The Choreography of Geodesics in SOL
Marc Troyanov
TL;DR
This work provides a self-contained geometric synthesis of SOL's geodesic flow by isolating a single invariant, the modulus $k\in[0,1]$, which classifies generic geodesics and governs their vertical oscillation, horizontal drift, and winding via Grayson cylinders. The authors solve the geodesic equations by quadrature, relate the period $T(k)$ and drift $H(k)$ to elliptic integrals, and show that the modulus is a complete invariant for geometric equivalence of generic geodesics, leading to a rendezvous phenomenon that constrains geodesic minimality to length $T(k)$. They also describe the cut locus (it is infinite for vertical/hyperbolic and equals $T(k)$ for generic; $\sqrt{2}\pi$ for horizontal) and prove a logarithmic growth law for large-ground-plane distances, $d(0,\lambda\dots)=4\log\lambda+O(1)$. The paper thereby offers a complementary, dynamical-geometric perspective to prior work by Grayson and Coiculescu--Schwartz, enriching the understanding of SOL's geodesic structure and its asymptotic geometry.
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.