Geometry and Topology Seminar

Thursday May 13, 14:00, room B3.03

Andrei Tetenov (Gorno-Altaisk)

The bending locus for Kleinian groups in space

Abstract: Let Λ be the limit set of a discontinuous group G of Möbius authomorphisms of 4-ball B⁴, and let C(Λ) be the hyperbolic convex hull of Λ. We consider elementary geometrical and topological properties of the bending locus BL of ∂C(Λ). We prove that if G is a group generated by a finite number of reflections then the quotient space BL/G ⊂ ∂C(Λ)/G is a finite union of totally geodesic surfaces S_i, with boundary ∂ S_i consisting of a finite number of closed geodesics. We also claim that the 1-skeleton BL⁽¹⁾ of the bending locus for a convex cocompact group G is nowhere dense in ∂C(Λ).

Thursday May 13, 15:30, room B3.03

Elmas Irmak (Bowling Green)

Mapping class groups and complexes of curves on orientable/nonorientable surfaces

Abstract: I will talk about the relation between the mapping class groups of surfaces and the automorphism groups and the superinjective simplicial maps of the complexes of curves on surfaces for both orientable and nonorientable surfaces. I will also talk about the proof that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface if g + n is at least 5 or at most 3, where g is the genus and n is the number of boundary components of the surface.

Thursday May 20, 15:00, room MS.04

Stephane Lamy (Warwick)

Normal subgroups in the Cremona group

Abstract: The Cremona group is the group of birational transformations of the plane. Following a construction by Manin it is possible to see this group as acting by isometries on an infinite dimensional hyperbolic space. Using these ideas we are able to produce many example of proper normal subgroups in the Cremona group. (Joint work with Serge Cantat)

Thursday May 20, 16:30, room MS.04

Oyku Yurttas (Liverpool)

Dynnikov coordinates and pseudo-Anosov braids

Abstract: Isotopy classes of orientation preserving homeomorphims on a finitely punctured disk are represented by braids. In this talk I will present a method for computing the dilatation of pseudo-Anosov braids using the Dynnikov coordinate system which is computationally much more efficient than the usual Thurston train track aproach. If time permits, I will talk about the relation between Dynnikov matrices and the train track transition matrix with an illustrative example that also shows the local dynamics around the unstable foliation in the boundary of Teichmuller space.

Thursday May 27, 15:00, room MS.04

Ivan Smith (Cambridge)

Floer cohomology and pencils of quadrics

Abstract: There is a classical relationship, in algebraic geometry, between a hyperelliptic curve and an associated pencil of quadric hypersurfaces. We investigate symplectic aspects of this relationship and their consequences in low-dimensional topology.

Thursday June 3, 15:00, room MS.04

Maciej Borodzik (Warsaw)

Morse theory for singular complex curves in C^2 and signatures of torus knots

Abstract: For a given complex curve C in ℂ² and a generic point z, we study the links S(z,r) ∩ C. Here S(z,r) denotes the standard 3-sphere in ℂ² with centre at z and radius r. As r varies, the links can change. We describe what happens when r crosses a singular point of C. Studying the changes of Tristram-Levine signatures of S(z,r) ∩ C, we obtain deep information about the topology and singularities of C.

Thursday June 10, 15:00, room MS.04

Nicolas Bergeron (Curie)

Torsion in the homology of 3-manifolds

Abstract: Compact 3-manifolds can have "a lot" of torsion in their homology. I will try to quantify what "a lot" means and sketch the proofs of two different kind of results: growth of torsion in abelian covers, and growth of torsion for some twisted local systems in residually finite covers. This is joint work with Akshay Venkatesh.

Thursday July 1, 15:30, room MS.04

David Futer (Temple)

The geometry of unknotting tunnels

Abstract: Given a knot K in S³, an unknotting tunnel for K is an arc τ from K to K, such that the complement of K and τ is a genus two handlebody. Fifteen years ago, Colin Adams asked a series of questions about how unknotting tunnels fit into the hyperbolic structure on the knot complement. For example: is τ isotopic to a geodesic? Can it be arbitrarily long, relative to a maximal cusp neighborhood? Does τ appear as an edge in the Epstein-Penner polyhedral decomposition?

Although the most general versions of these questions are still open today, I will describe fairly complete answers in the special case where K is created by a ``generic'' Dehn filling. As an application, there is an explicit family of knots in S³ whose tunnels are arbitrarily long. This is joint work with Daryl Cooper and Jessica Purcell.