Two-Generator Discrete Subgroups of Tree Automorphism Groups

In this paper, we give a complete and explicit classification of faithful HNN $2$-extensions, namely those whose actions on their Bass–Serre trees are faithful. We also construct infinite families of faithful two-generator graphs of groups realizing each admissible tree or unicyclic underlying graph. Together, these results indicate a natural threshold in the classification problem; in the small-geometry regime the groups admit an isomorphism-level classification, while beyond this barrier the natural classification is substantially coarser and is governed by quotient-graph data.