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., "Nielsen equalizer theory" (2011). Math & Computer Science Faculty Publications. 32.
Staecker, P. Christopher, Nielsen equalizer theory. Topology and its Applications 158, 2011, p. 1615–1625.