Publication

The Choreography of Geodesics in SOL

Marc Troyanov
2026

Abstract

We provide a self-contained geometric description of the geodesic flow in the threedimensional Lie group SOL, one of Thurston's eight model geometries. The geometry of geodesics is governed by a single invariant k ∈ [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 the 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.

Official source

https://infoscience.epfl.ch/entities/publication/c72c5b91-1b0f-4200-b7f5-2dbfd5f177de

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Marc Troyanov
2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/c72c5b91-1b0f-4200-b7f5-2dbfd5f177de

About this result

Related concepts (32)

Geodesic

In geometry, a geodesic (ˌdʒiː.əˈdɛsɪk,-oʊ-,-ˈdiːsɪk,_-zɪk) is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun geodesic and the adjective geodetic come from geodesy, the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry.

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa.

Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point.

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The Choreography of Geodesics in SOL

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