## Invited talks:

**Daniel Allcock:** *On the classification of reflective Lorentzian lattices*
An important source of Coxeter groups acting on hyperbolic
space is the automorphism groups of Lorentzian lattices, i.e., integer
quadratic forms of signature (n,1). Such a form is called reflective
if isisometry group is generated up to finite index by reflections.
It is known that there are only finitely many, and I have undertaken
to find them all; I am guessing there are between 1,000 and 10,000.
I will speak on parts of this work. It is motivated by the problem
of classifying the "Lorentzian Lie algebras" defined by Gritsenko and
Nikulin, which means the "best" kind of hyperbolic Kac-Moody algebra.

**Uri Bader:** *Old and new in Measurable Group Theory*
One of the main programs in Geometric Group Theory is the study of
the QI equivalence relation on groups.
In much the same way, Measurable Group Theory studies the ME (Measure
Equivalence) relation between groups.
In my talk I will define the notion of ME and explain how it
provides a neat platform to express a variety of classical results.
I will further present a collection of equivalence relations between
groups. Finally, I will report on some new progress, focusing mainly
on a recent result obtained with Furman and Sauer regarding
hyperbolic lattices.

**Igor Belegradek:** *Relative hyperbolicity via branched covers*
Let *M* be a finite volume complete negatively pinched manifold
and *S* be a codimension two, totally geodesic, immersed
submanifold with normal crossings. Allcock proved that the complement
*M-S* of *S* is aspherical, yet little is known about its
fundamental group.
In a joint work with Chris Hruska, we show that the fundamental
group of *M-S* is relatively hyperbolic if *S* is compact
embedded, or if *S* has a sufficiently thick regular
neighborhood. The proof
builds on ideas of Bowditch, and the model space for proving relative
hyperbolicity is the universal branched cover of *M* branched
over *S*, which is a non-proper CAT(-1) space.

**Mladen Bestvina:** *Recent developments on the geometry of Outer space*
I will survey our knowledge of the topology and geometry of
*Out(F*_{n}) and Outer space, particularly as they
compare with mapping class groups and Teichmuller space. In the
20+ years since the definition of Outer space by Culler and Vogtmann
we have succeeded (more or less) to understand the topology, and the
serious work on the geometry is in its early stages.

**Noel Brady:** *Higher Dehn functions of groups*
The Dehn function of a finitely presented group
expresses the area of loops (in a geometric model for the
group) as a function of the length of the loops.
satisfying higher finiteness conditions. These functions
express the volumes of *(k+1)*-ball fillings of
*k*-spheres (in suitable geometric models for the group)
as functions of the volumes of the spheres.

We discuss some results about higher dimensional
Dehn functions of groups.

**Martin Bridson:** *Subdirect products of groups*
I shall discuss several constructions, algebraic and geometric,
that involve subdirect products of groups. In particular I'll
outline a proof of the following theorem of
Bridson-Howie-Miller-Short: a subdirect product of finitely
presented groups is finitely presented if it virtually projects
onto each pair of factors (the VSP criterion). I'll use this
criterion to construct novel examples of finitely presented
subdirect products of surface groups. An algorithmic version
of the VSP criterion will be used to prove that the class of
finitely presented residually free groups is recursively enumerable.

**Pierre-Emmanuel Caprace:** *Kac-Moody groups from a Coxeter group viewpoint*
Résumé:
Kac-Moody theory provides notably examples of locally compact
(semi-)simples groups which possess discrete subgroups of finite
covolume.
While some of their properties are reminiscent of arithmetic groups in
positive characteristic, they generally behave very differently than
linear groups. A peculiar property of Kac-Moody lattices is that they
admit a Weyl group, which is always a Coxeter group (usually
non-amenable). The talk will illustrate this fact by presenting
certain properties of Kac-Moody groups which are best understood when
reduced to Coxeter group questions.

**Ruth Charney:** *Length functions of right-angled Artin groups*
Outer space for a free group *F*_{n} may be described
as the space of minimal actions of *F*_{n} on a tree.
By a theorem of Culler and Morgan, such an
action is uniquely determined by its length function, so an alternate
description of outer space is the space of length functions on
*F*_{n}.
Motivated by the search for a good "outer space" for right-angled Artin
groups, we prove an analogous theorem about length functions for
actions of 2-dimensional right-angled Artin groups on CAT(0)
cube complexes. (Joint work with Max Margolis)

**Tomas Delzant:** *On the complexity and Volume of hyperbolic 3-manifolds*
We compare the volume of a hyperbolic 3 manifold and the
complexity of its fundamental group.

**Alexander Dranishnikov:** *Systolic Category*
Lowner-Gromov type inequalities are used to define a manifold
invariant
called the 'systolic category'. We will discuss an analogy between the
systolic category and the Lusternik-Schnirelmann category.
and the relation of the former to the cup-length.

**Cornelia Druţu:** *Median spaces, properties (T) and Haagerup, applications
to mapping class groups*
Median spaces are non-discrete generalisations of CAT(0) cube complexes.
Both property (T) of Kazhdan and Haagerup property (a-T-menability) can be reformulated in terms of actions on median spaces. This and
the fact that
the asymptotic geometry of a mapping class group is median imply
results on homomorphisms into mapping class groups.
The talk is on joint work with I. Chatterji and F. Haglund (first part),
and on joint work with J. Behrstock and M. Sapir (second part).

