$SL_2(\Bbb Z)$ is generated by $M_1=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $M_2=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$.
Given a matrix $A\in SL_2(\Bbb Z)$ can there be two non-equivalent combinations of products of $M_1,M_2$ that produce $A$? Assume $M_2^{2n+1}$ represents the same symbol at every $n\in\Bbb N$ and $M_2^{2n}$ represents the same symbol at every $n\in\Bbb N$.
Given a matrix $A\in SL_2(\Bbb Z)$ can there be two non-equivalent combinations of products of $M_1,M_2$ of minimum length that produce $A$? By length I mean the minimum number of alternations needed in products of $M_1^a$ and $M_2^b$ for some $a,b\in\Bbb Z$.
Given a matrix $B\in\Bbb Z^{2\times 2}$ with determinant $>1$ we know that there are two $L,R\in SL_2(\Bbb Z)$ and an unique diagonal $D\in\Bbb SL_2(Z)$ with $D_{11}|D_{22}$ such that $D=LBR$ is a diagonal. We know $D$ is unique. Is $L,R$ unique as well (at least upto sign)?