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). Mathematics Faculty Publications. 32.
Staecker, P. Christopher, Nielsen equalizer theory. Topology and its Applications 158, 2011, p. 1615–1625.