TITLE: Are there unique geodesics in the NIL and SOL geometry?
QUESTION [11 upvotes]: Is there a unique geodesics between any two points in the NIL (resp. SOL) geometry?
If so, is there a nice way of parametrizing them? For example geodesics in $S^3$ can be parametrized using the embedding in $\mathbb{R}^4$ and $\sin , \cos$ functions. Geodesics in hyperbolic space can be parametrized using the hyperboloid model and the functions sinh,cosh.

REPLY [3 votes]: The geodesics in SOL that are not contained in a vertical (hyperbolic) foliation tend to come in families meeting periodically and are thus highly non unique. They are precisely described in my paper L’Horizon de SOL.  Expos. Maths 1998.
In this paper, the SOL geodesics are explicitly parametrized using elliptic functions.