**Scope of Study: My dissertation explores conjugacy
in groups that are near to be abelian. The study requires applications
of such classical combinatorial group theory results as Britton's Lemma
and Collins' Lemma. In Addition, a study of powers of rational matrices
and their associated polynomial is needed.**

**Findings and Conclusions: Algorithms
are developed for making decisions about complex zeros of rational polynomials
and estimates of the sizes of entries in powers of rational matrices.
These algorithms are carefully put together to yield the main result of
the thesis.**
**Theorem: The conjugacy problem is solved
for HNN extensions of finitely generated abelian groups.**