Nicolas Bergeron : A boundary criterion for cubulation.
I will describe a criterion in terms of the boundary
for the existence of a proper cocompact action of a wordhyperbolic group
on a CAT(0) cube complex.
This has applications towards arithmetic hyperbolic lattices and hyperbolic 3manifold groups.
By combining the theory of special cube complexes, the surface subgroup result of KahnMarkovic,
and Agol's criterion, this in particular shows that every subgroup separable closed hyperbolic 3manifold
is virtually fibered. This is joint work with Dani Wise.
Brian Bowditch : Median structures on groups.
A common theme in recent work on the mapping class groups
has been the existence of some kind of median structure.
We aim to give a brief survey of this, and suggest that
some results, such as the rank theorem, can be seen efficiently
in these terms. It seems to be a natural question to ask which
groups admit structures of this sort.
Emmanuel Breuillard : Word maps and the BanachTarski paradox.
In joint work with Bob Guralnick, Ben Green and Terence Tao, we establish that most Cayley graphs of finite simple
groups of bounded rank are expander graphs. A key part of the proof consists in establishing the existence of free
subgroups of semisimple groups with strong density properties. Such properties are related to the study of word maps
and to paradoxical decompositions of homogeneous spaces, in the spirit of the Banach Tarski paradox.
François Dahmani : Presentations of parabolic subgroups of relatively hyperbolic groups.
Joint work with V. Guirardel.
Given a finitely presented relatively hyperbolic group,
we prove that all its maximal parabolic subgroups are finitely presented.
The proof is constructive in the sense that one can compute such presentations
if one is given generators of the parabolic subgroups, and a solution to the word problem in the ambiant group.
Victor Gerasimov : Visibility and convexity for convergence group actions.
The purpose of my lectures is to demonstrate advantages of simultaneous
usage of geometric and topological ideas in studying actions with convergence property.
The construction of ''attractor sum'' is actually a glueing of
geometry and topology. The main applications concern the relatively hyperbolic groups.
However we introduce a wider class of ''geometric'' convergence actions
including the limits and the quotients of relatively hyperbolic actions.
For any finitely generated group G we define the ''initial geometric boundary''
∂_{ig}G
that covers the (relatively) hyperbolic boundaries, the Floyd boundaries
and the space of ends.
The action of G on ∂_{ig}G has convergence property.
For some groups we are able to calculate this ''space of fine ends''.
Anders Karlsson : Boundaries and dynamics of isometries.
Metrics, invariant or semi invariant,
under transformations arise in many contexts:
differential geometry, group theory, several
complex variables, operator theory, ergodic
theory etc. Halfspaces and their limits, called
stars, in various compactifications are notions
suitable to describe the dynamics of metric
preserving transformations. One fundamental type
of boundary dynamics is that of convergence group
actions, as it appears frequently and has
wellknown and strong consequences. This boundary
dynamics can be given a generalized description
in terms of the incidence structure at infinity
that the stars of a compactification give rise
to. Some examples and questions will be pointed
out, in particular to Teichmuller spaces and the first L^2 cohomology.
John Mackay : Generic conformal dimension estimates for random groups.
In Gromov's density model of random groups, at densities less than one
half, a random group is hyperbolic. It is known that the group also
infinite and that its boundary at infinity is homeomorphic to the Menger
curve, however this does not completely determine the quasiisometry
type of such groups.
I will outline recent work showing how Pansu's conformal dimension can
be used in the contexts of certain small cancellation groups to give a
more refined geometric picture of random groups at small densities.
Mahan Mj : Semiconjugacies of Convergence Actions.
Let G be a nonelementary (relatively) hyperbolic
group acting freely and properly discontinuously
on a hyperbolic metric space X. Let ∂G be the (relative) boundary
of G and ∂X be the boundary of X. Then G acts as a convergence
group on both ∂G and ∂X.
We shall discuss the following general question:
Does the action of G on X extend continuously to an equivariant map
between ∂G and ∂X?
A complete positive answer when X is hyperbolic
3space will be discussed. Such continuous
extensions (called CannonThurston maps) have
been studied by Cannon and Thurston,
Minsky, Bowditch, McMullen, Pal, Das and the author.
Dani Wise : Groups with a quasiconvex hierarchy.
Nonpositively curved cube complexes
have become increasingly central objects in
combinatorial and geometric group theory. We will
give a quick introduction focusing especially on
their identity as "higher dimensional graphs". We
will describe cubical smallcancellation theory,
and outline how to use nonpositively curved cube
complexes to resolve certain outstanding group
theoretical problems. These include Baumslag's
conjecture on the residual finiteness of
onerelator groups with torsion, as well as the
virtualfibering problem for Haken hyperbolic 3manifolds.
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