The degeneration of convex RP^2 structures on surfaces

Let M be a compact surface of negative Euler characteristic and let C(M) be the deformation space of convex real projective structures on M. For every choice of pants decomposition for M, there is a well known parameterization of C(M) known as the Goldman parameterization. In this paper, we study how some geometric properties of the real projective structure on M degenerates as we deform it so that the internal parameters of the Goldman parameterization leave every compact set while the boundary invariants remain bounded away from zero and infinity.Comment: 47 pages, 17 figures, Accepted for publication at PLM

The Goldman symplectic form on the PGL(V)-Hitchin component

This article is the second of a pair of articles about the Goldman symplectic form on the PGL(V )-Hitchin component. We show that any ideal triangulation on a closed connected surface of genus at least 2, and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the Hitchin component. Using this, we prove that a large class of flows defined in the companion paper [SWZ17] are Hamiltonian. We also construct an explicit collection of Hamiltonian vector fields on the Hitchin component that give a symplectic basis at every point. These are used to show that the global coordinate system on the Hitchin component defined iin the companion paper is a global Darboux coordinate system.Comment: 95 pages, 24 figures, Citations update

Deforming convex real projective structures

In this paper we define new flows—eruption flows and internal bulging flows—on the deformation space of convex real projective structures. These flows are associated to internal parameters associated to a pair of pants decomposition

Geometry of the Hitchin Component.

We construct a parameterization of the PSL(n,R) Hitchin component that generalizes the Fenchel-Nielsen coordinates on Teichmuller space. Using this parameterization, we study the degeneration of certain geometric quantities, such as length functions and topological entropy, that are associated to the representations in the Hitchin component.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113605/1/tengren_1.pd

Cusped Borel Anosov representations with positivity

We show that if a cusped Borel Anosov representation from a lattice $\Gamma \subset \mathsf{PGL}_2(\mathbb{R})$ to $\mathsf{PGL}_d(\mathbb{R})$ contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.Comment: 13 page

Regularity of limit sets of Anosov representations

In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a differentiable submanifold of projective space with Holder continuous derivatives. We also calculate the optimal value of the Holder constant in terms of the eigenvalue data of the Anosov representation.Comment: 58 pages, 1 figure, typos corrected in Version