Structure of the group generated by two specific symplectic matrices 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!
 A: 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 $p$.
A: After talking to Gabriela Weitze-Schmithuesen, I think that we can show the arithmeticity of $\langle A, B\rangle$ using the argument in Section 2 of this paper of Singh and Venkataramana here (http://www.ams.org/mathscinet-getitem?mr=3165424). 
Indeed, let us consider the permutation matrix $P=\left(\begin{array}{cccc} 
 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0
\end{array}\right)$ exchanging the second and fourth basis vectors and let us show that the conjugate $P\cdot \langle A, B\rangle\cdot P$ of $\langle A, B \rangle$ is arithmetic, i.e., it has finite-index in $Sp(4,\mathbb{Z})$. 
For this sake, we asked Sage to look words on $A$, $B$, $A^2$ and $B^2$ of size $\leq 10$ fixing the first basis vector, and we found that the matrices $x=P(A^2 B)^2(AB^2)^2P$, $y=PABA^2BA(AB^2)^2P$ and $z=PA^2BA^2(B^2A)^2BP$ are interesting because 
$$[y,x]=yxy^{-1}x^{-1} = \left(\begin{array}{cccc} 
 1 & 0 & 0 & 18 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1
\end{array}\right), \quad x^6[y,x] = \left(\begin{array}{cccc} 
 1 & 0 & 18 & 0 \\ 0 & 1 & 0 & 18 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1
\end{array}\right)$$
$$y^6[y,x]^{-1} = \left(\begin{array}{cccc} 
 1 & 18 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -18 \\ 0 & 0 & 0 & 1
\end{array}\right), \quad z^6 \beta^{-1} = z^6 (x^6 [y,x])^{-1} = \left(\begin{array}{cccc} 
 1 & 0 & 0 & 0 \\ 0 & 1 & -18 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1
\end{array}\right)$$ 
generate the positive root groups of $Sp(4,\mathbb{R})$ and, thus, $P\cdot\langle A, B \rangle\cdot P$ intersects the subgroup $U(\mathbb{Z})$ of unipotent upper triangular matrices of $Sp(4,\mathbb{Z})$ in a finite-index subgroup. 
Since we know that $\langle A, B\rangle$ is Zariski-dense (see my comment above to a question of Venkataramana), we can apply a result of Tits (http://www.ams.org/mathscinet-getitem?mr=424966) saying that Zariski dense subgroups of $Sp(4,\mathbb{Z})$ containing a finite-index subgroup of $U(\mathbb{Z})$ are arithmetic to get the desired conclusion.  
A: I don't know if it helps, but you can compose the representation of $\langle a, b \rangle \to \langle A,B \rangle$ with reduction (mod $q$) for any prime $q$, and you get a free kernel $K_{q}$ of $\langle a,b \rangle$ whose rank you can calculate with the Euler characteristics of CTC Wall. But you still have to figure out what $p(K_{q})$ looks like inside $\langle A,B \rangle$.
Note that the kernel of $p$ is the intersection of all the $K_{q}$, and is a free normal subgroup of infinite index of $\langle a,b \rangle$.
