Publications

A Coxeter n–orbifold is an n–dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order m, whose neighborhood is locally modeled … on \mathbbR^n modulo the dihedral group of order 2m generated by two reflections. For n ≥3, we study the deformation space of real projective structures on a compact Coxeter n–orbifold Q admitting a hyperbolic structure. Let e_+(Q) be the number of ridges of order greater than or equal to 3. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension e_+(Q)-n if n=3 and Q is weakly orderable, ie the faces of Q can be ordered so that each face contains at most 3 edges of order 2 in faces of higher indices, or Q is based on a truncation polytope.

Let S be a closed oriented surface of genus at least two. Gallo, Kapovich and Marden asked whether 2\pi–grafting produces all projective structures on S with arbitrarily fixed holonomy (the Grafting … conjecture). In this paper, we show that the conjecture holds true “locally” in the space \mathcalGL of geodesic laminations on S via a natural projection of projective structures on S into \mathcalGL in Thurston coordinates. In a sequel paper, using this local solution, we prove the conjecture for generic holonomy.

We explain how the generalized Milnor–Wood inequality of Burger and Iozzi for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates under the non-abelian … Hodge correspondence into an inequality for topological invariants of the corresponding Higgs bundles. In this manner, we obtain in a uniform way a universal Milnor–Wood inequality for Higgs bundles over complex-hyperbolic manifolds of arbitrary dimensions and with arbitrary Hermitian structure group. This complements results of Biquard, Bradlow, García-Prada, Gothen, Mundet, Rubio and Chaput, Koziarz, Maubon.

We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber … length compactification and apply our results to the theory of maximal and Hitchin representations.