Past Algebra Seminar

14 December 2012
Tsachik Gelander

I'll discuss some results about lattices in totally
disconnected locally compact groups, elaborating on the question:
which classical results for lattices in Lie groups can be extended to
general locally compact groups. For example, in contrast to Borel's
theorem that every simple Lie group admits (many) uniform and
non-uniform lattices, there are totally disconnected simple groups
with no lattices. Another example concerns with the theorem of Mostow
that lattices in connected solvable Lie groups are always uniform.
This theorem cannot be extended for general locally compact groups,
but variants of it hold if one implants sufficient assumptions. At
least 90% of what I intend to say is taken from a paper and an
unpublished preprint written jointly with P.E. Caprace, U. Bader and
S. Mozes. If time allows, I will also discuss some basic properties
and questions regarding Invariant Random Subgroups.

14 December 2012
Benjamin Klopsch

Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of Aut(G) on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev and Saitô for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice Γ in G is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of G is connected.  I will introduce all necessary notions and try to motivate and explain this result.

27 November 2012
Mark Wildon
Let G be a permutation group acting on a set Omega. For g in G, let pi(g) denote the partition of Omega given by the orbits of g. The set of all partitions of Omega is naturally ordered by refinement and admits lattice operations of meet and join. My talk concerns the groups G such that the partitions pi(g) for g in G form a sublattice. This condition is highly restrictive, but there are still many interesting examples. These include centralisers in the symmetric group Sym(Omega) and a class of profinite abelian groups which act on each of their orbits as a subgroup of the Prüfer group. I will also describe a classification of the primitive permutation groups of finite degree whose set of orbit partitions is closed under taking joins, but not necessarily meets. This talk is on joint work with John R. Britnell (Imperial College).
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.