Back to WTC Spring 2005

Azer Akhmedov
A new metric criterion for non-amenability

We will introduce a new metric criterion for non-amenability.
Then we will discuss an application of our technique to concrete
examples, which include Thompson's group F and free Burnside groups of
sufficiently large odd exponent.

Boundary representations for negatively curved groups -
irreducibility and rigidity
(joint work with Roman Muchnik)

Let M be  a compact negatively curved manifold, G be its fundamental
group and X its universal cover. Denote the boundary of X by B. B is
endowed with the Paterson-Sullivan measure. The associated unitary
representation $L^2(B)$ is called a boundary unitary representation.
Fixing G, but changing the metric on M, we get a different boundary
(given by a different measure on the same topological boundary), and a
different boundary representation.
We will explain the setting and indicate the proof of

Theorem 1: The boundary representations are irreducible.
Theorem 2: Two boundary representations are equivalent if and only if
the associated marked length spectrums are the same (up to a scalar
multiple).

The marked length spectrum is the assignment associate to a free
homotopy class of closed loops in M the length of a shortest geodesic
in it.
The proof of the theorem is based on the mixing property of the
geodesic flow on M.

Prescribing cusps of aspherical manifolds

Each cusp cross-section of a finite volume manifold
of bounded negative curvature is an infranilmanifold, yet
little is known on which infranilmanifolds can occur.
On the other hand, any zero-cobordant aspherical manifold
bounds an aspherical manifold obtained via relative strict
hyperbolization. I shall discuss group-theoretic properties
of these manifolds including relative hyperbolicity, splittings,
simplicial volume, and co-Hopf property.

Indira Chatterji
Homotopy idempotents on closed manifolds

The Geoghegan conjecture is that homotopy idempotents on closed manifolds of dimension at least 3 can be deformed into a map that has one
single fixed point. Earlier work of Geoghegan showed that it is equivalent to the Bass conjecture. We shall discuss the history and status of this conjecture. This is joint work with J. Berrick and G. Mislin.

Max Forester
First and second order isoperimetric exponents of groups

I will describe a simple construction of finitely presented groups having first or second order Dehn function of the form $x^{\alpha}$ for certain prescribed numbers $\alpha$. In particular we find that all rational numbers greater than $2$ occur, as both first and second order isoperimetric exponents. This is joint work with Noel Brady, Martin Bridson, and Ravi Shankar.

Daniel Groves
Aspects of relatively hyperbolic groups

We will discuss some recent work on relatively hyperbolic groups, particularly those with abelian or virtually abelian parabolics.  Some topics might be:  the automorphism group, equations, hyperbolic quotients, and others.

Michael Handel
Fixed Points of abelian actions on $R^2$ and $S^2$
(joint work with John Franks and Kamlesh Parwani)

We prove that if $\F$ is a finitely generated free abelian
group oforientation preserving $C^1$ diffeomorphisms of $\R^2$ which
leaves invariant a compact set then there is a common fixed point for
all elements of $\F.$ We also show that if $\F$ is a finitely generated
abelian subgroup of $\Diff^1_+(S^2)$ then there is a common fixed point
for all elements of an index two subgroup of $\F.$

John Holt
On the topology of the space of punctured-torus groups

We show that a certain geometric condition on a hyperbolic I-bundle over the once-punctured torus effects the local topology of the space of  punctured-torus groups near the holonomy representation.  We define what it means for a manifold N "not to wrap", and show that if N is an I-bundle over the once-punctured torus and does not wrap, then every sufficiently small neighbourhood of the holonomy of N in the deformation space is connected.

Richard Kent
Shadows of mapping class groups: capturing convex co-compactness

Farb and Mosher have developed a notion of convex co-compactness for subgroups of the mapping class group of a surface analogous to the notion of the same name in the theory of Kleinian groups. They have shown that this property is closely related to hyperbolicity of the associated surface group extension. There are several nice characterizations of convex co-compact Kleinian groups. We prove analogous characterizations for convex co-compact subgroups of the mapping class group. This is joint work with Chris Leininger.

Chris Leininger
Graphs of Veech groups

I'll discuss an extension of joint work with Alan Reid in which we prove a "Combination Theorem" for Veech subgroups of the mapping class group.  As a corollary of the original Combination Theorem (with Reid) we obtain closed surface subgroups of the mapping class group with "almost all" elements being pseudo-Anosov.  The new construction provides a broader class of groups exhibiting the same behavior, as well as new geometric information about the action of these groups on Harvey's complex of curves.

Jon McCammond
Artin groups and polytopes

Despite the fact that Artin groups are the natural
generalization of braid groups under the influence of Coxeter groups, they
remain poorly understood when compared to their nicely behaved relatives.
In this talk I will survey some recent progress in the area, focusing
particular attention on the connections relating Artin groups to
combinatorial objects such as Stasheff polytopes and quiver
representations.

Kevin Whyte
Quasi-isometric embeddings of symmetric spaces

Mostow's rigidity theorem is one of the motivations for many quasi-isometric rigidity results, and these results give a natural geometric generalization of the cocompact case of Mostow's result.   In a similar way, quasi-isometric embeddings give a natural generalization of Margulis'  Super-rigidity theorem.   Perhaps surprisingly, many of the obvious conjectures prove to be false.  I will discuss both these counter-examples and some rigidity results in special cases.  This is joint work with David Fisher.

Dani Wise
Special Cube Complexes

We describe a class of CAT(0) cube complexes
related to right-angled Artin groups. Applications are given towards
subgroup separability, Coxeter groups, embeddings into SL_n(Z),
and a linear version of Rip's short exact sequence.
This is joint work with Frederic Haglund