4-10 June 2017
Character varieties and knot symmetries
I will start introducing the basic definitions and properties of varieties of representations in SL(2,C) and character varieties, focussing in hyperbolic manifolds of finite volume. I plan to explain some classical applications and to develop few explicit examples. Then I will discuss how to use the variety of characters to distinguish different kinds of symmetries of hyperbolic knots, namely whether a symmetry of a knot has fixed points or not. This application is a joint work with Luisa Paoluzzi.
Alan Reid Arithmetic hyperbolic manifolds
A general method to construct finite volume or closed hyperbolic manifolds is any dimension is using arithmetic methods. In this mini course we provide an introduction to these methods, as well as describing some more up to date results about the geometry and topology of arithmetic hyperbolic manifolds.
Jean-Marc Schlenker, Ken Bromberg, Jeff Brock The renormalized volume of quasifuchsian manifolds, with applications
We will present the definition and main properties of the renormalized volume of quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds with incompressible boundary. We will then outline a few applications to the Weil-Petersson geometry of Teichmuller space.
Martin Bridgeman Schwarzian derivatives, projective structures, and the Weil-Petersson gradient flow for renormalized volume.
We consider complex projective structures and their associated
locally convex pleated surface.
We relate their geometry in terms of the L_2 and L_infinity norms the Schwarzian derivative.
We show that these give a unifying approach
that generalizes a number of well-known results for convex cocompact
hyperbolic structures including bounds on the Lipschitz constant
for the retract and the length of the bending lamination.
We then use these bounds to
study the Weil-Petersson gradient flow of renormalized volume
on the space CC(N) of convex cocompact hyperbolic structures
on a compact manifold N with incompressible boundary.
This leads to a proof of
the conjecture that the renormalized volume has infimum given by one-half
the simplicial volume of DN, the double of N.
Jeffrey Danciger Convex real projective structures and Anosov representations
We investigate the degree to which the geometry of a compact real projective manifold with boundary is reflected in the associated holonomy representation, a representation of the fundamental group in the projective general linear group PGL(n,R) which in general need not have any nice properties.
We show that if the projective manifold is strictly convex, then its holonomy representation is projective Anosov, a condition which generalizes the dynamical properties of convex cocompact representations in rank one (e.g. hyperbolic) geometry. Conversely, a strictly convex projective manifold may be constructed from a projective Anosov representation that preserves a properly convex set in projective space. Applications include new examples of both convex projective manifolds
and Anosov representations.
Colin Guillarmou Limit of the renormalized volume at the boundary of Teichmuller space
We define the renormalized volume on geometrically finite hyperbolic 3 manifolds, including those with non-maximal rank cusps, and we show that under some kind of degenaration of convex co-compact manifolds towards geometrically finite manifolds, the renormalized volume converge to the renormalized volume of the limit manifold. This was a joint work with S. Moroianu and F. Rochon.
Anna Lenzhen Limit sets of Teichmuller geodesic rays in the Thurston boundary of Teichmuller space
H. Masur showed in the early 80s that almost every Teichmuller ray converges to a unique point in PMF. It is also known since a while that there are rays that have more than one accumulation point in the boundary. I will give an overview of what is understood so far about the limit sets of Teichmuller rays. For example, I will mention recent joint work with K. Rafi and B. Modami where we give a construction of a ray whose limit set in PMF is an entire 2-simplex.
Sara Maloni Convex hulls of quasicircles in hyperbolic and anti-de Sitter space
Thurston conjectured that quasifuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a generalization of these conjecture to convex hulls of quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. (This is joint work in progress with Bonsante, Danciger and Schlenker.)
Yair Minsky Local connectivity in deformation spaces of hyperbolic 3-manifolds
The deformation space of a hyperbolic 3-manifold has complicated boundary, both topologically and geometrically. A detailed understanding of this boundary requires good control of the internal geometry of such manifolds, and their behavior under both algebraic and geometric limits. I will discuss a theorem, joint with Brock, Bromberg, Canary and Lecuire, showing that the boundary (for a manifold with incompressible boundary) is locally connected at all quasiconformally rigid points. This is a generalization of a previous result for acylindrical manifolds. The main tools involve the interaction of Thurston's skinning map with the curve-complex methods for studying Kleinian surface groups.
Kate Petersen Gonality and the Character Variety
The SL(2,C) character variety of a hyperbolic 3-manifold M is a complex variety that encodes a lot of information about M. I'll discuss how some algebro-geometric invariants of the character variety relate to the topology of M. In particular, I'll introduce the notion of gonality of an algebraic curve, and discuss how gonality behaves in families of Dehn fillings.
Maria Breatrice Pozzetti Basmajian-type inequalities for maximal representations
An injective homomorphism of the fundamental group of an hyperbolic surface in the symplectic group Sp(2n,R) is a maximal representation if it maximizes the so-called Toledo invariant. Maximal representations form interesting and well studied components of the character variety generalizing the Teichm\"uller space, that is encompassed in the case n=1. Given a hyperbolic surface S with totally geodesic boundary, Basmajian's equality allows to compute the length of the boundary as an explicit function of the lengths of the orthogeodesics: geodesic segments orthogonal to the boundary at both endpoints. In joint work with Federica Fanoni we provide a generalization of this result to the setting of maximal representations that allows us to describe geometric properties of the locally symmetric space associated to the representation.
Andrea Seppi Volume of Anti-de Sitter 3-manifolds
A celebrated theorem of J. Brock showed that the volume of the convex core of a quasi-Fuchsian manifold is coarsely equivalent to the Weil-Petersson distance between its two Bers parameters
in the Teichmuller space T(S) of the closed surface S. In this talk,
we will study the volume of maximal globally hyperbolic Anti-de Sitter 3-manifolds,
which are the Lorentzian equivalent of quasi-Fuchsian manifolds
and are again parameterized by T(S)xT(S) by a result of G. Mess.
We will show that the volume of the convex core is coarsely equivalent to
the optimal L^1-energy between hyperbolic surfaces, and
we will discuss the relation with
the Thurston distance and the Weil-Petersson distance.