Is there a unique geodesics between any two points in the [NIL (resp. SOL) geometry](http://en.wikipedia.org/wiki/Geometrization_conjecture#Nil_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](http://en.wikipedia.org/wiki/Hyperbolic_space) and the functions sinh,cosh.



Are there unique geodesics in the NIL and SOL geometry?
The geodesics between points are not unique in both cases. Moreover the following is true: if $M$ is a universal cover of a compact Riemannian manifold whose fundamental group is virtually solvable but not virtually abelian, then there are conjugate points on some geodesics in $M$ and hence geodesics between some points are not unique.


See Croke and Schroeder "[The fundamental group of compact manifolds without conjugate points](https://eudml.org/doc/140046)", Comment. Math. Helv. 61 (1986), no. 1, 161--175, [MR847526](https://mathscinet.ams.org/mathscinet-getitem?mr=847526), for the case when the metric is analytic, and Lebedeva "On the fundamental group of a compact space without conjugate points", [here](ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/2002/05-02.ps.gz), for the general case.


