Let $F_2=\langle a, b\rangle$ be the non-abelian free group on two generators. Let $\phi_1, \phi_2: F_2\to F_2$ be two group homormophisms. It may be vague, but I still want to ask the following question.
Is there any ``geometric" methods to test whether $\phi_i$'s are conjugate, i.e. $\phi_1(g)=h\phi_2(g)h^{-1}, \forall g\in F_2$ for some $h\in F_2$?
Here, for ``geometric methods", I do not have a good example in mind, but it roughly means some methods in geometric group theory by studying the coarse properties of there maps, maybe by studying good paths in the cayley graph of $F_2$ under these maps...
Of course, to check conjugacy, we only need to test it at the two generators, but I think this as an algebraic method.