2007 Abstracts and Timetable

Tomasz Elsner's page on Simplicial Nonpositive Curvature

Goulnara Arjantseva : Metrics on R. Thomson group F and uniform embeddings in a Hilbert space

Using the representation of F as a diagram group, we will prove the Haagerup property of F and show how to embed a Cayley graph of F into a Hilbert space with minimal possible distortion. We will also construct a natural CAT(0) cubical complex on which F acts.

Pierre-Emmanuel Caprace: Amenable groups and Hadamard spaces with a totally disconnected isometry group

The class of amenable locally compact groups enjoys remarkable closure properties with respect to algebraic operations, such as taking quotients or closed subgroups,* or forming group extensions. However, despite of this nice algebraic behaviour, the interaction between the amenability of a given group and the algebraic structure of that group is still not completely understood. It is well known that a connected locally compact group is amenable if and only if it is solvable-by-compact. We present a similar characterization for subgroups of a totally disconnected group acting properly and cocompactly on a locally compact Hadamard space.

Ruth Charney: Foldable cube complexes, hyperbolicity, and Artin groups There is a well-known condition, the "no-squares condition," which allows a CAT(0) cube complex to be deformed into a CAT(-1) complex. We show that for foldable cube complexes, a weaker condition suffices, and we apply this to prove that many Artin groups are (weakly) relatively hyperbolic with respect to their finite type parabolic subgroups.
(Joint with John Crisp.)

Indira Chatterji: Median spaces

Median spaces are metric spaces so that given any 3 points, there exists a unique median point, that is a point that is on 3 geodesics between those 3 points.
We will explain how median spaces are a natural generalisation of CAT(0) cube complexes and sketch how property (T) and the Haagerup property can be caracterized using median spaces.
This is joint work with C. Drutu and F. Haglund.

Tomasz Elsner : Flats in systolic spaces

Studying possible configurations of flats in a non-hyperbolic systolic complex with a geometric group action gives a lot of information about the complex and the group. Considering not only flats, but also flat minimal surfaces, we obtain a systolic version of the Flat Torus Theorem and systolic analogues of C.Hruska's results on CAT(0)-spaces with isolated flats (for systolic spaces one has a natural modification of the Isolated Flats Property). In particular, a cocompact systolic complex has isolated flats if and only if it contains no triplanes.

Frederic Haglund : Special Cubical Groups

Right-angled Coxeter groups act geometrically on a specific CAT(0) cube complex, the Davis complex. In this talk we study special (cubical) groups, that is subgroups of right-angled Coxeter groups that are convex cocompact on the Davis complex.
Firstly special groups enjoy excellent properties:

• quasi-convex subgroups of special groups are virtual retracts, hence separable
• a special group is hyperbolic iff it has no abelian subgroup of rank >1

Secondly the world of special groups is wider than one might expect:

• every Coxeter group is virtually special
• every compact arithmetic real hyperbolic manifold of simple type has a virtually special fundamental group

This is joint work with Dani Wise.

Jon McCammond : Instability in triangle-square complexes

One of the more natural candidates for a theory that would encompass both nonpositively curved cube complexes and simplicial nonpositive curvature is the study of (suitably restricted) complexes built out of direct products of regular simplices. And a natural test case for the theory is whether one can prove a (bi)automaticity result using a language that generalizes the natural languages in the two motivating cases. Rena Levitt and I have been investigating the 2-dimensional base case of such a grand unification program and the results are somewhat subtle and surprising. My talk will mostly focus on nonpositively curved 2-complexes built out of triangles and squares with some comments on their higher dimensional generalizations.

Graham A. Niblo : Amenability, Property A and CAT(0) cube complexes

Yu's property A is a generalisation of the notion of amenability for metric spaces and groups encoded via "approximate" weighted Folner sets. In joint work with Sarah Campbell, I showed that groups acting properly and co-compactly on a CAT(0) cube complex have sufficiently well controlled embeddings in Hilbert space that they must satisfy property A, but the proof was not constructive in that we did not define the weighting functions directly. In recent work with Brodzki, Guentner and Wright we were able to do so, showing that any finite dimensional CAT(0) cube complex has property A without any assumption on co-compactness, nor even local finiteness. In addition we used the method to show that if a group acts properly on a finite dimensional CAT(0) cube complex then the stabiliser of any point in the cubical boundary is amenable, and therefore virtually abelian, which parallels a classical result for actions on buildings.

Michah Sageev : CAT(0) cubical complexes and group theory

We will discuss CAT(0) cubical complexes and their connection to various other topics in geometric group theory. The plan is roughly as follows:

• A brief discussion of CAT(0) spaces in general and why we care about them
• Polyhedral CAT(0) spaces and the link condition
• CAT(0) cubical complexes and Gromov's criterion
• Examples
• A quick trip through Bass-Serre theory
• CAT(0) cubical complexes and codimension-1 subgroups (spaces with walls)
• Applications of the the previous topic may include: the Tits Alternative, small cancellation groups and Coxeter groups
• The l²-embedding and the relationship between CAT(0) cubical complexes and Property T and the Haagerup property
• Thompson's group (and diagram groups)

Jacek Świątkowski Systolic complexes

Simplicial nonpositive curvature is a purely combinatorial notio (applicable to simplicial complexes) which resembles metric nonpositive curvature. It has been introduced recently in my joint work with Tadeusz Januszkiewicz (and some aspects independantly by F. Haglund). Having no direct realtionship with metric nonpositive curvature, the notion has numerous similar consequences. For example, simplicially nonpositively curved complexes are aspherical, their fundamental groups are biautomatic (and thus semi-hyperbolic), and simplicially nonpositively curved complexes of groups are developable. Moreover, a slightly stronger variant of the concept yields Gromov-hyperbolicity.
Developability of simplicially nonpositively curved complexes of groups allows construction of examples, also in higher dimensions. This leads to solutions of open problems concerning existence of:

• hyperbolic Coxeter groups with arbitrary vcd,
• CAT(0) developments of simplicial billiard tables of any dimension,
• simple criterion for Gromov-hyperbolicity of simplicial complexes of arbitrary dimension, and many others.

Simplicial nonpositive curvature does not fit well to manifolds of dimension above 2. The corresponding spaces and groups in dimensions above 2 turn out to have rather exotic properties. Their further study seems worthwile, as well as looking for further applications and generalizations.
In the course I am going to introduce the concept of simplicial nonpositive curvature, prove its basic properties and consequences, describe construction of examples, discuss applications and show some exotic properties in high dimensions.