The Choreography of Geodesics in SOL

TL;DR

This paper provides a comprehensive geometric analysis of geodesic flows in the Sol geometry, revealing how geodesics behave based on an invariant, and characterizing key features like minimal segments and the cut locus.

Contribution

It offers a new geometric and dynamical perspective on geodesics in Sol, extending previous work with alternative methods and detailed descriptions.

Findings

Geodesics spiral around an axis with quantifiable amplitude, period, and drift.

Distances at the same altitude grow logarithmically.

Characterization of minimal geodesic segments and the cut locus.

Abstract

We provide a self-contained geometric description of the geodesic flow in the three-dimensional Lie group $\mathrm{Sol}$, one of Thurston's eight model geometries. The geometry of geodesics is governed by a single invariant $k\in[0,1]$, its modulus. Generic geodesics spiral around an axis, with well-defined amplitude $A(k)$, period $T(k)$, and horizontal drift $H(k)$. We characterize minimal geodesic segments and the cut locus, and obtain an asymptotic estimate showing that distances between points at the same altitude grow logarithmically. This work builds on previous work by Grayson and Coiculescu--Schwartz, but develops an alternative geometric and dynamical viewpoint.