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).