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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 this for Brin-Thompson group nV has unsolvable torsion problem. Thus both intersecting can be chosen to have decidable membership problem.

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.

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.

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 this for Brin-Thompson group nV has unsolvable torsion problem. Thus both intersecting can be chosen to have decidable membership problem.

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. 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 $G$. This 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. 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.

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.