Terminal Geometry of Rank-Two Nielsen Reduction on Simplicial Trees

Abstract

Let $X$ be a locally finite simplicial tree, and let $\Gamma =⟨{\gamma }_1,{\gamma }_2⟩\le \mathrm{Aut}(X)$ be a discrete subgroup generated by hyperbolic automorphisms acting without inversions, whose axes have finite edge intersection. We locate explicitly, using the Euclidean algorithm, the first stopping state of rank-two Nielsen reduction and determine the exact characteristic-set geometry of its terminal Nielsen pair. Write $m_1\ge m_2>0$ for the normalized translation lengths, $l$ for the axis-overlap length, and $d=\gcd (m_1,m_2)$. If $l=0$, the initial pair satisfies the Schottky criterion. If $l>0$, Euclidean quotients and remainders partition $0<l\le m_1+m_2-d$ into intervals locating the first stopping state and its ordered Nielsen generators. Interior points yield terminal Schottky pairs. At nonfinal right endpoints, an auxiliary translated-axis overlap $l_0$ distinguishes Schottky terminal pairs from pairs containing a nontrivial elliptic generator. At the final endpoint and for larger overlaps, a nontrivial elliptic terminal generator occurs. We compute the terminal translation lengths and the relative geometry of axes or fixed subtrees in every case. Each terminal pair is Nielsen-equivalent to the initial pair and generates $\Gamma$. If both terminal generators are hyperbolic, they form a Schottky pair; otherwise exactly one is nontrivial elliptic. By local finiteness and discreteness, this element has finite order, so $\Gamma$ contains nontrivial torsion and is not free of rank two.

arXiv commentv2: Substantially revised, expanded, and retitled; abstract and introduction rewritten, with improved exposition, a running example, and fuller discussion of related work. The first stopping state and terminal characteristic-set geometry, including equality cases, are treated in greater detail. Weighted-tree material (Section 4 of v1) is reserved for subsequent work. 39 pages, 4 figures