Abstracts

Biringer - Random samplings of locally symmetric spaces

Suppose that M_i is a sequence of closed locally symmetric spaces modeled on a fixed symmetric space X, e.g. a sequence of hyperbolic n-manifolds. We introduce a tool to understand how the geometry of M_i near a random sample point develops as i tends to infinity. Applications include a control on the growth of the Betti numbers of M_i when X has higher rank.

Boileau - Homology Spheres, L-spaces and taut foliations

We discuss Ozsvath and Szabo's conjecture that an aspherical Z-homology 3-sphere has a non-trivial Heegaard-Floer homology, and the existence of a taut foliation on such a manifold. In particular we show that an aspherical graph Z-homology 3-sphere always admits a taut foliation which is transverse to the fibres in each Seifert piece. It is a work with Steve Boyer.

Bonsante - A cyclic flow on Teichmuller space

In a recent work with Mondello and Schlenker, we introduced a new family of deformations of hyperbolic structures which can be regarded as a smooth version of Thurston hyperbolic earthquakes. We proved that such deformations determine a period flow which is Hamiltonian for the Weil-Petersson metric. In fact we proved that the usual earthquake flow can be realized as the limit of this cyclic flow. In the talk I'll introduce this family of deformations,  making a comparison with usual earthquake flow, and showing some possible applications.

Brock - Fat, exhausted, integer homology spheres

Since Perelman's groundbreaking proof of the geometrization conjecture for three-manifolds, the possibility of exploring tighter correspondences between geometric and algebraic invariants of three-manifolds has emerged.  In this talk, we address the question of how homology interacts with hyperbolic geometry in 3-dimensions,  providing examples of hyperbolic integer homology spheres that have large injectivity radius on most of their volume.  (Indeed such examples can be produced that arise as (1,n)-Dehn filling on knots in the three-sphere). Such examples fit into a conjectural framework of Bergeron, Venkatesh and others providing a counterweight to phenomena arising in the setting of arithmetic Kleinian groups.  This is joint work with Nathan Dunfield.

Cooper - Degenerations and transitions of sub-geometries of projective geometry

A geometric transition is a path of geometric structures on a manifold which changes type. The most basic example is a path of constant curvature Riemannian metrics on a surface (with cone points) so that the curvature changes from positive to negative. This talk discusses a more general version of this phenomenon from the point of view of projective geometry.

We will discuss a framework for analyzing which geometric transitions can be achieved by a path of projective structures on some manifold M. More specifically, given representations of two homogeneous spaces (X,G) and (X',G') in projective space, we study the question of whether or not there exists a path of projective structures which changes type from (X,G) to (X',G').

This is joint with Anna Wienhard and Jeff Danciger.

Danciger - Degenerations of hyperbolic cone manifolds

We study collapsing hyperbolic cone manifolds in dimension 3. These collapses occur at the boundary of hyperbolic Dehn surgery space. Viewing the hyperbolic structures as real projective structures, it can happen that the underlying projective geometry does not collapse, but rather degenerates to a different type of geometry. We discuss several examples including the degenerations from hyperbolic to Euclidean and Nil, as well as the degeneration to Half-pipe geometry, which arises naturally in the transition from hyperbolic to AdS. Finally, we discuss joint work with Daryl Cooper and Anna Wienhard describing exactly which geometries can arise as degenerations in this way.

Futer - Cusp geometry of fibered 3-manifolds

Let F be a surface with punctures, and suppose that φ : F → F is a pseudo-Anosov homeomorphism fixing a puncture p of F. Then the mapping torus of φ is a hyperbolic 3-manifold M_φ, which contains a maximal cusp corresponding to the puncture p. We show that the geometry of the maximal cusp can be predicted, up to explicit multiplicative error, by the action of φ on the complex of essential arcs of in the surface F.

This result is motivated by an analogous theorem of Brock, which predicts the volume of M_φ in terms of the action of φ on the pants graph of F. However, in contrast with Brock’s theorem. our result gives effective estimates, and is proved using completely elementary methods. This is joint work with Saul Schleimer.

Gabai - On the topology of ending laminations space

The ending lamination space is a naturally defined space associated to a surface of negative Euler characteristic that is important in hyperbolic geometry and geometric group theory. We will discuss recent progress on characterizing the topology of these beautiful and mysterious spaces.

Hodgson - Counting hyperbolic 3-manifolds with a given volume

The work of Thurston and Jorgensen shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. We will look at the question of how N(v) varies with v.

We show that there is an infinite sequence of closed hyperbolic 3-manifolds that are uniquely determined by their volumes. The proof uses work of
Neumann-Zagier on the change in volume during hyperbolic Dehn surgery together with some elementary number theory.

We also describe examples showing that the number of hyperbolic link complements with volume v can grow at least exponentially fast with v.

(This is joint work with Hidetoshi Masai, Tokyo Institute of Technology)

Kahn - The good pants homology and the Ehrenpreis conjecture

The Ehrenpreis conjecture states that given any two closed Riemann surfaces Y and Z of genus greater than 1, and any K > 1, there are finite covers Ŷ and Ẑ of the two surfaces, and a K-quasiconformal map between them. In joint work with Vladimir Markovic, we prove the Ehrenpreis conjecture by showing that any closed hyperbolic surface Y has a cover Ŷ that is made of "good pants" that have been assembled in a good way, and then show that any two ''good panted surfaces'' have common covers that are close in the Teichmüller metric.

We find these good panted covers by showing that the set of good pants (which have boundary lengths close to a given large R) is evenly distributed around every good geodesic. We can assemble a formal sum of good pants to form a good panted surface provided that the formal sum is balanced---that there are an equal number of pants on the two sides of every good geodesic. We define the "good pants homology" to analyze the obstruction to correcting an imbalance, and we show that the good pants homology is equal to the standard homology, which implies that the obstruction is trivial.

