INdAM Meeting
Geometric topology in Cortona
Interactions of quantum topology and hyperbolic geometry

A conference in honor of Riccardo Benedetti for his 60th birthday

3-7 June 2013





Practical information

Monday 3 June Tuesday 4 June
9.30 Boileau Murakami
10.30 - Coffee break - - Coffee break -
11.00 Costantino Marche
12.30 - Lunch - - Lunch -
15.00 Hoffman Danciger
16.00 - Coffee break - - Coffee break -
16.30 Baseilhac Souto
Wednesday 5 June
9.00 Petronio
10.00 - Coffee break -
10.30 Van Der Veen
11.45 Brock
13.00 - Lunch -
14.00 Free afternoon
20.00 - Social dinner -
Thursday 6 June Friday 7 June
9.30 Bonahon Lackenby
10.30 - Coffee break - - Coffee break -
11.00 Kalfagianni Roger
12.30 - Lunch - - Lunch -
15.00 Leininger Alessandrini
16.00 - Coffee break - - Coffee break -
16.30 Agol Schlenker


Ian Agol The virtual Haken conjecture (presentation)

We'll discuss the proof of a conjecture of Wise that cubulated hyperbolic groups are virtually special. This has as a corollary that closed hyperbolic 3-manifolds have large fundamental group and are virtually fibered, in conjunction with results of Kahn-Markovic and Bergeron-Wise.

Daniele Alessandrini Length spectra for surfaces of infinite type

I will describe a parametrization of the length spectrum Teichmuller space for surfaces of infinite topological type. In some cases this space is homeomorphic to the ordinary Teichmuller space, but under some condition it is much bigger, and it contains the ordinary Teichmuller space as a no-where dense subset.

Stephane Baseilhac Analytic families of quantum hyperbolic invariants (presentation)

This is a joint work with R. Benedetti. We will describe sequences of rational functions defined on refinements of the PSL(2,C)-character varieties of 3-manifolds. These functions, which organize the quantum hyperbolic invariants, have clean relations with the Chern-Simons functional in the case of cusped hyperbolic 3-manifolds. Hopefully they will shed lights on the asymptotical behaviour of the quantum hyperbolic invariants.

Michel Boileau Z-homology spheres, taut foliations and left order on groups

We will discuss the existence of a taut foliation on an aspherical $\mathbb Z$-homology 3-sphere, in relation with the notion of L-spaces, introduced by Ozsvath and Szabo, and the notion of left order on its fundamental group. We will consider the case of a graph manifold, and some other examples of Z-homology 3-spheres obtained by splicing knot exteriors. This is a joint work with Steve Boyer.

Francis Bonahon Representations of the Kauffman skein algebra of a surface (presentation)

A Kauffman bracket on a surface is a certain invariant of knots and links in a thickening of the surface. Kauffman brackets can also be interpreted as non-commutative geometry versions of group homomorphisms from the fundamental group of the surface to SL(2, C). I will discuss a new construction of Kauffman brackets on a surface. Time permitting, I will offer some speculations connecting this construction to the topological quantum field theory of Kashaev-Baseilhac-Benedetti. This is joint work with Helen Wong.

Jeff Brock Fat, exhausted, hyperbolic knots

We describe the latest in a general framework for producing models for hyperbolic 3-manifolds with bounded combinatorial descriptions. As an application, we describe a particular family of knot complements in the 3-sphere that Benjamini-Schramm converge to hyperbolic 3-space. Since (1,n) fillings of these knots are homology 3-spheres, this gives an appealing specific construction of homology 3-spheres Benjamini-Schramm converging to hyperbolic 3-space. This talk will describe work with Minsky, Namazi and Souto, as well as work with Dunfield.

Francois Costantino On an analytic family of representations of mapping class groups

We will first recall the notion of family of representations of a discrete group depending analytically on a parameter and a general approach to the construction of such families due to A. Vallette. We will then apply this construction to the case of mapping class groups of punctured surfaces by using some objects (the 6j-symbols of Uq(sl2)) coming from quantum topology. The family of representations that we obtain has many interesting properties and in particular the path of representations corresponding to the real values of the parameter interpolate between two remarkable unitary representations of mapping class group well known to geometers. This is join work with Bruno Martelli.

Jeffrey Danciger Complete flat Lorentzian three-manifolds

A complete flat Lorentzian three-manifold is the quotient of the (2+1)-dimensional Minkowski space by a discrete group acting properly by affine O(2,1) transformations. In the interesting cases, the group acting is a free group and the quotient manifold is called a Margulis space-time. I will describe work in progress toward classifying the topology of Margulis space-times. In particular, when the O(2,1) part of the group action does not contain parabolics, we prove that the quotient manifold is a handle-body. The proof depends on a new properness criterion for free groups acting on Minkowski space and draws on ideas from anti de Sitter (AdS) geometry. This is joint work with Francois Gueritaud and Fanny Kassel.

