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

TL;DR

This paper analyzes the complexity of the conjugacy problem in amalgamated free products of groups, showing that for most elements, the conjugacy search problem is decidable under certain conditions.

Contribution

It introduces a stratification of elements in amalgamated free products and demonstrates decidability of the conjugacy search problem for the generic case.

Findings

Conjugacy search problem is decidable for generic elements.

Stratification helps understand complexity in amalgamated products.

Decidability depends on properties of groups A, B, and C.

Abstract

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 $A,B,C$.