|9/11 (Tue)||9/12 (Wed)||9/13 (Thu)||9/14 (Fri)|
Title: The virtual Haken conjecture
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question.
Part of the result relies on joint work with Groves and Manning.
Title: Coarse median spaces
We describe a notion of a ``coarse median space''. Roughly speaking, this is a geodesic metric space equipped with a ternary relation, satisfying the properties of a median algebra up to bounded distance. It can be applied to finitely generated groups via their Cayley graphs. Many naturally occuring groups have such a structure. For example, hyperbolic groups are precisely the coarse median groups ``of rank one''. More interestingly, it follows from work of Behrstock and Minsky, that the mapping class groups also have a coarse median. Moreover, the property is preserved under taking products, and under relative hyperbolicity etc. One can derive various consequences from the existence of such a structure. For example, we can recover the rank theorem for the mapping class group, due to Behrstock and Minsky and to Hamenstadt. We will describe various other related results.
Title: Fixed point theorems in infinite dimensional Teichmuller spaces
We consider the action of the Teichmuller modular group on the Teichmuller space. In particular, we investigate several phenomena of the orbits which appear only when the Teichmuller space is infinite dimensional. In this talk, we focus our attention on the fixed point theorem and introduce related results. We also consider the asymptotic version of the fixed point theorem, which asserts that every finite subgroup of the asymptotic Teichmuller modular group has a common fixed point in the asymptotic Teichmuller space.
Title: On the ring of Fricke characters of free groups
We study a descending filtration of the ring of Fricke characters of a free group consisting of ideals on which the automorphism group of a free group naturally acts. Then by using it, we define a descending filtration of the automorphism group of a free group, and investigate a relation between it and the Andreadakis-Johnson filtration.
This is a joint work with Takao Satoh.
Title: Ideal triangulations, angle structures and invariants for cusped hyperbolic 3-manifolds
It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a ``geometric'' triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a necessary condition for the triangulation to be geometric. In particular, every hyperbolic knot or link complement in the 3-sphere has such a triangulation.
We will also discuss the question of when ideal triangulations with a strict angle structure are actually geometric, and answer a question on ``veering triangulations'' posed by Agol.
Finally, we will describe a new topological invariant for cusped hyperbolic 3-manifolds defined using ideal triangulations.
(This includes joint work with Hyam Rubinstein, Henry Segerman, Stephan Tillmann, Ahmad Issa and Stavros Garoufalidis.)
A calculation of a complex volume from a quandle shadow coloring
(joint work with Yuichi Kabaya)
A quandle is an algebraic system whose definition is motivated in knot theory. A shadow coloring is an assignment of elements of a quandle to arcs and regions of a link diagram on a rule. In this talk, we see that we can compute the complex volume (i.e., the volume and the Chern-Simons invariant) of a hyperbolic link from a certain shadow coloring of its diagram with the aid of a work of W. Neumann.
Title: Conformally flat Lorentzian parabolic manifold
A Lorentzian parabolic structure contains Lorentzian flat structure and Fefferman-Lorentz structure. We give a classification of compact Lorentzian flat 4-manifolds in which there is a compact Lorentzian flat nilmanifold which cannot admit any conformally flat Fefferman-Lorentz parabolic structure or vice versa.
Title: Noncompact Euclidean Cone Manifolds
Title: Rigidty, flexibility and bounded cohomology
We review some of recent developments of rigidity and flexibility in the character variety of locally symmetric spaces through various techniques like bounded cohomology, representation theory and hyperbolic geometry.
Title: Epimorphisms between knot groups, special values of colored Jones polynomials and Mahler measure for Jones polynomial.
When an epimorphism exits between knot groups, it gives a partial order on the set of knots. Recently there are lots of studies related this partial order, or epimorphisms. Along these directions some inequality on special values of colored Jones polynomials appears from the view point of the volume conjecture. In this talk I will explain the motivation and the back ground and mention some example of torus knot and 2-bridge knot by using a computer. If we have the time, I would like to do some another conjecture on Mahler measures of Jones polynomials.
This is a partially joint work with Masayuki Niimura and Kiyohito Kuwahara.
Title: Exceptional surgeries on alternating knots
We will report on our recent project toward the complete classification of the exceptional surgeries on hyperbolic alternating knots.
This is a joint work with Kazuhiro Ichihara.
Title : Orthospectra and identities
The orthospectra of a hyperbolic manifold with geodesic boundary consists of the lengths of geodesic arcs perpendicular to the boundary.
