Abstract:

The Thurston compactification of Teichmuller space describes the limit of a sequence of hyperbolic structures on a surface as a geometric object—namely a measured lamination. Convex projective structures generalize hyperbolic structures, and there have been various proposed analogues for the Thurston compactification. In this talk we make the case that the objects at infinity, in directions "orthogonal" to Teichmuller space, are singular, flat, Finsler metrics whose unit balls are equilateral triangles. A defining feature of the Thurston compactification is that a boundary point records limits of ratios of curve lengths, or equivalently ratios of logarithms of eigenvalues. Our main theorem (in progress) shows that ratios of logarithms of eigenvalues of appropriate sequences of convex projective structures are similarly captured by the Finsler metrics we define. Specifically, we use the Labourie-Loftin parametrization of convex projective structures by cubic differentials on Riemann surfaces and describe what happens when the Riemann surface structure converges and the cubic differential diverges. We will show how our theorem makes testable predictions in a specific example.