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!



Structure of the group generated by two specific symplectic matrices
Your representation $p$ is *not* faithful, since we have
$$
 (ABA^{-1}BA^{-1}BAB^{-1})^3 \ = \ 1.
$$
In particular, this means that
$$
 (aba^{-1}ba^{-1}bab^{-1})^3 \ = \
 \left(\begin{array}{rr}%
 -24587&42408\\%
 15048&-25955\\%
 \end{array}\right)
$$
lies in the kernel of $p$.


