Terminal Geometry of Rank-Two Nielsen Reduction on Simplicial Trees

Let $X$ be a locally finite simplicial tree, and let $\Gamma=\langle\gamma_1,\gamma_2\rangle\leq\aut(X)$ be a discrete subgroup generated by two hyperbolic automorphisms acting without inversions, whose axes have finite edge intersection. We give an explicit Euclidean localization of the first stopping stage in rank-two Nielsen reduction and determine the exact characteristic-set geometry of the resulting terminal Nielsen pair. The continued fraction expansion of the ratio $m_1/m_2$ of the initial translation lengths determines a finite partition of the range of the axis-overlap $l$. Away from the partition endpoints, the terminal pair is Schottky. At nonfinal endpoints, an additional overlap length determines whether the terminal pair is Schottky or contains a nontrivial elliptic generator. At or beyond the final endpoint, the terminal pair contains a nontrivial elliptic generator. If the terminal pair is Schottky, then $\Gamma$ is a discrete free group of rank two whose nontrivial elements are all hyperbolic. If one terminal generator is nontrivial elliptic, then Local finiteness and discreteness imply that $\Gamma$ contains nontrivial torsion and is not free of rank two. We determine in every case the translation data and relative positions of the axes or fixed subtrees of the terminal generators.