Spring Lecture Series 2003
Title: Representations of the symmetric group
Abstract:
A classical problem in representation theory is to classify the
representations of the symmetric group. There is a beautiful solution over
a field of characteristic zero. The behavior in characteristic p is
related to the behavior of the Hecke algebra when the parameter q is a
primitive pth root of unity. I will discuss a topological approach to
understanding this problem and its relation to the classical problem.
Title: Curvature, complexity and subgroups
Abstract:
In this talk I'll discuss recent results about the nature
of the groups that cluster around the concept of non-positive
curvature in group theory. Most of these results concern
issues of complexity and subgroup structure.
Much of the discussion will focus on free-by-cyclic groups,
whose subtle diversity is both illuminated and illuminating
in this context.
If time allows, I shall also discuss the complexity of
certain families of Andrews-Curtis trivialisable balanced presentations.
Title: Circular groups and planar groups
Abstract:
We show that many natural classes of groups of homeomorphisms of the
plane are (abstractly) isomorphic to groups of homeomorphisms of the
circle. There are connections with the theory of flows on 3-manifolds,
(generalized) braid groups, and Zimmer's conjecture.
Title: Laminations of 3-manifolds and groups of homeomorphisms of the circle.
Abstract:
If M is an atoroidal 3-manifold with a taut foliation,
Thurston showed that pi_1(M) acts faithfully on a circle. In this
talk, I will discuss why certain other classes of essential
laminations also give rise to circle actions. (Joint work with
D. Calegari).
Title: Surface subgroups of Coxeter groups
Abstract:
We characterize those Coxeter groups that contain a surface
subgroup. It follows from this characterization that a Coxeter group
either contains a surface group or is virtually free. In particular,
Gromov's question as to whether a 1-ended word hyperbolic group contains a
hyperbolic surface group has an affirmative answer in the class of Coxeter
groups.
(Joint work with Darren Long and Alan Reid.)
Title: Arithmetic rational homology spheres
Abstract:
We will discuss how common it is that an
arithmetic hyperbolic manifold or orbifold is a rational homology sphere.
Title: There are no unexpected tunnel number one knots of genus one
Abstract:
We show that the only knots that are tunnel number one and genus one are those that are already known: 2-bridge knots obtained
by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Muņoz and by Morimoto-Sakuma.
The principal new tools are a useful way of defining width for a 3-valent graph in S3, and a controlled way of loading the knot onto a
neighborhood of such a graph. We analyze how the knot loading allows the graph to be thinned and show that eventually either the graph
contains an unknot or the knot tunnel can be pushed onto the Seifert surface. In either of these circumstances the result (known as the
Goda-Teragaito Conjecture) was already known.
Title: Low dimensional topology and aspects of the first order theory of a free group
Abstract:
We study sets of solutions to equations over a free group,
projections of such sets, and the structure of elementary sets defined
over a free group. The structre theory we obtain enable us to answer
some questions of A. Tarski's, and classify those finitely generated
groups that are elementary equivalent to a free group. Connections
with low dimensional topology, a generalization to (Gromov) hyperbolic
groups, and further aspects of the first order theory of free (and
hyperbolic) groups will also be discussed.
Title: Smallish knots
Abstract:
Let \Sigma be a non-Haken irreducible 3-manifold. A knot K in
\Sigma is said to be smallish if its exterior is irreducible
and contains no bounded essential surface whose boundary
components are meridian curves. I will describe recent
progress in an on-going joint project with Culler, Dunfield
and Jaco directed at proving the conjecture that every
non-Haken irreducible \Sigma contains a smallish knot. The
conjecture, in addition to its intrinsic interest, forms part
of a program for proving the Poincare Conjecture.