Interpolating between splittings over cyclic subgroups

Abstract

Given a finitely generated and torsion-free group Γ, it may be possible to express Γ as a product of smaller groups. In particular, we consider graph of groups decompositions of Γ — henceforth called splittings — wherein Γ is expressed as a collection of groups that are amalgamated over shared subgroups, and the underlying structure presented as a graph. In this thesis, we present a universal process for interpolating between two distinct splittings X1 and X2 of a group Γ to obtain a third splitting X3, also of the group Γ. While we present our results in the general case, we end the thesis by specifically considering cyclic one-edge cases — where the graph of groups has only one edge, representing a cyclic subgroup — and the three possible combinations of such one-edge splittings. Each of these three combinations yields a unique structure for the resulting interpolated splitting X3.