What is known about the following decision problem?


Given two finite sets in a finitely generated group G,
decide whether the subgroups generated by them have trivial intersection.


Is this problem decidable for a free non-abelian group G?



Decision problem on triviality of intersection of two subgroups
Let $F$ be a free group of rank $2$. If the intersection triviality problem is decidable for $F\times F$, then using the Mikhailova construction, it would be decidable given a finitely presented group $G=\langle X\mid R\rangle$ and a word $w$ in $X$ whether $w$ has infinite order in $G$. This latter problem was proved undecidable in G. Baumslag, W. Boone, and B. Neumann, Some unsolvable problems about elements and subgroups of groups. Math. Scand. 7 (1959), 191–201.


Indeed, embed $F\_X\times F\_X$ in $F\times F$. The subgroup $E=\{(u,v)\in F\_X\times F\_X\mid u=v\ \text{in}\ G\}$ is finitely generated. It intersects $\langle (1,w)\rangle$ trivially if and only if $w$ has infinite order in $G$. 


So intersection triviality is undecidable for $F\times F$, although this group has decidable word problem.


I assume this is what the OP was getting at in his/her comment.


**Added** In G. Arzhantseva, J.F. Lafont, and A. Minasyan, Isomorphism versus commensurability for a class of finitely presented groups, examples of finitely presented groups with solvable word problem are given for which it is undecidable if an element has infinite order. Belk and Bleak proved the Brin-Thompson group nV has unsolvable torsion problem. Thus both intersecting subgroups can be chosen to have decidable membership problem.


