Geometry and Topology Seminar

Thursday January 14, 15:00, room B1.01

Stephen Tawn (Warwick)

The Hilden subgroup of the braid group

Abstract: The Hilden, or Wicket, subgroup of the braid group on 2n string can be defined in several ways. As the group of motions of n hoops in halfspace, as the stabiliser of n hoops under the action of the braid group on tangles, or as a mapping class group. Using the mapping class point of view, we will construct a simply-connected complex with an action by the Hilden group. We will then use this action to show that the Hilden group has a finite presentation.

Thursday January 21, 15:00, room B1.01

Anne Thomas (Oxford)

Lattices with surface subgroups

Abstract: Let I_{p,q} be Bourdon's building, the unique simply-connected 2-complex such that each 2-cell is a regular right-angled hyperbolic p-gon, and the link at each vertex is the complete bipartite graph K_{q,q}. We determine all triples (p,q,g) such that the automorphism group of I_{p,q} admits a lattice Γ so that the quotient of I_{p,q} by Γ is a compact surface of genus g. In these cases, the surface group naturally embeds in Γ. This is joint work with David Futer.

Thursday January 28, 15:00, room B1.01

Alexander Coward (Oxford)

Upper bounds on Reidemeister moves

Abstract: (Joint work with Marc Lackenby.) Given any two diagrams of the same knot or link, we provide an explicit upper bound on the number of Reidemeister moves required to pass between them in terms of the number of crossings in each diagram. This provides a new and conceptually simple solution to the equivalence problem for knot and links.

Thursday February 4, 15:00, room B1.01

Andrei Tetenov (Gorno-Altaisk)

The structure and rigidity of self-similar Jordan arcs

Abstract: Suppose the Jordan arc γ is the invariant set for a digraph system S of contraction similarities. Then either the arc γ is an invariant set for some multizipper and admits non-trivial deformations or γ is a straight line segment, the system S does not satisfy the weak separation property, and the self-similar structure (γ,S) is rigid.

Thursday February 11, 15:00, room B1.01

Dmitri Panov (Imperial)

Hyperbolic geometry and symplectic manifolds with c₁=0

Abstract: We will show how to use 4-dimensional hyperbolic geometry to construct symplectic manifolds of dimension 6 with c₁=0. In particular using Davis manifold we construct a first known example of simply connected symplectic 6-fold with c₁=0 that does not admit a compatible Kahler structure. If the time permits we will describe further examples with arbitrary large Betti numbers together with higher-dimensional analogues of this construction, producing non-Kahler symplectic Fano manifolds. This is a joint work with Joel Fine.

Thursday February 18, 15:00, room B1.01

Hee Jung Kim (Max-Planck)

Double point surgery and configurations of surfaces in 4-manifolds

Abstract: We introduce a new operation, double point surgery on a configuration of surfaces in a 4-manifold, and use it to construct configurations that are smoothly knotted, without changing the topological type or the smooth embedding type of the individual components of the configuration. Taking branched covers, we produce smoothly exotic actions of Z_m ⊕ Z_n on simply connected 4-manifolds with complicated fixed-point sets.

Thursday February 25, 15:00, room B1.01

Andrzej Zuk (Paris)

L² Betti numbers of closed manifolds

Abstract: TBA

Thursday March 4, 15:00, room B1.01

Caroline Series (Warwick)

Top terms of trace polynomials in Kra's plumbing construction

Abstract: (Joint work with Sara Maloni.) Kra's plumbing construction manufactures a surface S by `plumbing' together a suitable family of triply punctured spheres. This gives a natural pants decomposition of S, together with a projective structure for which the associated holonomy representation ρ depends on the `plumbing parameters' τ_i. In particular Trace ρ(γ), for γ ∈ π₁(S), is a polynomial in the τ_i.

Simple curves on S can be described in terms of their Dehn-Thurston coordinates relative to the pants decomposition. After explaining the construction, we show that if γ is simple there is a very simple formula relating the coefficients of the top terms of ρ(γ) and its Dehn-Thurston coordinates.

Maloni spoke in last year's seminar: in this talk we present a combinatorial proof which applies to an arbitrary pants decomposition and which involves a rather interesting result on matrix products.

Thursday March 11, 15:00, room B1.01

Daniele Alessandrini (Strasbourg)

On the compactification of Teichmuller-like parameter spaces

Abstract: The construction of compactifications of Teichmuller spaces in the approach of Morgan and Shalen has close relationships with tropical geometry. By studying the general properties of the logarithmic limit set of real semi-algebraic sets, it is possible to generalize their construction and to understand some of its properties. When applied to Teichmuller spaces, this gives the compactification of Thurston, and the natural piecewise linear structure of the boundary appears automatically, showing clearly how this structure is related with the semi-algebraic structure of the interior part. It is also possible to construct a compactification of the parameter spaces of strictly convex projective structures on a closed n-manifold. In this case objects from tropical geometry also appear in the interpretation of the boundary points.

Thursday March 11, 16:15, room B3.03

Jim Howie (Heriot-Watt)

Dehn surgery and prime factors

Abstract: A 3-manifold obtained by Dehn surgery on a knot is generically - but not always - irreducible. The Cabling Conjecture of Gonzalez-Acuna and Short asserts that a reducible manifold can be obtained only in a specified way, in which case the number of prime factors is precisely 2. I shall discuss the weaker conjecture that Dehn surgery can never produce a manifold with more than 2 prime factors.

Thursday March 18, 15:00, room D1.07

Laurent Bartholdi (Göttingen)

Amenability of groups and algebras

Abstract: Amenability of groups (a notion introduced by von Neumann in his study of the Banach-Tarski paradox) is a far-reaching generalization of "finiteness". It leads to a fundamental dichotomy of the "landscape" of groups, and I will describe some features of amenable / non-amenable groups, providing typical or fundamental examples along the way.

I will explain interesting related questions in the world of algebras, and give an application to the theory of cellular automata.