Schottky pairs on Trees via Continued Fractions and Axial Geometry

TL;DR

This paper provides a geometric criterion for when two hyperbolic automorphisms of a tree generate a free, discrete subgroup, using continued fractions and axis overlap, extending to weighted trees and connecting to the three-gap theorem.

Contribution

It introduces a complete criterion based on geometric invariants and continued fractions for free subgroup generation in trees, including weighted cases and irrational ratios.

Findings

Criterion depends on translation lengths and axis overlap

Extension to weighted trees with positive real lengths

Connection to the three-gap theorem in irrational cases

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.