Past Algebra Seminar

20 November 2012
Martin Bridson

In our 2004 paper, Fritz Grunewald and I constructed the first
pairs of finitely presented, residually finite groups $u: P\to G$
such that $P$ is not isomorphic to $G$ but the map that $u$ induces on
profinite completions is an isomorphism. We were unable to determine if
there might exist finitely presented, residually finite groups $G$ that
with infinitely many non-isomorphic finitely presented subgroups $u_n:
P_n\to G$ such that $u_n$ induces a profinite isomorphism. I shall
discuss how two recent advances in geometric group theory can be used in
combination with classical work on Nielsen equivalence to settle this

6 November 2012
Peter Symonds

 We present a new, more conceptual proof of our result that, when a finite group acts on a polynomial ring, the regularity of the ring of invariants is at most zero, and hence one can write down bounds on the degrees of the generators and relations. This new proof considers the action of the group on the Cech complex and looks at when it splits over the group algebra. It also applies to a more general class of rings than just polynomial ones.

30 October 2012
Dr Chris Bowman

The Kronecker coefficients describe the decomposition of the tensor product of two Specht modules for the symmetric group over the complex numbers. Surprisingly, until now, no closed formula was known to compute these coefficients. In this talk, I will report on joint work with M. De Visscher and R. Orellana where we use the Schur-Weyl duality between the symmetric group and the partition algebra to find such a formula.

23 October 2012
Nick Gill

I describe recent work with Pyber, Short and Szabo in which we study the `width' of a finite simple group. Given a group G and a subset A of G, the `width of G with respect to A' - w(G,A) - is the smallest number k such that G can be written as the product of k conjugates of A. If G is finite and simple, and A is a set of size at least 2, then w(G,A) is well-defined; what is more Liebeck, Nikolov and Shalev have conjectured that in this situation there exists an absolute constant c such that w(G,A)\leq c log|G|/log|A|. 
I will present a partial proof of this conjecture as well as describing some interesting, and unexpected, connections between this work and classical additive combinatorics. In particular the notion of a normal K-approximate group will be introduced.

5 June 2012
Professor S. Rees
I’ll report on my recent work (with co-authors Holt and Ciobanu) on Artin groups of large type, that is groups with presentations of the form G = hx1, . . . , xn | xixjxi · · · = xjxixj · · · , 8i < ji for which both sides of the ‘braid relation’ on xi and xj have length mij 2 N [1 with mij  3. (In fact, our results still hold when some, but not all possible, relations with mij = 2 are allowed.) Recently, Holt and I characterised the geodesic words in these groups, and described an effective method to reduce any word to geodesic form. That proves the groups shortlex automatic and gives an effective (at worst quadratic) solution to the word problem. Using this characterisation of geodesics, Holt, Ciobanu and I can derive the rapid decay property for most large type groups, and hence deduce for most of these that the Baum-Connes conjec- ture holds; this has various consequence, in particular that the Kadison- Kaplansky conjecture holds for these groups, i.e. that the group ring CG contains no non-trivial idempotents. 1