On explicit embeddings of ℚ into finitely presented groups

Abstract Explicit embeddings of the group ℚ into a finitely presented group 𝒬 and into a 2-generator finitely presented group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mi mathvariant="script">Q</m:mi> </m:msub> </m:math> T_{\mathcal{Q}} are given. The embeddings constructed reflect questions mentioned by Johnson, Bridson, and de la Harpe in the literature in the late 1990s about the possibility of such explicit embeddings for ℚ. The technique used here is based on the methods with integer-valued sequences used by Higman, and on methods with specific free constructions of groups, including free products of groups with amalgamation, HNN-extensions of groups, and an auxiliary technical structure called the ∗-construction.