Measured foliations and train tracks are fundamental tools for understanding the geometry of hyperbolic surfaces and 3–manifolds. In particular, a hyperbolic 3–manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants—geometric data encoded by the ending laminations on its boundary [1]. Both compressible and incompressible boundaries can occur. However, in the compressible case—when a surface bounds a handlebody—not every lamination can arise as an ending lamination; only those that are incompressible can.

This phenomenon was studied by Masur and Kerckhoff in the 1980s, who showed that the mapping class group of a handlebody acts properly discontinuously on an open subset of the space of projective measured laminations [2]. This open set, known as the Masur domain, has full measure with respect to the Thurston measure and consists entirely of incompressible laminations [3]. In this talk, we follow Masur and Kerckhoff’s approach to reproduce this result. Also, we will see that it contains a beautiful application of train track theory.

Reference:
[1] J. F. Brock, R. D. Canary, and Y. N. Minsky, The classification of Kleinian surface groups II: The Ending Lamination Conjecture, Annals of Mathematics (2) 176 (2012), no. 1, 1–149.
[2] H. Masur, Measured foliations and handlebodies, Ergodic Theory and Dynamical Systems 6 (1986), no. 1, 99–116.
[3] S. P. Kerckhoff, The measure of the limit set of the handlebody group, Topology 26 (1987), no. 3, 353–362.