VOGTMANNFEST
June 21 − June 25, 2010

Speakers:

Goulnara Arzhantseva
   University of Geneva
   Title: Coarse non-amenability and coarse embeddings
Abstract: The concept of coarse embedding was introduced by Gromov in 1993. It plays an important role in the study of large-scale geometry of groups and the Novikov higher signature conjecture. Guoliang Yu's property A is a weak amenability-type condition that is satisfied by many known metric spaces. It implies the existence of a coarse embedding into a Hilbert space. We construct the first example of a uniformly discrete metric space with bounded geometry which coarsely embeds into a Hilbert space, but does not have property A. This is a joint work with Erik Guentner and Jan Spakula.

Mladen Bestvina
   University of Utah
   Title: Outer space and the work of Karen Vogtmann
Abstract: I will recall the definition of Culler-Vogtmann's Outer space and review some of the work of Karen Vogtmann on Out(F_n).

Brian Bowditch
   University of Warwick
   Title: Models of 3-manifolds and hyperbolicity
Abstract: We give an outline of how various geometric models have been used to understand hyperbolic 3-manifolds in relation Teichmuller space. It is possible to characterise models in terms Gromov hyperbolicity. The situation is well understood in the bounded geometry case, and we suggest ways in which this can be generalised.

Thomas Brady
   Dublin City University
   Title: Climbing elements in finite Coxeter groups
Abstract: Suppose we have a fixed total order on the reflections of a finite Coxeter group W. We will call an element w in W a climbing element if w has a reduced expression whose corresponding sequence of reflections is increasing in the total order. We give a characterisation of climbing elements for certain fixed total orders. This is joint work with Aisling Kenny and Colm Watt.

Martin R Bridson
   Oxford University
   Title: On the geometry and rigidity of Out(F_n)
Abstract: In this talk, I'll explain the main ideas and results in the papers that I have written with Karen Vogtmann about Out(F_n). Manifestations of rigidity provide a unifying theme, as do comparisons with mapping class groups and SL(n,Z).

Ruth Charney
   Brandeis University
   Title: Automorphisms of right-angled Artin groups
Abstract: Automorphism groups of free groups have many properties in common with automorphism groups of free abelian groups. Interpolating between these are the automorphism groups of right-angled Artin groups. In joint work with Karen Vogtmann, we have shown that many of these properties hold for all such groups. I will give an overview of our techniques and results.

Indira Chatterji
   Universite d'Orleans
   Title: An explicitly finitely presented very lazy group
Abstract: We give an explicit finite presentation of a group normally generated by SL(∞, Z). As a consequence of a work by Bridson-Vogtmann, this group cannot act by homeomorphisms on any contractible finite dimensional manifold. This is joint work with Martin Kassabov.

James Conant
   University of Tennessee
   Title: The Cohomology of Out(F_n) and the Eichler-Shimura Isomorphism
Abstract: By work of Kontsevich, the rational cohomology of Out(F_n) can be studied via the cohomology of a certain infinite dimensional Lie algebra. The abelianization of this Lie algebra becomes quite useful in extracting cohomological information, and indeed, to this end, Morita made a conjecture about the abelianization many years ago. Recently we have discovered a general method for computing the abelianization, and it turns out that there is much more than what Morita had guessed. I will explain this result and show how the Eichler-Shimura isomorphism, which connects the cohomology of SL(2,Z) with modular forms, can be used to establish the next piece of the abelianization beyond Morita's. (Joint work with Martin Kassabov and Karen Vogtmann)

Yves de Cornulier
   University of Rennes
   Title: On Lie groups whose Dehn function is polynomial
Abstract: We describe the class of connected Lie groups (in particular, polycyclic groups) whose Dehn function is polynomial. In particular, it includes some groups with non-simply connected asymptotic cone. This is joint work with R. Tessera.

Cornelia Drutu
   Mathematical Institute, Oxford
   Title: Divergence and order of thickness
Abstract: The divergence is an invariant initially defined and used by Gersten to classify Haken and non-positively curved manifolds. I shall explain how it connects to the order of thickness and illustrate this with an example of compact CAT(0) space with fundamental group having divergence polynomial of order n and order of thickness n. Other applications are estimates of divergence for Out(F_n), and Teichmueller space. This is joint work with Jason Behrstock.

Benson Farb
   University of Chicago
   Title: Representation theory and homological stability
Abstract: We introduce the idea of representation stability for a sequence of representations V_n of groups G_n. One main goal is to expand the important and well-studied concept of homological stability so that it will apply to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood-Richardson rule, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (Lefschetz representations associated with rank-selected posets), to counting problems in analytic number theory. This is joint work with Tom Church.
Mark Feighn
   Rutgers University
   Title: Definable and Negligible Subsets of Free Groups
This is joint work with Mladen Bestvina. There is a condition called negligibility on subsets of a free group F and we have conjectured that definable (in the sense of the first order theory of F) subsets of F either are negligible or have negligible complement. A positive resolution of the conjecture would answer some old questions of Mal'cev and Rips-Sela. We discuss recent progress on this conjecture.

