MONDAY-------------
9:40 - 10:40 Shahar Mozes: Invariant measures and divisibility
Abstract: In a joint work with Manfred Einsiedler we study a
relationship between the dynamical properties of the action a maximal
diagonalizable group A on certain arithmetic quotients G/Gamma where
G is a Lie group and Gamma<G a lattice, and arithmetic properties of
the lattice. In particular, given a finite set of odd primes with at
least two elements we consider the semigroup of all integer
quaternions that have norm equal to a product of powers of primes from
the set. For this semigroup we use measure rigidity theorems to prove
that the set of elements that are not divisible by a given quaternion
from the semigroup has subexponential growth.
4:30 - 5:30 Jason Manning: Relatively hyperbolic Dehn filling
Abstract: Dehn filling is a classical tool in 3-dimensional topology,
in which a 3-manifold with torus boundary is "filled" to obtain a
closed 3-manifold. Thurston showed that if the bounded 3-manifold has
interior with a hyperbolic metric, then so do most fillings. I'll
talk about what it means to "fill" a relatively hyperbolic pair
(G,P), and indicate how this procedure can be performed in a way which
preserves much of the geometry and group theory of (G,P). I'll also
try to indicate how this procedure is applied in an important step of
Ian Agol's proof of the Virtual Haken Conjecture. (More details will
be given in a subsequent informal talk.)
This is joint work with Daniel Groves and Ian Agol.
TUESDAY-------------
9:40 - 10:40 Iddo Samet: TBA
4:30 - 5: 30 Bill Thurston: TBA
Abstract: TBA
WEDNESDAY-------------------
9:40-10:40 Jim Conant: Hairy graphs and the homology of out(F_n)
.
Joint with Martin Kassabov and Karen Vogtmann
Abstract: We develop Kontsevich's graph homology approach to the rational
homology of the group Out(F_n). We introduce a graph homology theory,
called hairy graph homology, involving graphs with univalent vertices
labeled by elements of a symplectic vector space V. We construct a map
from the free graded commutative algebra on hairy graph cohomology to
the homology of Out(F_n), and show this map detects all known homology
classes in Out(F_n) (and by a related construction, all known classes
in Aut(F_n).) Hairy graph homology is highly non-trivial. It contains
odd polynomials over V, related to Morita's trace map, and also
contains spaces of classical modular forms. This gives rise to
embeddings, for example, of the second exterior power of the space of
cusp forms into cycles in the chain complex for Out(F_n). If this
embedding on the chain level is also an embedding upon passing to
homology, it would contradict a recent conjecture of Church, Farb and
Putman, which predicts homology in fixed codimension should stabilize
as n increases.
THURSDAY----------------
9:40-10:40 John Pardon: Totally disconnected groups (not) acting on
three-manifolds
Abstract: Hilbert's Fifth Problem asks whether every topological group
which is a manifold is in fact a (smooth!) Lie group; this was solved in the
affirmative by Gleason and Montgomery--Zippin. A stronger conjecture is
that a locally compact topological group which acts faithfully on a
manifold must be a Lie group. This is the Hilbert--Smith Conjecture,
which in full generality is still wide open. It is known, however (as a
corollary to the work of Gleason and Montgomery--Zippin) that it suffices
to rule out the case of the additive group of $p$-adic integers acting
faithfully on a manifold. I will discuss a solution in dimension three.
The proof uses tools from low-dimensional topology, for example
incompressible surfaces, minimal surfaces, and a property of the mapping
class group.
4:30-5:30 Lars Louder: Relative hyperbolicity and hierarchies for
finitely presented groups
Abstract: A hierarchy of a finitely generated group is a tree of
groups obtained by repeatedly passing to one-ended factors of vertex
groups of nontrivial (minimal) graphs of groups decompositions over
slender edge groups. We show that hierarchies of finitely presented
groups relative to slender subgroups having infinite dihedral
quotients are finite. This is joint work with Nicholas Touikan.
FRIDAY-----------
9:40-10:40 Sebastian Hensel: Realisation and Dismantlability
Abstract: The Nielsen Realisation problem asks if a finite subgroup of
the mapping class group of a surface can be realised as a group of
isometries of a hyperbolic metric. Similarly, one can ask if a finite
subgroup of Out(F_n) can be realised by a group of isometries of a
metric graph. The answer to both questions is yes, due to Kerckhoff in
the surface case, and Culler-Khramtsov-Zimmermann in the graph case.
In this talk I will present joint work with Damian Osajda and Piotr
Przytycki which gives a new and simplified proof of the realisation
theorem for Out(F_n), and the mapping class groups of punctured
surfaces.
The arguments are combinatorial in flavor and rely on the notion of
dismantlable graphs.
3:15-4:15 Alexandra Pettet: On fully irreducible elements of the outer automorphism group of a free group
Abstract: The outer automorphism group Out(F_n) of a non-abelian free
group F_n of rank n shares many properties with linear groups and
mapping class groups Mod(S) of surfaces, although the techniques for
studying Out(F_n) are often quite different from the latter two.
Motivated by analogy, I will describe some results concerning the
fully irreducible elements of Out(F_n), which are analogous to the
pseudo-Anosov elements of Mod(S).