on Saturday, May 3, 2003

at the Cornell Topology Festival

May 20, 2003

This year’s Cornell Topology Festival devoted about one third of its
talks to the area of Geometric Group Theory, with another large cohort of speakers
talking about low-dimensional topology and knot theory. In addition to their
individual talks, the speakers were asked to conduct a panel discussion. In
particular, each speaker was asked to prepare a description of some recent mathematics
which had caught his or her interest, outside of the speaker’s own work.
**Karen Vogtmann**, one of the Cornell organizers of the Festival
as well as a speaker, served as moderator for the discussion. She asked speakers
to limit their remarks to “one blackboard’s length”.

In this summary of the panel discussion, a panelist’s name is indicated in bold-face type, whereas names of mathematicians whose work receives significant mention are indicated in italic.

Not surprisingly, most panelists chose to remark about work that highlighted some connection between geometric group theory / low-dimensional topology and other areas of mathematics.

**Dror Bar Natan** of Hebrew University in Jerusalem and the University
of Toronto had begun the conference with a talk on the connection between knot
theory and certain algebraic structures, namely, quandles and Lie algebras.
He further emphasized role that the “algebraic sciences” could play
in illuminating knot theory by leading offthe panel discussion with some remarks
on *Khovonov*’s categorification of the Jones polynomial. “Khovonovification”,
as Bar Natan refers to the theory, associates to a knot a complex of vector
spaces which yield homological invariants stronger than the Jones polynomial.
Bar Natan pointed out that despite its promise for insight into knot theory,
as yet mathematicians understand very little about the subject.

**Joan Birman** of Columbia University then described recent work
on mapping class groups by *C. Leininger* which may provide information
about a conjecture of Lehmer involving certain monic polynomials. Leininger
has analyzed a construction due to *W. Thurston* of subgroups in the
mapping class group of a surface generated by positive multi-twists and has
given a graph theoretic characterization of when such subgroups are free. Leininger
shows that the dilatation of a pseudo-Anosov map contained in a subgroup generated
by two positive multi-twists is bounded below by the constant known as Lehmer’s
number (which we define below). Moreover, Leininger constructs a particular
pseudo-Anosov map which factors as a product of two positive multi-twists whose
dilatation factor realizes Lehmer’s number. Lehmer’s number is theMahler
measure of a certain monic integer polynomial of degree 10, and it is conjectured
that Lehmer’s number is a lower bound for the Mahler measure of all such
monic polynomials.

**Benson Farb** of the University of Chicago added to Birman’s
discussion by pointing out a connection between between Leininger’s work
and certain Coxeter groups as studied by *C. McMullen*. There is a natural
surjection from a certain hyperbolic Coxter group onto the mapping class group
of a genus 5 surface for which the image of the Coxeter element is Leininger’s
example realizing Lehmer’s number, indicating some connection between
the eigenvalues of elements of Coxeter groups and pseudo-Anosov dilatation factors.

Farb went on to describe a deep theorem of *A. Nabutovsky* and *S.
Weinberger* which provides infinitely many examples of local minimum values
of the diameter functional from the space of a certain class Riemannian metrics
on a manifold of dimension 5 or greater into the real numbers. Their proof gives
a connection between the geometry near local minima of the diameter functional
and the complexity of the word problem for groups.

**Fred Cohen**, from the University of Rochester gave an outline
of a recent work of *Ib Madsen* and *Michael Weiss* in which they
compute the integral cohomological structure of the stable mapping class group.
He emphazised that this was a very promising line of research and further asked
what happens to their construction if one tries to apply it to the automorphism
group of a free group, as those are known to share similarities with mapping
class groups. For a more detailed account of Cohen's remarks, click
here to download a PDF file.

In fact, the work of Madsen and Weiss builds on techniques previously developed
by Madsen and one of the panelists,**Ulrike Tillmann** of Oxford
University. Tillmann added to Fred Cohen’s remarks by explaining some
theorems of a book by *Kiyoshi Igusa*, as well as a result by *Allen
Hatcher* on the stable homology of the automorphism group of a free group.
Both rely on the homotopy machinery developed for the study of high dimensional
manifolds and use techniques and results from Waldhausen K-theory. In the case
of the mapping class group, the stable homotopy theoretic approach highlighted
in Cohen’s discussion has by now completely determined the stable homotopy
of mapping class groups. For a more detailed account of Tillmann's remarks,
click here to download a PDF file.

