Centralizers in fundamental groups of graphs of groups

Two classic constructions in group theory are amalgamated free products, first introduced by O. Schreier in 1927, and HNN extensions, first introduced by G. Higman, B. Neumann, and H. Neumann in 1949. Both play central roles in topics such as algebraic topology and group theory.; The fundamental group of a graph of groups was first introduced by J. P. Serre in 1977 as a generalization of both of these constructions. Serre's method was to use the action of a group G on a graph theoretic tree T. It turns out that information on the stabilizers of the edges and vertices of T under the action of G and the fundamental group of the quotient graph G T was enough to completely recover G from the action. Conversely, any graph of groups gives rise to an inversion-free action of a group on a tree.; In this paper, we are interested in commuting elements in the fundamental group of a graph of groups. We start by examining the classic cases of free products, amalgamated free products, and HNN extensions. We then move on to the general setting of the fundamental group of a graph of groups, using both algebraic and topological methods to obtain a description of the centralizers of elements. Finally, we examine some consequences of our results for centralizers of subgroups of the fundamental group of a graph of groups. In the Appendix, we display explicit calculations of the centralizers of elements in GL ₂$Z$, which can be written as the amalgamated free product of two finite dihedral groups.; The major results of the paper are the following: (1) We give a complete and explicit description of centralizers of elements and subgroups of free products. (2) We give several conditions under which centralizers in the fundamental group of a graph of groups are finitely generated. (3) We take two classic propositions about the structure of commuting subgroups of amalgamated free products or HNN extensions and generalize them to the case of the fundamental group of a graph of groups.