We discuss time complexity of The Conjugacy Problem in HNN-extensions of
groups, in particular, in Miller's groups. We show that for "almost all", in
some explicit sense, elements, the Conjugacy Problem is decidable in cubic
time. It is worth noting that the Conjugacy Problem in a Miller group may have
be undecidable. Our results show that "hard" instances of the problem comprise
a negligibly small part of the group

Borovik, Alexandre V.