TITLE: Is it decidable whether or not a collection of integer matrices generates a free group?
QUESTION [29 upvotes]: Suppose we have integer matrices $A_1,\ldots,A_n\in\operatorname{GL}(n,\mathbb Z)$.  Define $\varphi:F_n\to\operatorname{GL}(n,\mathbb Z)$ by $x_i\mapsto A_i$.  Is there an algorithm to decide whether or not $\varphi$ is injective?

REPLY [5 votes]: Some comments on the question  'Is it decidable whether or not a
collection of integer matrices generates a free group?'
Given a finite set of matrices S over a field, the problem of
testing whether the group H generated by S contains a free
non-abelian subgroup is decidable. An algorithm solving the
problem as well as its implementation (in Magma) available. Notice
that the algorithms does not construct a free non-abelian subgroup
in H but justifies its existence. As to testing freeness of
finitely generated linear groups then the problem has quite a long
history. I may recommend as a start point the paper by John Dixon
Can.J Math, v. 37, n. 2, 1985, 238-259 (see p. 240 there, and then
follow the references).