**Gilbert Levitt**, from Universit´e Paul Sabatier in France
explained the concept of random groups, introduced by *M. Gromov*, and
outlined some related questions and results. “What are the properties
of a randomly chosen finitely presented group? The answer to this question of
course depends on the method used for the random choice.” wrote Etienne
Ghys in his recent contribution on the subject for the S´eminaire Bourbaki.
“Gromov’s work highlights the existence of finitely presented groups
with astonishing properties.” Levitt particularly focused on the question
of groups with two generators and asked what can be said about such a group
for “most” choices of relations. He gave a result of Gromov which
characterizes when the group is trivial or else word hyperbolic, according to
a fixed “density” value in the chosen “randomness” or
density model.

**John Meier** from Lafayette College shared a few quotes about
group theory, including the following excerpt from an article in “The
Morning Call”, a local newspaper, on April 20, 2003: “Topology is
the study of geometric groups.” Meier continued the discussion of finitely
presented groups with astonishing properties, by emphasizing work by *Daniel
Farley*. Starting from a very simple diagram, Farley can describe fairly
complicated groups such as Thompson’s groups. Moreover Farley highlights
some geometrical properties of those groups, called diagram groups, such as
non-positive curvature. A recent theorem of *Victor Guba* and *Mark
Sapir* gives a new approach to the study of diagram groups, as well as a
topological interpretation.

**Justin Roberts** of the University of California at San Diego
described recent work by *Y. Taylor* and *C. Woodward* on quantum
6j-symbols, which are the simplest example of quantum invariants of three-dimensional
manifolds and can be thought of as somehow being invariants of the tetrahedra
which make up the manifold. To quote Woodward himself, their work sheds light
on “the relationship between non-Euclidean geometry...and the representation
theory of quantum groups”. Taylor and Woodward use spherical tetrahedra
to produce an explicit formula for these 6j-symbols. Roberts noted that they
also have a version of the formula in the case of hyperbolic tetrahedra. Taylor
and Woodward’s formulas generalize Roberts’ own work on classical
6j-symbols.

Up to this point in the panel discussion, various speakers had alluded to the
recent announcement by *G. Perelman* of a possible solution to the famous
*Geometrization Conjecture of W. Thurston*, but had declined to address
the issue. However, **Dylan Thurston** of Harvard University took
the plunge and began with a short description of the conjecture itself as well
as a very general overview of how Perelman’s possible solution builds
on *R. Hamilton*’s program for using Ricci flows as an approach
to the Geometrization Conjecture. Thurston then asked the audience to entertain
the “If Perelman is correct” scenario, and presented some problems
for mathematicians in the field to consider in that case. For example, he suggested
a search for a combinatorial model for Ricci flow. He also described a connection
with the Nabotovsky-Weinberger result mentioned previously in the panel discussion
by Farb and suggested that the methods used in the 5 (and higher)-dimensional
case ought to be compared to the 3-manifold situation. Finally, he asked the
following: what is the complexity of recognizing the 3-sphere? Is it NP-complete?
In support of this guess, he cited a theorem due to *I. Agol, J. Hass*,
and *W. Thurston* which states that the problem of finding the genus
of a knot is NP-hard.

Finally, **Alain Valette** of Universit´e de Neuchˆatel
outlined recent work by *Yehuda Shalom* in which he proves that several
well-known algebraic or analytic features such as Betti numbers are actually
geometric when restricted to some classes of amenable groups, in the sense that
they are quasi-isometry invariants. Conversely, Shalom also showed that some
features of certain groups which are, a priori, geometric in nature, such as
having a uniformly embedded amenable group, yield a lower bound on the rational
cohomological dimension.

Karen Vogtmann then thanked the speakers for their participation in the panel, and declined her share of the blackboard because of the length of the session. She proposed that further discussion take place at the Topology Festival picnic, which received the unanimous approval of the audience.