I would like to spend the first third of this talk explaining how the Ehrenpreis conjecture reduces to the study of the good pants homology, and then the remaining two-thirds on an overview of the main lemmas of the good pants homology, including the Algebraic Square Lemma, the Four-Part Itemization Lemma, and the Rotation Lemma.

Kassel - Representations of surface groups and Lorentzian space-forms in dimension 3

I will explain how 3-dimensional Lorentzian space-forms of negative curvature arise from pairs of representations of surface groups into PSL₂(R), where one representation is "more contracting" than the other. I will discuss several applications of this contraction property, in particular to the construction of fundamental domains and to the spectral theory of the (Lorentzian) Laplacian. This is joint work with François Guéritaud and with Toshiyuki Kobayashi.

Leininger - Mapping class groups, Kleinian groups and convex cocompactness

For mapping class groups there is a notion of convex cocompactness, due to Farb and Mosher, defined by way of analogy with the concept of the same name in Kleinian groups. On the other hand, there are certain Kleinian groups which can themselves naturally be thought of as subgroups of mapping class groups.  After describing some of the background, I will discuss a direct relationship between convex cocompactness in the two settings for this special class of groups.  This is joint work with Spencer Dowdall and Richard Kent.

Minsky - Hyperbolic 3-manifolds of bounded type

We describe a class of hyperbolic 3-manifolds built out of compact pieces with gluings satisfying a "bounded combinatorics" condition. For this class we can give a uniform family of bilipschitz models, and prove a rigidity theorem that applies in the infinitely-generated case.  Another corollary is a topological characterization of hyperbolic 3-manifolds with bounded Heegaard genus and lower bound on injectivity radius.
Joint work with Jeff Brock, Hossein Namazi and Juan Souto

Masur - Winning sets of Diophantine measured foliations

In the 1960's W.Schmidt invented a game now called a Schmidt game to be played in Rⁿ.  Associated are what are called winning sets which have various nice properties; one of which is full Hausdorff dimension. The main motivating example of a winning set which Schmidt considered is  the subset of reals with bounded continued  fraction expansion. Classically these are the reals that are badly approximable by fractions. They also correspond in the moduli space H²/SL(2,Z) to hyperbolic geodesics that stay in a compact set. One can formulate  a similar condition for a measured foliation on a higher genus surfaces to be badly approximated by simple closed curves. These correspond to Teichmuller geodesics that stay in a compact subset of the  corresponding moduli space. These are called Diophantine foliations. After giving the background on winning sets I will discuss the theorem, joint with Jon Chaika and Yitwah Cheung that the set of Diophantine foliations is Schmidt winning as a subset of PMF, Thurston's sphere of measured foliations.

Purcell - The geometry of unknotting tunnels

An unknotting tunnel is an arc in a 3-manifold M with torus boundary, such that the complement of the tunnel in M is a handlebody.  Classically, one can "unknot" a knot or link by pulling its diagram along an unknotting tunnel.  In 1995, Adams, and Sakuma and Weeks, asked three questions concerning the geometry of unknotting tunnels in a hyperbolic 3-manifold:  Are they geodesic?  Do they have bounded length?  Are they canonical?  While the answer to the first question is still open, we will describe fairly complete answers to all three questions in the case where M is created by a "generic" Dehn filling.  As an application, there is an explicit family of knots in the 3-sphere whose tunnels are arbitrarily long.  This is joint with Daryl Cooper and David Futer.

Reid - Profinite rigidity and flexibility

This talk will discuss to what extent residually finite groups are determined by their profinite completions. In addition we discuss what properties of residually finite groups are "seen" by their profinite completions.

Series - Limiting on the Maskit slice

Let S be a surface together with a set of pants curves. A Maskit group is a 3-manifold group on the boundary of quasi-fuchsian space QF in which all the curves in a fixed pants decomposition are pinched, so that one side of the conformal boundary is a union of triply punctured spheres. The Maskit slice M is the space of all such groups up to conjugation.

In this talk we discuss convergence of certain slices of QF to M.

Recall that a pleating variety in a slice of the representation variety is the locus on which the projective  bending lamination of a component of the convex hull boundary is fixed. Both in M and in our chosen slices of QF, the pleating variety of a fixed rational projective lamination is a line, called a pleating ray. Conjecturally, the collection of all rays foliate the slice.

As the bending angle along a ray goes to zero in QF, its limiting position and direction of can be described in terms of Kerckhoff's lines of minima in Teich(S). In M on the other hand, as the bending angle along a ray goes to zero, the groups diverge in the representation variety. As shown by myself and Sara Maloni, the asymptotic direction of the ray in M can be neatly expressed in terms of the Dehn-Thurston coordinates of the bending lamination.

By carefully studying the convergence of the slices, we elucidate the relationship between these two apparently disjoint results.

Souto - Hyperbolic knots complements and metrics on manifolds with large spectral gap

We prove that every manifold M of dimension at least 3 admits a sequences of Riemannian metrics with bounded geometry, whose volume tends to infinity and whose Cheeger constants are uniformly bounded from below by a positive number. We use this to construct a sequence of hyperbolic knot complements with volume tending to infinity and whose Cheeger constants are again uniformly bounded from below. This is joint work with Marc Lackenby.

Wienhard - Anosov representations, domains of discontinuity and applications

I will give examples of Anosov representations and explain a construction of domains of discontinuity for these representations. I will then discuss several applications of this construction to obtain geometric structures and to obtain proper actions on homogeneous spaces. This is joint work with Olivier Guichard.