Generic complexity of the Conjugacy Problem in HNN-extensions and algorithmic stratification of Miller's groups

TL;DR

This paper analyzes the complexity of the Conjugacy Problem in HNN-extensions, especially Miller's groups, showing that for most elements it is decidable in cubic time, despite some instances being undecidable.

Contribution

It provides a detailed complexity analysis of the Conjugacy Problem in Miller's groups and introduces an algorithmic stratification highlighting the typical tractability.

Findings

Most elements have a decidable conjugacy problem in cubic time.

Hard instances are negligibly small in the group.

The problem can be undecidable for certain elements.

Abstract

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.