ICTS

A recent work by Mazzeo-Swoboda-Weiss-Witt describes a stratum of the frontier of the space of SL(2,C) surface group representations in terms of 'limiting configurations' which solve a degenerated version of Hitchin's equations on a Riemann surface. We interpret these objects in (a mapping class group invariant way in) terms of the hyperbolic geometric objects of shearings of pleated surfaces. The effect is to refine the Morgan-Shalen compactification for this stratum. Much of the talk is devoted to defining the various objects. (Joint with Andreas Ott, Jan Swoboda, and Richard Wentworth).

A CP^1-structure on a surface is the geometric structure modeled on the Riemann sphere, and it’s holonomy representation is a homomorphism from the fundamental group of the surface into PSL(2, C). We discuss about certain degenerations of CP^1-structures when their holonomy representations converge and their conformal structures are pinched along loops.

Let f be a homomorphism from the fundamental group of a compact orientable surface X to the multiplicative group of nonzero complex numbers. If X is given a complex structure, f produces a holomorphic line bundle. If f does not factor through U(1), Ghys asked whether there is a complex structure on X such that the holomorphic line bundle is trivial. We prove that there is indeed such a complex structure on X (joint work with Sorin Dumitrescu and Mahan Mj).

TBA

The holomorphic isometry group of complex hyperbolic space is PU(n,1). In this talk I will discuss representations of triangle groups in PU(2,1). Schwartz gave a conjectural picture of the situation when the reflections in the sides of the triangle have order 2. Some of his conjectures have been proved (or disproved) but most remain open. The situation is more complicated when the reflections have higher order (which can happen for complex reflections). I will give a survey of recent work in this area and discuss some open problems.

Thurston's hyperbolic Dehn surgery theorem says that if M is a cusped hyperbolic three dimensional manifold then almost all Dehn fillings of M admit a hyperbolic structure. However, the hyperbolic Dehn filling is impossible for dimension bigger than three. In this talk, I will give the first examples of cusped hyperbolic four dimensional manifolds whose Dehn fillings admit a convex real projective structure. Joint work with Suhyoung Choi and Ludovic Marquis.

Let S be a closed, orientable, connected surface of genus at least 2. We prove that any ideal triangulation on S determines a symplectic trivialization (with respect to the Goldman symplectic form) of the tangent bundle of the Hitchin component. One can then consider the parallel flows with respect to the flat structure given by this trivialization. We give a geometric description of all such flows in terms of explicit deformations of Frenet curves, and prove that all such flows are Hamiltonian. Applying this to a particular ideal triangulation allows us to find a maximal family of Poisson commuting Hamiltonian flows on the Hitchin component. This generalizes the well-known fact that on Teichmüller space, the twist flows along a pants decomposition of S is a maximal family of Poisson commuting Hamiltonian flows. This is joint work with Zhe Sun and Anna Wienhard.

TBA

Ultra-parallel complex hyperbolic triangle groups are groups of isometries of the complex hyperbolic plane generated by 3 complex reflections in complex geodesics which do not intersect pair-wise. I will discuss discreteness and non-discreteness results for certain types of such groups. This is joint work with A. Monaghan and J. Parker and with S. Povall.

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between the boundaries of proper, geodesically complete CAT(-1) spaces, we define an extension $F : X \to Y$ of $f$, called the circumcenter extension of $f$, which is shown to be a $(1, \log 2)$-quasi-isometry, which is locally $1/2$-Holder continuous. If $X, Y$ are complete, simply connected Riemannian manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$, then the circumcenter extension is a $(1, (1 - 1/b) \log 2)$-quasi-isometry. Moreover if $g : \partial Y \to \partial X$ is the inverse of $f$, then the circumcenter extensions $F : X \to Y$ and $G : Y \to X$ of $f$ and $g$ are inverses of each other, and are $\sqrt{b}$-bi-Lipschitz. This is proved by constructing a family of extensions $F_p : X \to Y, 1 \leq p \leq \infty$ of $f$, called the hyperbolic $p$-barycenter extensions of $f$, and analyzing their behaviour as $p$ tends to $\infty$. As a corollary we obtain that if two closed, negatively curved manifolds have the same marked length spectrum, then they are bi-Lipschitz homeomorphic.