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

GENERIC COMPLEXITY OF THE CONJUGACY PROBLEM IN HNN-EXTENSIONS AND ALGORITHMIC STRATIFICATION OF MILLER'S GROUPS