TITLE: What does the free action of a surface group on an R-tree look like?
QUESTION [7 upvotes]: Morgan and Shalen "Free action of surface groups on R-trees" 1989 shows that surface groups (genus at least 2) act freely on some real trees (R-trees). Their proof seems to be non-constructive, requiring the Baire category theorem. But the object they are constructing sounds quite simple. Their Proposition 17 says:

On every closed non-exceptional hyperbolic surface, there exists a measured geodesic lamination whose leaves and complementary regions are all simply connected.

Does anyone have a picture of this lamination on the genus 2 surface? And, using that, a picture of the R-tree that the genus 2 surface group acts freely on?

REPLY [11 votes]: Here is a picture of a train track, which approximates a lamination:

Another picture: 

The leaf space of this lamination (obtained by collapsing each
leaf and complementary polygon to a point) is an $\mathbb{R}$-tree.
As you contract the leaves to points, you might get a picture
looking somewhat like this:

If you fill in with the boundary circle, you get a dendrite which might help you visualize an $\mathbb{R}$-tree: