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.

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.

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.

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.

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.

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.

(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 $\F.$

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.

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.

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.

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.

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.

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