Forthcoming events in this series


Tue, 30 Apr 2013
17:00
L2

'Amalgamated products of free groups: from algorithms to linguistic.'

Elizaveta Frenkel
(Moscow)
Abstract

In my talk I shall give a small survey on some algorithmic properties of amalgamated products of finite rank 
free groups. In particular, I'm going to concentrate on Membership Problem for this groups. Apart from being algorithmically interesting, amalgams of free groups admit a lot of interpretations. I shall show how to 
characterize this construction from the point of view of geometry and linguistic.  

Tue, 05 Mar 2013
17:00
L2

"Galois problems in Schubert Calculus, and related problems"

Prof Iain Gordon
(Edinburgh)
Abstract

I will discuss some recent developments in Schubert calculus and a potential relation to classical combinatorics for symmetric groups and possible extensions to complex reflection groups.

Tue, 26 Feb 2013
17:00
L2

Relatively hyperbolic groups, mapping class groups and random walks

Alessandro Sisto
(Oxford)
Abstract

I will discuss similarities and differences between the geometry of
relatively hyperbolic groups and that of mapping class groups.
I will then discuss results about random walks on such groups that can
be proven using their common geometric features, namely the facts that
generic elements of (non-trivial) relatively hyperbolic groups are
hyperbolic, generic elements in mapping class groups are pseudo-Anosovs
and random paths of length $n$ stay $O(\log(n))$-close to geodesics in
(non-trivial) relatively hyperbolic groups and
$O(\sqrt{n}\log(n))$-close to geodesics in mapping class groups.

Tue, 12 Feb 2013
17:00
L2

Rigidity of group actions

Alex Gorodnik
(Bristol)
Abstract

We discuss the problem to what extend a group action determines geometry of the space. 
More precisely, we show that for a large class of actions measurable isomorphisms must preserve 
the geometric structure as well. This is a joint work with Bader, Furman, and Weiss.

Tue, 29 Jan 2013
17:00
L2

Intersections of subgroups of free products.

Yago Antolin Pichel
(Southampton)
Abstract

I will introduce the notion of Kurosh rank for subgroups of 
free products. This rank satisfies the Howson property, i.e. the 
intersection of two subgroups of finite Kurosh rank has finite Kurosh rank.
I will present a version of the Strengthened Hanna Neumann inequality in 
the case of free products of right-orderable groups. Joint work with  A. 
Martino and I. Schwabrow.

Tue, 15 Jan 2013
17:00
L2

Homological dimension of soluble groups and some new complement and supplement theorems.

Peter Kropholler
(Southamapton)
Abstract

The homological dimension of a group can be computed over any coefficient ring $K$.
It has long been known that if a soluble group has finite homological dimension over $K$
then it has finite Hirsch length and the Hirsch length is an upper bound for the homological
dimension. We conjecture that equality holds: i.e. the homological dimension over $K$ is
equal to the Hirsch length whenever the former is finite. At first glance this conjecture looks
innocent enough. The conjecture is known when $K$ is taken to be the integers or the field
of rational numbers. But there is a gap in the literature regarding finite field coefficients.
We'll take a look at some of the history of this problem and then show how some new near complement
and near supplement theorems for minimax groups can be used to establish the conjecture
in special cases. I will conclude by speculating what may be required to solve the conjecture outright.

Fri, 14 Dec 2012
16:00
L3

Some results and questions concerning lattices in totally disconnected groups

Tsachik Gelander
(Jersulem)
Abstract

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.

Fri, 14 Dec 2012
14:15
L3

Deformations and rigidity of lattices in soluble Lie groups

Benjamin Klopsch
(RHUL and Magdeburg)
Abstract

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.

Fri, 14 Dec 2012
13:00
L3

Cayley graphs of Fuchsian surface groups versus hyperbolic graphs

Caroline Series
(Warwick)
Abstract

Most results about the Cayley graph of a hyperbolic surface group can be replicated in the context of more general hyperbolic groups. In this talk I will discuss two results about such Cayley graphs which I do not know how to replicate in the more general context.

Tue, 27 Nov 2012
17:00
L2

'Orbit coherence in permutation groups'

Mark Wildon
(Royal Holloway)
Abstract

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).

Tue, 20 Nov 2012
17:00
L2

"Nielsen equivalence and groups whose profinite genus is infinite"

Martin Bridson
(Oxford)
Abstract

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
question.

Tue, 06 Nov 2012
17:00
L2

Group actions on rings and the Cech complex.

Peter Symonds
(Manchester)
Abstract

 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.

Tue, 30 Oct 2012
17:00
L2

A closed formula for the Kronecker coefficients.

Dr Chris Bowman
Abstract

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.
Tue, 23 Oct 2012
17:00
L2

The width of a group

Nick Gill
(Open University)
Abstract

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.

Tue, 16 Oct 2012
17:00
L2

Superrigidity for mapping class groups?

Prof Juan Souto
(British Columbia)
Abstract

There is a well-acknowledged analogy between mapping class
groups and lattices in higher rank groups. I will discuss to which
extent does Margulis's superrigidity hold for mapping class groups:
examples, very partial results and questions.

Tue, 09 Oct 2012
17:00
L2

Rank Gradient of Artin Groups and Relatives

Nikolay Nikolov
(University of Oxford)
Abstract

We prove that the rank gradient vanishes for mapping class groups, Aut(Fn) for all n, Out(Fn), n > 2 and any Artin group whose underlying graph is connected. We compute the rank gradient and verify that it is equal to the first L2-Betti number for some classes of Coxeter groups.

Tue, 05 Jun 2012
17:00
L2

Artin groups of large type: from geodesics to Baum-Connes

Professor S. Rees
(Newcastle)
Abstract

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 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

Tue, 15 May 2012
17:00
L2

'More words on words'

Aner Shalev
(Jerusalem)
Abstract

In recent years there has been extensive interest in word maps on groups, and various results were obtained, with emphasis on simple groups. We shall focus on some new results on word maps for more general families of finite and infinite groups.

Tue, 01 May 2012
17:00
L2

Reflection group presentations arising from cluster algebras

Professor R. Marsh
(Leeds)
Abstract

 Finite reflection groups are often presented as Coxeter groups. We give a
presentation of finite crystallographic reflection group in terms of an
arbitrary seed in the corresponding cluster algebra of finite type for which
the Coxeter presentation is a special case. We interpret the presentation in
terms of companion bases in the associated root system. This is joint work with 
Michael Barot (UNAM, Mexico)