Series: Mathematics Colloquium

Date: Thursday, November 29, 2001

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Edward Formanek

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Simon Thomas, Rutgers University

Title: The Asymptotic Geometry of Finitely Generated Groups

Abstract: 

In this talk, I will discuss some of the basic notions of Geometric
Group Theory.  The talk is aimed at a general mathematical audience
and there are essentially no prerequisites.  Geometric Group Theory
constitutes the third wave of combinatorial group theory.  In the
first wave, combinatorial group theorists worked directly with group
words.  Then came the realisation that more fun could be had and more
progress made if they instead pretended to be doing something else.
In the second wave, they pretended to be doing very low dimensional
topology.  In the third wave, following the example of Gromov, they
are pretending to be doing geometry; i.e., they are regarding finitely
generated groups as metric spaces and studying their ``large-scale''
or ``asymptotic'' geometries.  For example, if an observer moves
steadily away from the Cayley graph of a finitely generated group,
then any finite configuration will eventually become indistinguishable
from a single point; but he may observe certain finite configurations
which closely resemble earlier configurations.  The asymptotic cone is
a topological space which encodes all of these recurring finite
configurations.  Unfortunately the actual construction of an
asymptotic cone involves a number of non-canonical choices, and it was
not clear whether the resulting asymptotic cone depended on these
choices.  Towards the end of this talk, answering a question of
Gromov, I will present an example of a finitely generated group which
has two non-homeomorphic asymptotic cones.