We give an easily checkable algebraic condition which implies that two elements of a finitely generated free group are members of distinct doubly-twisted conjugacy classes with respect to a pair of homomorphisms. We further show that this criterion is satisfied with probability 1 when the homomorphisms and elements are chosen at random.
Topology and its Applications
Staecker, Christopher P., "Typical elements in free groups are in different doubly-twisted conjugacy classes" (2010). Math & Computer Science Faculty Publications. 23.
Staecker, P. Christopher. 2010. Typical elements in free groups are in different doubly-twisted conjugacy classes. Topology and its Applications 157 (10-11), 1736-1741.