I was having a discussion with my friend, who is a physics major, but nevertheless enjoys taking math courses. He believes that a non-trivial homomorphism as I describe below cannot exist. My gut tells me that that is not true, but I am not able to find an example.
Could anyone define a group homomorphism from a matrix group that is a subgroup of $GL_n(\mathbb{Z}[x, x^{-1}])$ to a matrix group that is a subgroup of $GL_n(\mathbb{Z})$?
There exists a surjective homomorphism $GL_n(\mathbb{Z}[x,x^{-1}])\rightarrow GL_n(\mathbb{Z})$ given by the evaluation of $x$ at $1$, i.e. given a matrix $A\in GL_n(\mathbb{Z}[x,x^{-1}])$ whose coefficients are integral polynomials $A_{ij}$ of $x$ and $x^{-1}$, one maps $A$ to the matrix $[a_{ij}]$ with $a_{ij}=A_{ij}(1,1)$.
Now since any homomorphism $GL_n(\mathbb{Z}[x,x^{-1}])\rightarrow GL_n(\mathbb{Z})$ is defined by the image of $x$ in $GL_n(\mathbb{Z})$, the real question is:
What are those $a\in GL_n(\mathbb{Z})$ such that $x\rightarrow a$ is extended to a homomorphism $GL_n(\mathbb{Z}[x,x^{-1}])\rightarrow GL_n(\mathbb{Z})$?
My guess: all elements of $GL_n(\mathbb{Z})$.