Zero-in-the-corner problems for matrix groups

The upper-left-corner problem and upper-right-corner problem for
matrix semigroups with integer entries are known to be undecidable. Here we ask
the analogous problems for matrix groups and prove their undecidability for
rational entries, by reducing to the undecidability of the word problem for
groups. To reach this aim, we answer a question of Dixon from 1985 by proving
the undecidability of the stabilizer problem for matrix groups.