Geometry and Topology Seminar

Thursday 31 January 2008 at 13:40, room B1.01

Vladimir Kisil (Leeds)

Erlangen program at large: starting with the group SL(2,R)

Abstract: The Erlangen program collects dust at a respectful place in the Mathematical History Museum on the shelves of the glorious XIX century. Taking the group SL(2,R) as an example we demonstrate that it can be still in use. Erlangen approach provides useful insights much beyond the traditional geometry: e.g. in analytic function theory, functional calculus and non-commutative geometry.

Thursday 7 February 2008 at 13:40, room B1.01

Gabriele Mondello (Imperial)

Triangulated Riemann surfaces and the Weil-Petersson Poisson structure

Abstract: The Teichmüller space T(S) of a Riemann surface with boundary is a real-analytic manifold that parametrises hyperbolic metrics on S with geodesic boundary up to isotopy. It comes endowed with a Poisson structure, defined (by Weil) using Petersson's pairing of modular forms.
On the other hand, the geometry of hyperbolic hexagons ensures that the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at the boundary perpendicularly are coordinates on T(S).
One can compute the Weil-Petersson Poisson structure on T(S) in this system of coordinates and prove that it limits pointwise to the piecewise-linear Poisson structure already defined by Kontsevich on the arc complex of S.
Something similar happens with the cellularisation of T(S) using ribbon graphs: in an appropriate sense, it interpolates between Penner-Bowditch-Epstein's and Harer-Mumford-Thurston's triangulations of the moduli space of Riemann surfaces with punctures.

Thursday 14 February 2008 at 17:10, room B1.01

Alastair Fletcher (Nottingham)

Asymptotic Teichmüller space

Abstract: For a hyperbolic Riemann surface M, the asymptotic Teichmüller space AT(M) is a quotient of Teichmüller space. The Bers embedding induces a biholomorphic mapping from AT(M) onto a bounded open set in a quotient space of holomorphic quadratic differentials $Q(M)/Q_{0}(M)$. Using this asymptotic Bers embedding and the result that if M is a Riemann surface of infinite analytic type then $Q(M)/Q_{0}(M)$ is isomorphic to $l^{\infty}/c_{0}$, it can be shown that if M and N are two such Riemann surfaces, then AT(M) and AT(N) are locally bi-Lipschitz equivalent.

Thursday 21 February 2008 at 15:00, room A1.01

Anna Pratoussevitch (Liverpool)

Higher spin structures on Riemann surfaces

Abstract: A higher spin structure on a Riemann surface is a line bundle with m-th tensor power isomorphic to the cotangent bundle. In the talk I will show how the space of m-spin structures on a Riemann surface can be interpreted as a finite affine space of Z/mZ-valued functions on the fundamental group of the surface.
Using this result one can describe the topology of the moduli space: any connected component of the space of m-spin structures on compact Riemann surfaces with finitely many punctures and holes is homeomorphic to a quotient of the vector space R^d by a discrete group action.

Thursday 28 February 2008 at 16:00, room B1.01

Arthur Bartels (Imperial)

Topological rigidity and word-hyperbolic groups

Abstract: The Borel conjecture asserts that aspherical manifolds are topologically rigid, i.e., every homotopy equivalence between such manifolds is homotopic to a homeomorphism. This conjecture is strongly related to the Farrell-Jones conjectures in algebraic K- and L-theory. We will give an introduction to these conjectures and discuss the proof of the Borel conjecture for high-dimensional aspherical manifolds with word-hyperbolic fundamental groups.

Thursday 6 March 2008 at 16:00, room B1.01

Jessica Purcell (Brigham Young / Oxford)

Volume change under Dehn filling

Abstract: Any 3-manifold can be obtained by gluing solid tori into components of a link complement in the 3-sphere. This process is called Dehn filling. When the original manifold and the Dehn filled manifold both admit hyperbolic structures, Thurston showed that the volume of the Dehn filled manifold is strictly less than that of the original. Recently, we found an explicit lower bound on the amount the volume decreases based on the length of the slope of the Dehn filling. We will discuss this bound as well as several applications to knots and links in the 3-sphere. This is joint work with David Futer and Efstratia Kalfagianni.

Thursday 13 March 2008 at 16:00, room B1.16

Cornelia Drutu (Oxford)

Property (T) and a-T-menability from the median spaces viewpoint, applications to the mapping class groups

Abstract: Both Kazhdan's property (T) and a-T-menability turn out to be related to actions of groups on median spaces and on spaces with measured walls. These relationships allow to study the connection between Kazhdan property (T) and the fixed point property for affine actions on L^p spaces, on one hand. On the other hand, they allow to discuss conjugacy classes of subgroups with property (T) in Mapping Class Groups. The latter result is due to the existence of a natural structure of measured walls on the asymptotic cone of a Mapping Class Group. The talk is on joint work with I. Chatterji and F. Haglund (first part), and J. Behrstock and M. Sapir (second part).