 0&-1\\% 1&-1\\% \end{array}\right), \ \ \ b \ = \ \left(\begin{array}{rr}% 1&-3\\% 1&-2\\% \end{array}\right) $$ in ${\rm SL}(2,\mathbb{Z})$ such that $p(a) = A$ and $p(b) = B$.
As it turns out, $G = \langle a \rangle * \langle b \rangle$ is the free product of two copies of $\mathbb{Z}/3\mathbb{Z}$ (since $\{a,a^2\}$ and $\{b,b^2\}$ play ping-pong with some cones in $\mathbb{R}^2$). Moreover, Vincent and I believe that the group $\langle A, B \rangle$ is thin because some numerical experiments with non-trivial words on $A, A^2$ and $B, B^2$ of length $< 25$ seem to indicate that the representation $p$ might be faithful (and, thus, $<A,B>$ would be thin as ${\rm Sp}(4,\mathbb{Z})$ doesn't contain finite-index subgroups isomorphic to free groups).
Nevertheless, after trying a couple of standard tricks (e.g., testing the injectivity of $p$ on finite-index free subgroups of $G$ or playing ping-pong in $\mathbb{R}^4$, its exterior powers [and $p$-adic variants], etc.), Vincent and I are still unable to establish the thinness of $\langle A, B \rangle$ and/or the faithfulness of $p$, so that we would be thankful to any help with these problems!
