Abstracts of Talks at the Cornell Topology Festival, May 5-7, 2000

STEPHEN BIGELOW

BRAID GROUPS ARE LINEAR

The braid groups B_n were introduced by Artin in 1925 as a tool for studying knots. Since then, they have been given many equivalent definitions, and found a wide variety of applications. I will discuss some representations of braid groups by groups of matrices, and outline the proof that one such representation is faithful for all braid groups.

MARTIN BRIDSON

SUBGROUPS OF SEMIHYPERBOLIC GROUPS

In this lecture I shall discuss recent progress on the question of which groups occur as fundamental groups of compact non-positively curved (orbi)spaces. In the same vein, I shall present results that exemplify the difference between this class of groups and the class of groups that occur as finitely presented subgroups of groups in this class. Within this context, I shall discuss various classes of free-by-cyclic groups, aperiodic tilings, and the following concepts of dimension.

 

Let G be a group that is the fundamental group of a compact non-positively curved (orbi)space. Define dim_cc⁰ G to be the minimum dimension of such a space. One might consider semisimple actions instead of cocompact ones and hence study dim_ss⁰ G.

CHRIS CONNELL

VOLUME GROWTH RIGIDITY

When can a locally symmetric metric g on a manifold M be characterized among all metrics by a single number? Besson, Courtois and Gallot showed this to be the case when M is closed and g has rank one. I will present some similar rigidity results in different contexts with applications to geometry and topology. For instance, we will take a look at some dynamical consequences for the geodesic flow and determine the minimum volume for certain manifolds. I will mainly concentrate on the first theorem in this direction for a class of higher rank symmetric spaces in joint work with Benson Farb.

ROSS GEOGHEGAN

SL(2) ACTIONS ON THE HYPERBOLIC PLANE

Take a positive integer m and consider the action of SL(2, Z[1/m]) on the hyperbolic plane by Moebius transformations. More generally, consider a not-necessarily-discrete action of a discrete group G on a non-positively curved (i.e. CAT(0)) space M. Are there interesting topological invariants of such an action - invariants which distinguish one from another?

 

I will describe a new kind of "controlled topology" invariant of such an action - a measure of how highly connected the action is. In particular, for the SL(2, Z[1/m]) case it will turn out that the action is (s - 2)-connected but not (s - 1)-connected, where s is the number of different primes which divide m. The most interesting feature of the proof of this is the role played by the Borel subgroup of upper triangular matrices. (Joint work with Robert Bieri.)

THOMAS MARK

CURVE-COUNTING AND SEIBERG-WITTEN INVARIANTS OF THREE-MANIFOLDS

We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold with first Betti number greater than 1 to an invariant that "counts" gradient flow lines--including closed orbits--of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the Seiberg-Witten invariants of 3-manifolds by making use of a "topological quantum field theory."

JOHN ROE

AMENABILITY AND ASSEMBLY MAPS

This talk is about the Baum-Connes conjecture, which is an analytic counterpart to the Borel conjecture about topological rigidity and its generalizations to isomorphism conjectures about various kinds of assembly maps. The Baum-Connes conjecture has implications in geometry, in unitary representation theory, and in pure algebra. In the last year some results which are close to being definitive have been established about the Baum-Connes conjecture: on the positive side, one now knows that some sort of generalized amenability is sufficient to imply the conjecture, and, on the negative side, one now knows that when amenability fails in a strong enough way, there are counterexamples (though so far only to certain extended versions of the conjecture, not to its original statement). I will give a survey of the work of Gromov, Higson, Lafforgue, Roe, Skandalis, Tu and Yu which led to these recent results, and I will speculate on the future for assembly isomorphism conjectures in analysis and in topology.

MARK SAPIR

SOME APPLICATIONS OF HIGMAN EMBEDDINGS

This is joint work with A. Olshanskii.

 

In 1961, Higman proved that every finitely generated recursively presented group can be embedded into a finitely presented group. We shall discuss applications of different variations of Higman embeddings to several outstanding problems in group theory and topology.

KEVIN WHYTE

LARGE SCALE GEOMETRY OF GRAPHS OF GROUPS

Gromov proposed that finitely generated groups be studied as geometric objects. We discuss the geometry of several classes of graphs of groups. In particular, we study the geometry of accessible groups, and the geometry of the Baumslag-Solitar groups. We will see that similar looking graphs of groups can have widely varying rigidity properties, and that subtle large scale dynamics are enter in an essential way.