The Conjugacy Problem in Amalgamated Products I: Regular Elements and Black Holes

We discuss the time complexity of the word and conjugacy search problems for free products $G=A{\star }_{C}B$ of groups $A$ and $B$ with amalgamation over a subgroup $C$. We stratify the set of elements of $G$ with respect to the complexity of the word and conjugacy problems and show that for the generic stratum the conjugacy search problem is decidable under some reasonable assumptions about groups $A,B,C$.