Is there an algorithm to determine whether a specified set of $k$ invertible $n \times n$ matrices generate a free group $F_k$?
We have to make some assumption on the matrix entries: it seems reasonable to go with algebraic entries in $\mathbb{C}$, but I'd be interested in any results in this vein, such as integer entries, entries in other fields, etc.
Partial results for low $n$ and $k$ are of course also interesting: even $n=2, k=2$ seems challenging!
The case $n = 2, k=2$ integer coefficients is decidable (the magic words are "Poincare polygon theorem" - the group is a subgroup of $SL(2, \mathbb{Z})$ so a fuchsian group, so the geometric machinery works. For higher $n$ it is believed that the question is undecidable, but this is open, to the best of my knowledge. Other algebraic number fields reduce to the integer case.