Groups and Geometry in the South East
This is a series of meetings, with the aim of bringing together the geometric group theorists in the South East of England. The meetings are sponsored by mathematicians from the Universities of London, Oxford and Southampton, and organised by Martin Bridson and Henry Wilton. We have been awarded LMS Scheme 3 funding.
In 2013-14, the meetings will (tentatively) be as follows:
- 25 October 2013, UCL
- 6 December 2013, Southampton
- February 2014, location tbc
- 9 May 2013, Oxford
Here are some details of our next meeting.
Friday 6th December, 2013
Location: Building 67, room E1001, University of Southampton (maps and arrival information here)
1.15pm A polynomial upper bound on Reidemeister moves for each knot type
Marc Lackenby (Oxford)
For each knot type K, we establish the existence of a polynomial pK with the following property. Any two diagrams of K with n and n' crossings respectively differ by a sequence of at most pK(n) + pK(n') Reidemeister moves. As a consequence, the problem of deciding whether a knot is of type K is in the complexity class NP. This result generalises earlier work which dealt with the case when K is the unknot, for which we may take pK(n) to be (231n)11.
2.30pm Small cancellation groups and conformal dimension
John MacKay (Bristol)
The boundary at infinity of a hyperbolic group has a natural invariant called its conformal dimension, introduced by Pansu. This analytic invariant of the boundary can be studied using lp-cohomology of the group. I will discuss how recent work of Bourdon, Kleiner and others combines with ideas of Ollivier and Wise to give new insights to the geometry of small cancellation groups; in particular, to certain random groups.
4pm Coarse embeddings of graphs and groups: monsters versus beauty
Goulnara Arzhantseva (Vienna)
The concept of coarse embedding was introduced by Gromov in 1993. It plays a crucial role in the study of large-scale geometry of groups and the Novikov higher signature conjecture. Coarse amenability, also known as Guoliang Yu's property A, is a weak amenability-type condition that is satisfied by many known metric spaces. It implies the existence of a coarse embedding into a Hilbert space. In this expository talk, we discuss the interplay between infinite expander graphs, coarse amenability, and coarse embeddings. We present several 'monster' constructions in the setting of metric spaces of bounded geometry.
This research was partially supported by my ERC grant ANALYTIC no. 259527.