The conjugacy problem for graph products with central cyclic edge groups

Abstract

A graph product is the fundamental group of a graph of groups. Amongst the simplest examples are HNN extensions and free products with amalgamation. Graph products with cyclic edge groups inherit a solvable conjugacy problem from their vertex groups under certain conditions, the most important of which imposed here is that all the edge group generators in each vertex group are powers of a common central element. Under these conditions the conjugacy problem is solvable for any two elements not both of zero reduced length in the graph product, and for arbitrary pairs of elements in HNN extensions, tree products and many graph products over finite-leaf roses. The conjugacy problem is not solvable in general for elements of zero reduced length in graph products over graphs with infinitely many circuits.