Victor Gerasimov
   Univeridade Federal de Minas Gerais, Belo Horizonte
   Title: Geometry of convergence group actions
Abstract: Sometimes dynamics yields geometry. So, every minimal convergence group action on a perfect metrisable compactum which is cocompact on triples is isomorphic to the action of a hyperbolic group on its boundary [Bowditch]. To prove this one needs to express geometric notions in terms of the topology and of the action. Further, every minimal convergence action of a finitely generated group on a perfect metrisable compactum which is cocompact on pairs is isomorphic to the action of a relatively hyperbolic group on its boundary [Yaman]. Any dynamically quasiconvex subgroup of a hyperbolic group is geometrically quasiconvex [Bowditch]. The isometry group of a proper hyperbolic metric space M acts on its Gromov compactification [Gromov] with convergence property. The problem whether any convergence action has a "geometric" nature is still open. For a wide class of "geometric" convergence actions (it includes the class of geometrically finite actions, the class of actions of a group on its Floyd completion; it is closed under taking quotients, pullbacks and inverse limits; actually we have no examples of non-geometric actions) we derive a "hyperbolic-like" theory motivated by the "delta-hyperbolic" geometry. One of the basics of the theory is a lemma by A. Karlsson [Ka03]: "a sufficiently far geodesic is arbitrary small with respect to the Floyd metric." We generalize this lemma replacing `geodesic' by `alpha-distorted curve' (a generalization of quasi-geodesics). We apply the theory to pull back a `relatively hyperbolic structure' along a quasi-isometric map (even along an "alpha-distorted map"). We study relations between dynamical quasiconvexity, geometrical relative quasiconvexity and quasiconvexity with respect to Floyd metrics. In particular we prove the conjecture that the dynamical and the geometrical quasiconvexities are equivalent for relatively hyperbolic groups [Osin]. We complete the generalization of the Floyd theorem to relatively hyperbolic groups. If the acting group is not finitely generated the traditional methods that use Cayley graphs are not applicable because the change of generating set is not a quasi-isometry. We omit this difficulties using "beetwinness relations" instead of (quasi-)geodesics in Cayley graphs. This approach does not require restrictions on the group. The acting group can be even uncountable and the compact space acted upon can be non-metrisable. We also prove that every relatively hyperbolic group (in the dynamical sense) is a union of a "coherent" family of finitely generated relatively hyperbolic groups. Moreover, it is relatively finitely generated. Finally, to illustrated our methods we give short proofs of some known results.


Etienne Ghys
   ENS Lyon
   Title: Uniformization of Riemann surfaces : a glimpse of history
Abstract: The uniformization theorem has a long and complicated history. For algebraic curves, it was formulated by Klein and Poincaré around 1881. I would like to give an outline of this history, mainly focusing on Poincaré's approach which leads to an "almost convincing proof" in August 1881.
Vincent Guirardel
   Universite Paul-Sabatier, Toulouse
   Title: Strata and stabilizers of trees in the boundary of outer-space
Abstract: Consider T an R-tree in the boundary of outer-space. We introduce a notion of "admissible" subtree of T that somewhat mirrors the existence of strata for a relative train-track map. There are only finitely many orbits of admissible subtrees, and they are canonical. We deduce a structure result for the stabilizer of T in Out(F_N) that relates it to McCool groups, i.e. stabilizers of sets of conjugacy classes in Out(F_N).

Ursula Hamenstaedt
   Universität Bonn
   Title: Geometric properties of Outer space and Out(F_n)
Abstract: Motivated by results for the mapping class group, we give a construction of families of quasi-geodesics in Outer space (for the bilipschitz metric) and Out(F_n) and discuss some applications.

Michael Handel
   CUNY, Lehman College
   Title: Subgroup classification in Out(F_n)
Abstract: (Joint work with Lee Mosher) Our main theorem is that for any subgroup H of Out(F_n), either H has a finite index subgroup that fixes the conjugacy class of some proper, nontrivial free factor of F_n, or H contains a fully irreducible element. Progress toward a relative version of this result will also be discussed.

Allen Hatcher
   Cornell University
   Title: Stable Homology by Scanning: Variations on a Theorem of Galatius
Abstract: Galatius' theorem determines the homology of the automorphism group of a free group in a stable range of dimensions by studying spaces of embedded graphs in high-dimensional Euclidean spaces. This talk will be about modifications and enhancements of these techniques that can be used to determine, among other things, the stable homology of the mapping class group of a 3-dimensional handlebody, which is a subgroup of the mapping class group of a surface. The modifications also add some transparency to the proof of Galatius' theorem itself.

Seonhee Lim
   Seoul National University
   Title: Counting orbits of geometrically finite groups acting on hyperbolic spaces
Abstract : We will see two problems, one in rigidity theory and one in number theory, that look different but uses similar techniques, namely group action on a hyperbolic space and measures induced on the boundary of the space. This is a joint work with Hee Oh.

Alex Lubotzky
   Hebrew University
   Title: The dynamic of Aut(Fn)-action of presentations and representations
Abstract: We will survey and compare various results on the action of Aut(Fn) on Hom(F_n,G) when G is a finite group, a compact group or a semisimple (real or p-adic) Lie group. This comparison suggests some questions and conjectures. .

Lee Mosher
   Rutgers-Newark
   Title: Distorted and undistorted subgroups of Out(F_n) (joint with Michael Handel)
Abstract: We shall prove that if [A] is the conjugacy class of a proper, nontrivial, rank r free factor A of F_n then its stabilizer subgroup Stab[A] is undistorted in Out(F_n) if and only if r=n-1. We shall also generalize the r=n-1 case by proving that if [T] is the conjugacy class of a free splitting of F_n (a minimal, simplicial F_n tree with trivial edge stabilizers) then its stabilizer subgroup Stab[T] is undistorted in Out(F_n).

Saul Schleimer
   University of Warwick
   Title: On train track splitting sequences
Abstract: We will give a structure theorem for train track splitting sequences. This, and a theorem of Masur and Minsky, proves that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce the notions of induced tracks, efficient position, and wide curves.

Zlil Sela
   Hebrew University

   Title: JSJ decompositions and varieties over (free) semigroups.
Abstract: We study sets of solutions to systems of equations over a free semigroup. To do that we construct analogues of the JSJ decomposition over groups, and a Makanin-Razborov diagram that encodes the entire set of solutions. Parametric families of such varieties are analyzed as well.