Past Topology Seminar

I will describe a new family of groups exhibiting wild geometric and computational features in the context of their Conjugacy Problems. These features stem from manifestations of "Hercules versus the hydra battles."

This is joint work with Martin Bridson.

The arc complex is a combinatorial moduli space, very similar to the curve complex. Using the techniques of Masur and Minsky, as well as new ideas, I'll sketch the theorem of the title. (Joint work with Howard

Masur.) If time permits, I'll discuss an application to the cusp shapes of fibred hyperbolic three-manifolds. (Joint work with David Futer.)

We are planning to have dinner at Chiang Mai afterwards.

If anyone would like to join us, please can you let me know today, as I plan to make a booking this evening. (Chiang Mai can be very busy even on a Monday.)

The complement of a knot or link is a 3-manifold which admits a geometric

structure. However, given a diagram of a knot or link, it seems to be a

difficult problem to determine geometric information about the link

complement. The volume is one piece of geometric information. For large

classes of knots and links with complement admitting a hyperbolic

structure, we show the volume of the link complement is bounded by the

number of twist regions of a diagram. We prove this result for a large

collection of knots and links using a theorem that estimates the change in

volume under Dehn filling. This is joint work with Effie Kalfagianni and

David Futer

Both Kazhdan and Haagerup properties turn out to be related to actions

of

groups on median spaces and on spaces with measured walls.

These relationships allows 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).

Recent joint work with Greg McShane has answered the following question: Which curves can be short in a given cell of the decomposition of Teichmueller space? The answer involves a new combinatorial structure called "screens on fatgraphs" as we shall describe. The techniques of proof involve Fock's path-ordered product expansion of holonomies, Ptolemy transformations, and the triangle inequalities. This is a main step in giving a combinatorial description of the Deligne-Mumford compactification of moduli space which we shall also discuss as time permits.

I will present a general formalism for understanding coloured operads of different flavours, such as cyclic operads, modular operads and topological field theories. The talk is based on arXiv:math/0701767.

Roughly speaking, a quasiregular map is a possibly-branched covering map with bounded distortion. The theory of such maps was developed in the 1970s to carry over to higher dimensions the more geometric aspects of the theory of complex analytic functions of the plane. In this talk I shall outline the proof of rigidity theorems describing the quasiregular self-maps of hyperbolic manifolds. These results rely on an extension of Sela's work concerning the stability of self-maps of hyperbolic groups, and on older topological ideas concerning discrete-open and light-open maps, particularly their effect on fundamental groups. I shall explain how these two sets of ideas also lead to topological rigidity theorems. This talk is based on a paper with a similar title by Bridson, Hinkkanen and Martin (to appear in Compositio shortly). http://www2.maths.ox.ac.uk/~bridson/papers/QRhyp/

  In this talk I will show how one can sometimes "uncomplete" the p-completed classifying space of a finite group, to obtain the original (non-completed) classifying space, and hence the original finite group. This "uncompletion" process is closely related to well-known local-to-global questions in group theory, such as the classification of finite simple groups. The approach goes via the theory of p-local finite groups. This talk is a report on joint work with Bob Oliver.