Resource for Measured Foliations and Whitehead Equivalence

Question

I am having trouble making the so-called "Whitehead equivalence" explicit.

It is quite easy to draw a picture of what a Whitehead move is, given a foliation, as a kind of limit of isotopies of the foliation. What I would like to more carefully understand is precisely what is happening to the transverse measure. The goal is to understand why the map that takes a measured foliation to a map from the space of isotopy classes of closed curves to the real numbers is constant under Whitehead equivalence (which is why, I suppose, measured foliations are considered as Whitehead equivalence classes).

Can anyone point out a resource? The more explicit the better. With respect, I have been reading the book by A. Fathi, F. Laudenbach, and V. Po´enaru ("Thurston's Work on Surfaces"), and while it is quite useful, it does not make this point clear.

Thanks!

Answer 1

I don't know of a source other than FLP for understanding the Thurston compactification via measured foliations, but you might find it easier to understand this compactification using measured geodesic laminations. The nice thing about measured geodesic laminations is that you don't need clumsy operations like Whitehead moves to make sense of them (and there is a bijective correspondence between measured foliations and measured geodesic laminations, so they contain the same information). Two nice sources for this theory are Casson-Bleiler's book and Bonahon's paper

MR0931208 (90a:32025) Bonahon, Francis(1-SCA) The geometry of Teichmüller space via geodesic currents. Invent. Math. 92 (1988), no. 1, 139–162.

Probably Bonahon's paper is what you want -- it explicitly constructs the Thurston compactification using measured geodesic laminations and has references to places that give full details about the correspondence between measured foliations and measured geodesic laminations.

Another nice source for laminations is Bonahon's unfinished manuscript "Simple Closed Curves on Surfaces", available here.