  In this thesis, we study the combinatorics of cyclically fully commutative
elements in Coxeter groups of type $A$ as it relates to conjugacy. In
particular, we introduce the notion of cylindrical heaps and ring equivalence
in order to state our main result, which says that two cyclically fully
commutative elements of a Coxeter group of type $A$ are conjugate if and only
if their corresponding cylindrical heaps are ring equivalent.
