Are there some nontrivial group homomorphisms from $SL_n(\mathbb{Z})$ to $GL_{n-1}(\mathbb{Z})$ for $n\geq3$ except the determinant? Or This should be a natural question and any references are welcomed.
PS. A similar question has the answer 'NO' for a finite field $F_p$ instead of $\mathbb{Z}$? This should be a natural question and any references are welcomed\mathbb{Z}$as explained below. 2 added 23 characters in body Are there some nontrivial group homomorphisms from$SL_n(\mathbb{Z})$to$GL_{n-1}(\mathbb{Z})$for$n\geq3$? n\geq3$ except the determinant? Or a similar question for a finite field $F_p$ instead of $\mathbb{Z}$? This should be a natural question and any references are welcomed.
# Are there some nontrivial group homomorphisms from $SL_n(\mathbb{Z})$ to $GL_{n-1}(\mathbb{Z})$ for $n\geq3$?
Are there some nontrivial group homomorphisms from $SL_n(\mathbb{Z})$ to $GL_{n-1}(\mathbb{Z})$ for $n\geq3$? Or a similar question for a finite field $F_p$ instead of $\mathbb{Z}$? This should be a natural question and any references are welcomed.