Conjugacy in Miller’s groups

In 1971 C. F. Miller associated to every finitely presented group G a free-by-free group [Formula: see text] known as the Miller Machine, whose conjugacy problem is closely related to the conjugacy and word problems of G. We quantify this relationship, and look to fully understand the conjugacy problem of [Formula: see text]; namely, we reduce the conjugacy problem in [Formula: see text] to a strong form of list conjugacy in G, which we term iso-computational list conjugacy. As an application, we show that if G is finite, the conjugacy problem for [Formula: see text] is in [Formula: see text].