Geometry and Topology Seminar

Thursday October 13, 15:00, room MS.04

Henry Wilton (UCL)

One-ended subgroups of graphs of free groups

Abstract: A longstanding question in geometric group theory is the following. Suppose $G$ is a hyperbolic group where all subgroups of infinite index are free. Is $G$ the fundamental group of a surface? This question is still open for some otherwise well-understood classes of groups. In this talk, I will explain why the answer is affirmative for graphs of free groups with cyclic edge groups. I will also discuss the extent to which these techniques help with the harder problem of finding surface subgroups.

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

Armindo Costa (Warwick)

Topology of random complexes

Abstract: The probabilistic method consists of proving existence of mathematical objects through models that produce these objects with positive probability. Its successful use in combinatorics (pioneered by P.Erdos) led to the theory of random graphs.

A model for random simplicial complexes was suggested recently by Linial-Meshulam and Meshulam-Wallach. Their model is a higher-dimensional analogue of the Erdos-Renyi model for random graphs. Natural problems within this model include understanding the expected topology of random complexes and understanding the expected properties of fundamental groups of random complexes.

In this talk I will survey the main results in this recent subject and, if time allows, the connections to known models of random groups.

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

Will Cavendish (Princeton)

Finite-sheeted covers of 3-manifolds and the cohomology of solenoids

Abstract: The study of finite-sheeted covering spaces of 3-manifolds has been invigorated in recent years by the resolution of several long-standing conjectures by Kahn-Markovic, Agol and Wise. In this talk, I will discuss how using this work one can reformulate some of the central open questions in the field in terms of objects called solenoids. These objects are formed by taking inverse limits of families of finite-sheeted covering spaces of a compact manifold $M$, and they can be thought of as pro-finite analogues of covering spaces of $M$. While such an object can in general be quite complicated, I will show in this talk that if $M$ is a compact aspherical 3-manifold, then the solenoid given by taking the inverse limit of the family of all finite-sheeted connected covering spaces of $M$ has the Cech cohomology of a disk. I will then talk about the relevance of this result to concrete questions about finite-sheeted covers.

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

Lars Louder (Michigan)

Simple loop conjecture for limit groups

Abstract: There are non-injective maps from surface groups to limit groups that do not kill any simple closed curve. As a corollary, there are non-injective all-loxodromic representations of surface groups in $\SL(2,\CC)$ that do not kill any simple closed curve. This answers a question of Minsky. There are also examples, for any $k$, of non-injective all-loxodromic representations of surface groups killing no curves with self intersection number at most $k$.

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

Kai Ishihara (Imperial)

An algorithm for finding parameters of tunnels

Abstract: Cho and McCullough gave a numerical parameterization of the collection of all tunnels of all tunnel number one knots and links in the 3-sphere. In this talk, I will explain their parameterization. I will also give an algorithm for finding the parameter of given tunnel by using its Heegaard diagram.

Thursday November 17, 15:00, room MS.04

Bruce Westbury (Warwick)

How to draw a planar graph

Abstract: This is an expository talk on the circle packing theorem of Koebe-Andreev-Thurston. I will state the theorem, outline a proof using non-linear convex optimisation, and discuss some applications.

Thursday November 24, 15:00, room MS.04

Jelena Grbic (Manchester)

Decompositions of looped co-$H$-spaces and applications to topology and algebra

Abstract: The motivating problem of this talk is the study of the homotopy rigidity of the functor $\Sigma\Omega$. To do so new decompositions of looped co-$H$-spaces are needed. I shall start by recalling some classical unstable homotopy theoretical results in this field and thereafter state new achievements. New functorial decompositions arise from an algebraic analysis of functorial coalgebra decompositions of tensor algebras that depends on the modular representation theory of the symmetric group.

Time permitting, as a valuable example, I hope to give an application of these decompositions to the theory of quasi-symmetric functions.

The whole programme has been carried over the last 10 years (in various combinations) by Paul Selick, Stephen Theriault, Jie Wu and myself.

Thursday December 1, 15:00, room MS.04

Matthew Rathbun (Imperial)

Tunnel-number one, fibered links

Abstract: All fibered links can be constructed from the unknot by a sequence of operations called plumbing (and perhaps de-plumbing) along Hopf bands. Interestingly, if a fibered link has an unknotting tunnel that happens to lie in the fiber, then plumbing a Hopf band along the tunnel results in a new fibered link that is again tunnel number one. Natural questions are whether this restricted plumbing can always be performed, and whether this is sufficient to construct all tunnel one, fibered links. I will answer the first question affirmatively, and discuss progress towards answering the second.

Thursday December 8, 15:00, room MS.04

Sebastian Hensel (Bonn)

Sphere systems and the geometry of $\Out(\FF_n)$

Abstract: By a theorem of Laudenbach, the outer automorphism group $\Out(\FF_n)$ of a free group is a cofinite quotient of the mapping class group of a suitable 3-manifold $M_n$. This point of view allows us to study the geometry of $\Out(\FF_n)$ with methods inspired from surface mapping class groups. In this talk, we present an application of this strategy: we show that the natural inclusion of the mapping class group of a surface of genus $g$ with one puncture into $\Out(\FF_{2g})$ is undistorted. The results are joint work with Ursula Hamenstädt.