Neil Hoffman Rigorous interval computations of hyperbolic tetrahedron shapes (presentation)

The study of hyperbolic 3-manifolds has been greatly influenced by Jeff Weeks' software package Snappea and its progeny (SnapPy, Snap, Orb, etc.) I will describe a new method to rigorously verify an approximate solution to the gluing equations and detail some its advantages over existing techniques. This is joint work with Masahide Kashiwagi, Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu.

Efstratia Kalfagianni Jones polynomials and incompressible surfaces (presentation)

I will report on joint work with Dave Futer (Temple) and Jessica Purcell (BYU): Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement; in particular, the surface is a fiber if and only if a particular coefficient vanishes. Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our approach is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We employ normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). Since state graphs have previously appeared in the study of Jones polynomials, our setting and methods create a bridge between the polynomials and the geometry of the surface complement.

Marc Lackenby Polynomial upper bounds on Reidemeister moves (presentation)

For each knot type K, we establish the existence of a polynomial pK with the following property. Any two diagrams of K with n and n' crossings respectively differ by a sequence of at most pK(n) + pK(n') Reidemeister moves. As a consequence the problem of deciding whether a knot is of type K is in the complexity class NP. This result generalises earlier work which dealt with the case when K is the unknot, for which we may take pK(n) to be (231n)11.

Chris Leininger Dynamics on free-by-cyclic groups (presentation)

The combined work of Thurston and Fried shows that the monodromies associated to different fibrations of a hyperbolic 3-manifold are intimately related to each other in important topological, geometric, and dynamical ways. After reviewing this set-up, I will describe recent results provi- ding a similar picture for the monodromies associated to splittings of a free-by-cyclic group. In particular, when the free-by-cyclic group is the mapping-torus group of a fully irreducible atoroidal automorphism of Fn, we will see that the monodromies of all nearby splittings are also fully irreducible. Furthermore, the ranks of the kernels and the stretch factors of the monodromies are restrictions of convex, homogeneous functions on an appropriate cone in the first cohomology.

Julien Marche Global-local rigidity for representations of Dehn fillings of knot complements

I will describe some questions on character varieties of 3-manifolds related to the asymptotic behavior of Witten-Reshetikhin-Turaev invariants. One of them is the following global-local rigidity: for a given manifold, show that all irreducible representations in SL2 are locally rigid.

Hitoshi Murakami Survey of the volume conjecture (presentation)

I will give a survey of the volume conjecture, which states that the asymptotic behavior of the colored Jones polynomial would give the volume of the knot complement. I will also talk about its generalizations.

Carlo Petronio Extra structures on 3-manifolds via extra structures on spines (presentation)

In the late 1990's Riccardo had the idea of endowing the spine dual to an ideal triangulation of a 3-manifold with the structure of branched surface, that had already proved useful in the context of dynamical systems. Starting from this idea, and employing constructions of Ishii and Christy, Riccardo and I were then able to provide combinatorial encodings of several types of extra structures on 3-manifolds (such as combings, framings, and spin structures) by introducing extra combinatorial structures on (suitable) spines. After reviewing this theory I will illustrate two recent developments, the latter of which is joint work with Riccardo:

  1. An encoding of vector fields with generic tangency to the boundary, by means of a certain weakening of the notion of spine;
  2. An encoding of spin structures via arbitrary triangulations, by means of a certain weakening of the notion of branching.

Julien Roger Some aspects of quantum Teichm├╝ller theory at a root of unity

In this talk, we will discuss some aspects of the representation theory of the quantum Teichmuller space at a root of unity, emphasizing the connections with geometry. Following the approach of Bai-Bonahon-Liu, we will focus on the construction of a projective vector bundle over the moduli space whose fibers are endowed with an action of the quantum Teichmuller space. Specifically, our goal is to describe a possible extension of this bundle to the Deligne-Mumford compactification of moduli space.

Jean-Marc Schlenker Cone singularities in anti-de Sitter geometry (presentation)

Anti-de Sitter (AdS) 3-manifolds can be considered as Lorentzian analogs of hyperbolic 3-manifolds. They have attracted some attention recently. We will sketch the analogy (discovered by Mess) between quasifuchsian hyperbolic manifolds and globally hyperbolic AdS 3-manifolds, and explain recent results and questions on AdS manifolds with cone singularities. The Lorentzian nature of the metric means that new phenomena occur for AdS manifolds, compared with hyperbolic 3-manifolds.

Juan Souto Distributional limits of Riemannian manifolds

We investigate distributional limits (or equivalently Benjamini-Schramm limits) of sequences of Riemannian manifolds with bounded curvature which satisfy certain condition of quasi-conformal nature. We then apply our results to somewhat improve Benjamini's and Schramm's original result on the recurrence of the simple random walk on limits of planar graphs. For instance, as an application give a proof of the fact that for graphs in an expander family, the genus of each graph is bounded from below by a linear function of the number of vertices.

Roland Van Der Veen Quantum invariants of planar graphs

Upon replacing knots by planar graphs many new techniques for analysing quantum invariants become available. We will give a survey focusing on the colored HOMFLY polynomial and its relations to hyperbolic structures.