Let \(S = S_{g, b}\) be a surface of genus \(g\) with \(b\) boundary components. Suppose that \(3g-3+b > 1\). The curve complex \(\mathcal C(S)\) is the flag complex whose vertices are (isotopy classes of) non-null homotopic and nonperipheral simple closed curves. A set of \(k\) distinct isotopy classes defines a \(k\)-simplex if any pair can be represented by disjoint curves. Given a sequence \(\{\gamma_n\}\) of curves, we can ask about the geometry of how this sequence sits inside the curve complex. A natural question is: when does this sequence approximate a geodesic in the curve complex?
A part of the main result of this paper gives a sufficient condition for this sequence to be an infinite quasi-geodesic. Exploiting connections between the geometry of the curve complex and the Teichmüller and Weil-Petersson geometries of Teichmüller space, several interesting corollaries are obtained about the behaviors of limit sets of Teichmüller and Weil-Petersson geodesics. In particular, the authors construct Teichmüller geodesics with non-simply connected limit sets, and give sufficient for recurrence of Weil-Petersson geodesics in moduli space in terms of their ending laminations.
For Part I, see [the second author et al., “Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation”, Preprint, arXiv:1312.2305].