**Tom Farrell:** *Bundles with negatively curved fibers*
This talk is a report on joint work with Pedro Ontaneda. We have
been studying smooth bundles whose closed manifold fibers are equipped
with negatively curved Riemannian metrics. Two such bundles with
the same base *B* are equivalent if there is such a bundle with
base *Bx[0,1]* whose restrictions to *Bx0* and *Bx1*
are fiberwise isometric to the given bundles. We obtain
results about the forget extra structure map from this theory to
ordinary smooth bundle theory.

**Mark Feighn:** *he Recognition Theorem for Out(F*_{n}) and applications
We characterize elements of Out(F__{n}) in terms of dynamic
invariants and describe some applications.

This is joint work with Michael Handel.

**Frédéric Haglund:** *Negatively curved spaces with embedded hyperplanes*
In his seminal article on hyperbolic groups, M. Gromov
defined a combinatorial-geometric structure called CAT(0) cube
complexes. A fundamental example of CAT(0) cube complex was associated
by M. Davis with any right-angled Coxeter system, so that the Coxeter
group acts geometrically on the complex.

When a compact space *M* has (immersed) hyperplanes,
its fundamental group acts on a CAT(0) cube complex (by the work of
M. Sageev). We show the following : when the hyperplanes of *M*
are embedded and the space *M* is negatively curved then
the fundamental group *π*_{1}(M) acting on the CAT(0)
cube complex (virtually) extends to a right-angled Coxeter group
acting on its Davis' complex

This is joint work with D. Wise.

**Ursula Hamenstädt:** *
Geometry of the mapping class group*
We explain how to construct a quasi-convex bicombing
for the mapping class group and derive some applications to the structure of subgroups.

**Michael Kapovich:** *Rank 2 nondiscrete affine buildings*
In his work classifying spherical and affine buildings,
J.Tits proved that every (irreducible) thick affine building of rank
at least 3 is associated with an algebraic group over a field. In
particular, it follows that the finite Weyl groups of such buildings
have to be crystallographic. We complete this picture by constructing
rank 2 thick nondiscrete affine buildings associated with an arbitrary
finite dihedral group.

This is a joint work with Arkady Berenstein.

**Guido Mislin:** *Bounded cohomology of Lie groups*
We show that each integral Borel cohomology class of a connected
Lie group *G* can be represented by a Borel bounded cocycle
if and only if the radical of *G* is linear. This leads to
a generalization of Gromov's boundedness theorem on characteristic
classes of flat bundles.

This is a joint work with Indira Chatterji, Christophe Pittet,
and Laurent Saloff-Coste.

**Thomas Schick:** *Refinements of the Atiyah conjecture
about integrality of L*^{2}-Betti numbers
*L*^{2}-Betti numbers are invariants of
compact manifolds obtained from the square integrable differential
forms on the universal covering via the use of functional analysis
(von Neumann traces), introduced by Atiyah and studied by Gromov,
Lück and many others. By definition, they
are non-negative real numbers.

However, for torsion-free fundamental groups the
*L*^{2}-Betti numbers are conjectured always to be
integers (know for many groups). By work of Linnell this is
equivalent to the fact that a certain overring *DG* of the group
ring *QG* is a skew field.

Also for groups with torsion there is a precise conjecture
on the possible values of *L*^{2}-Betti numbers.
The overring *DG* in this case can not be a skew field (as
*QG* contains non-trivial zero divisors). We present work about a
refinement of the Atiyah conjecture, and we explain how this
is equivalent to a precise structure result for the ring *DG*.

We prove this refined Atiyah conjecture for certain classes
of groups, including new cases of the original Atiyah conjecture.

This is joint work with Peter Linnell and Anselm Knebusch.

**Karen Vogtmann:** *Automorphism groups of right-angled Artin groups*
Right-angled Artin groups (RAAGs) are finitely-generated
groups which are completely described by the fact that some of the
generators commute; thus they interpolate between free groups at one
extreme and free abelian groups at the other. RAAGs and their
subgroups are the source of many important examples and
counterexamples in geometric group theory.
In this talk I will describe joint work with Ruth Charney on the outer
automorphism groups of RAAGs, which can be though of analogously as
interpolating between the outer automorphism groups of free groups and
the general linear groups *GL(n,Z)*. We are particularly
interested in determining which properties shared by *Out(F_n)*
and *GL(n,Z)* are in fact true for all automorphism groups of
RAAGs.

**Kevin Wortman:** *Dehn functions of arithmetic groups*
I'll talk about partial results in determining the Dehn
functions of arithmetic subgroups of semisimple Lie groups. (Joint with
Mladen Bestvina and Alex Eskin.)

**Robert Young:** *The Dehn function of SL(n,Z)*
The Dehn function is a group invariant which connects
geometric and combinatorial group theory; it measures
both the difficulty of the word problem and the area
necessary to fill a closed curve in an associated space
with a disc. The behavior of the Dehn function for high-rank
lattices in high-rank symmetric spaces has long been an open
question; one particularly interesting case is *SL(n,Z)*.
Thurston conjectured that *SL(n,Z)* has a quadratic
Dehn function when *n≥4*.

This differs from the behavior for *n=2* (when the Dehn
function is linear) and for *n=3* (when it is
exponential). In this talk, I will discuss some of the
background of the problem and sketch a proof that
the Dehn function of SL(n,Z) is at most quartic when *n≥5*.