Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension {3g-3}, for example.We also study the limit sets in the Thurston boundary of Teichmüller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the 1-skeleton of the simplex of projective classes of measures visiting every vertex.

Limit Sets of Weil–Petersson Geodesics

In this paper we prove that the limit set of any Weil–Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil–Petersson geodesics with minimal non-uniquely ergodic ending laminations and limit set a circle in the Thurston compactification.

CANNON–THURSTON MAPS FOR KLEINIAN GROUPS

We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

Prescribing the behavior of Weil–Petersson geodesics in the moduli space of Riemann surfaces

We study Weil–Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space of Riemann surfaces. This class of geodesics is rich enough to provide for examples of closed WP geodesics in the thin part of the moduli space, as well as divergent WP geodesic rays with minimal filling ending lamination. Some ingredients of independent interest are the following: A strength version of Wolpert's Geodesic Limit Theorem proved in Sec. 4. The stability of hierarchy resolution paths between narrow pairs of partial markings or laminations in the pants graph proved in Sec. 5. A kind of symbolic coding for laminations in terms of subsurface coefficients presented in Sec. 7.