TITLE: Teichmuller geodesics vs. geodesics in the hyperbolic plane
QUESTION [7 upvotes]: Geodesics in $\mathbb H^2$ have the following properties: 

For every two points in the plane there exists a unique geodesic joining them. 
Every geodesic determines exactly two points on the boundary of $\mathbb H^2$. 
Conversely, every pair of points on $\partial \mathbb H^2$ determine a unique geodesic 
Any two different geodesics that travels at bounded distance are asymptotic. 

How many of the above properties are true for Teichmuller geodesics on the Teichmuller space 
of a closed surface with Thurston's boundary? Could you also provide references, please?

REPLY [5 votes]: This subject is addressed in Kasra Rafi's very nice paper (particularly relevant to point 4).