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 2012-13, the meetings will (tentatively) be as follows:
- 19 October 2012, UCL
- 7 December 2012, Southampton
- 22 February 2013, UCL
- 10 May 2013, Oxford
Here are some details of our next meeting.
Friday 10th May, 2013
Location: Room RI.0.48, the Gibson Building, Oxford
1.15pm Homological dimension from an algebraic perspective
Peter Kropholler (Southampton)
A closer look at what we know about the homological (or weak) dimension of a group over various coefficient rings. This talk will survey the territory and include some discussion of the modern developments from ring theory and category theory.
2.30pm The solution to Siegel's problem on hyperbolic lattices
Gaven Martin (Massey University, Auckland, visiting Oxford)
We outline the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram.
This solves (in three dimensions) the problem posed by Siegel in 1945.
Siegel solved this problem in two dimensions by deriving the signature formula identifying the 2-3-7 triangle group as having minimal co-area. There are strong connections with arithmetic hyperbolic geometry in the proof and the result has applications identifying three-dimensional analogues of Hurwitz's 84g-84 theorem as Siegel's result does.
4pm Ordering the space of finitely generated groups
Laurent Bartholdi (Göttingen)
Consider the following relation `emulates' between finite generated groups: G emulates H if, for some generating set T in H and some sequence of generating sets Si in G, the marked balls of radius i in (G, Si) and (H, T) coincide.
This means, informally, that any group-theoretical statement that can be computed in a finite portion of H can be computed in G.
Given a nilpotent group G, we characterize the groups that are related to G by the `emulation' relation: it consists, essentially, of those groups which generate the same variety of groups as G.
The `emulation' relation is transitive, so defines a preorder on the set of isomorphism classes of finitely generated groups. We show that a partial order can be imbedded in this preorder if and only if it is realizable by subsets of a countable set under inclusion.
We study the groups that emulate free groups. This lets us show that every countable group imbeds in a group of non-uniform exponential growth. In particular, there exist groups of non-uniform exponential growth that are not residually of subexponential growth and do not admit a uniform imbedding into Hilbert space.
This is joint work with Anna Erschler.