We discuss the properties of the orthospectra, asymptotics, multiplicities in relation to the other spectra (eigenvalues of the Laplacian, lengths of closed geodesics) and identities due to Basmajian, Bridgeman and Calegari. We will give a proof that the identities of Bridgeman and Calegari are in fact the same.
Title: Networking Seifert surgeries on knots
We will make a survey of the Seifert Surgery Network which is introduced to describe relationships among Seifert surgeries on knots in the $3$--sphere and get a global picture of them. One problem of the network is whether there is a path from each Seifert surgery to a Seifert surgery on a torus knot, the most basic Seifert surgery. We also discuss a geometric aspect of the network.
This is ongoing work with Arnaud Deruelle and Katura Miyazaki; partly joint also with Mario Eudave-Mu¥~noz.
Title: Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions
For a Heegaard splitting or a bridge decomposition along a surface S, we consider a subgroup of the mapping class group of S generated by Dehn twists around "meridians" on S. We shall show that this group has a natural free product decomposition, and that there is an open subset in the projective lamination space of S on which this group acts properly discontinuously.
This is joint work with Makoto Sakuma and partially with Brian Bowditch.
Title: Computations of Euler characteristics of graph homologies in low weights
This is a joint work with Shigeyuki Morita and Masaaki Suzuki.
We report our explicit computations of the Euler characteristics of graph homologies for commutative, Lie and associative cases up to certain low weights. Then we discuss their applications to characteristic classes of moduli spaces by using a theorem of Kontsevich.
Title: Simple loops on 2-bridge spheres in even Heckoid orbifolds for 2-bridge links
For a 2-bridge link $K$ and a positive integer $n$, the even Heckoid orbifold of index $n$ for $K$ is the $3$-orbifold with underlying space the exterior of $K$ and with cone axis the lower tunnel of index $n$. Let $S$ be a $4$-punctured sphere in the even Heckoid orbifold obtained from the $2$-bridge sphere of $K$. We give a complete answer to the following problems. (1) For an essential simple loop on $S$, when is it null-homotopic in the orbifold, peripheral or torsion? (2) For two essential simple loops on $S$, when are they homotopic in the orbifold? We will also discuss applications of these results to a variation of McShane's identity and to the end invariants of $SL(2,C)$-representations of punctured torus groups, introduced by Bowditch and Tan-Wong-Zhang.
This is a joint work with Donghi Lee.
Title: Some recent developments in the length spectrum Teichmüller theory
The length spectrum Teichmüller space is defined by looking at the behavior of the hyperbolic lengths of simple closed geodesics on Riemann surfaces. The length spectrum Teichmüller space is equipped with the "length spectrum distance" while the traditional Teichmüller space has the Techmüller distance. In the past decade, recognized are differences between those Teichmüller spaces as metric spaces. In this talk, we will present some recent progress in the length spectrum Techmüller theory.
Title: Diffeomorphism groups of aspherical 3-manifolds
We discuss the homotopy type of aspherical geometric 3-manifolds and topics related to the Smale conjecture for Seifert fibered spaces with infinite fundamental group.
Title: Small dilatation pseudo-Anosovs living in the magic 3-manifold.
Finding the minimal dilatations of pseudo-Anosovs is one of the fundamental problems on the study for surface automorphisms. Farb, Leininger and Margalit show that small dilatation pseudo-Anosovs live in a finite set of fibered hyperbolic 3-manifolds. The magic 3-manifold $N$ gives us many examples of small dilatation pseudo-Anosovs (including known minimal examples) and interesting examples live in particular surgeries of $N$. We give the upper bounds of the minimal dilatations for many surfaces and investigate asymptotic behavior varying the genera and punctures.
This is a joint work with Eiko Kin.
Title: A dilogarithm identity on moduli spaces of curves
We will talk about a new identity for closed hyperbolic surfaces which involves the dilogarithm of the lengths of simple closed geodesics on the surface, and also relate it to some previously known identities by Basmajian, McShane and Bridgeman.
This is joint work with Feng Luo.
Title: The diagonal slice of SL(2,C)-character variety of free group of rank two
Let F_2 = be the free group of rank two. It is well known that SL(2,C)-character variety of F_2 can be identified with C^3 by the identification [¥rho] -> (tr¥rho(A), tr¥rho(B), tr¥rho(AB)), where ¥rho is a representation of F_2 into SL(2,C). In this talk, we study "the diagonal slice", that is, the subset of the character variety where tr¥rho(A) = tr¥rho(B) = tr¥rho(AB). We (computationally) compare the "Bowditch set" and the discreteness locus. We also compute "rays" in the diagonal slice, and discuss some properties.
(This is a joint work with Caroline Series and Ser Peow Tan.)