CLASSIFYING LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$

Abstract

We show that low height-achieving non-simple geodesics on a low-index cover of the modular surface can be classified into seven types, according to the topology of highest arcs. The lowest geodesics of the signature (0;2,2,2,∞)-orbifold [Formula: see text] are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x2 + y2 + z2 = 3xyz, and have heights [Formula: see text]. Geodesics considered by Crisp and Moran have heights [Formula: see text]; they conjectured that these heights, which lie in the "mysterious region" between 3 and the Hall ray, are isolated in the Markoff Spectrum. As a step in resolving this conjecture, we characterize the geometry on [Formula: see text] of geodesic arcs with heights strictly between 3 and 6. Of these, one type of geodesic arc cannot realize the height of any geodesic.