Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as possible entries?

Edit: after the remark by Mark Sapir that it is a famous open problem (which for me was already sufficient as an answer), I changed my question into the following ones, so now maybe it is more suitable for staying on MO without being closed.

What is known about this problem? Which other parts of mathematics is it connected to?