Schottky pairs on Trees via Continued Fractions and Axial Geometry
Yukun Du, Sa'ar Hersonsky
TL;DR
The paper provides a complete geometric criterion for when two hyperbolic tree automorphisms generate a free discrete subgroup, using only translation lengths and axis overlap, organized via the continued-fraction expansion of the length ratio. It introduces a Nielsen-reduction framework that mirrors Euclidean division to classify all possible configurations, yielding explicit cases where a Schottky pair exists. In the irrational-ratio setting, the non-free configurations correspond precisely to gap lengths from the three-gap theorem, revealing a Diophantine mechanism governing freeness. The results extend to weighted trees, preserving the same structural classification and linking axis-intersection geometry with continued fractions and gap-length phenomena in p-adic-like contexts.
Abstract
We give a complete criterion for when two hyperbolic automorphisms of a tree generate a free, discrete subgroup. The decision depends only on three geometric invariants: the translation lengths of the generators and the length of overlap of their axes. This data is organized using the continued-fraction expansion of the translation-length ratio. We extend the result to weighted trees, allowing arbitrary positive real translation lengths under local finiteness. In the irrational case, the exceptional configurations are shown to correspond precisely to the gap lengths in the three-gap theorem.