Geometric Group Theory Workshop at MathFest
August 12, 2004, from 3:15 to 5:15
Geometric group theory freely exploits techniques from various branches of mathematics (often geometry and topology) in order to study the structure of discrete, infinite groups. This workshop consists of four talks intended to introduce central aspects of geometric group theory to a broad audience. If you come in the door knowing the definition of a group, by the end of this session you will know everything there is to know about geometric group theory!*
This is an Invited Paper Session at the MAA's MathFest in Providence, RI.
The Talks
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Ted Turner (Albany)
Curvature in dimension two: a case study in geometric group theory
Geometric group theory is an approach to the study of discrete groups using ideas from geometry. We will discuss what is meant by `curvature' and `dimension' for discrete groups and describe the very different properties of groups of dimension two that have negative curvature as opposed to those that have non-positive curvature.
Kim Ruane (Tufts) Infinite groups and geometry - a match made in heaven!
Peanut butter and jelly, popcorn and a movie, beer and pizza - some of the more familiar matches made in heaven. Add to this list: In dimension one, free groups and trees; in dimension two, surface groups and hyperbolic space; and in all dimensions free abelian groups and Euclidean spaces. And this is just the start! There are several important theorems that illustrate just how beautifully these two fields compliment each other. In this talk, we will discuss some of these theorems to help introduce techniques used in Geometric Group Theory.
Jennifer Taback (Bowdoin) Dead end words and other anomalies
Geometric group theory is a fascinating field of mathematics which utilizes geometric descriptions and properties of groups to obtain both algebraic and geometric consequences. For example, given a group G, with a finite set of generators S, there is a canonical way to create a "picture" of this group, called a Cayley graph. If you change the set of generators, you might get a different picture! I will give some examples of Cayley graphs of groups, and of interesting phenomena that can occur in Cayley graphs. Examples of groups I will discuss are the Baumslag-Solitar groups and the lamplighter group. An element of the lamplighter group can be envisioned as an infinite string of light bulbs, some of which are illuminated, and a cursor which tells you the light bulb you are considering.
Ken Brown (Cornell) Amenability of groups
The concept of amenability was introduced into group theory in 1929 by von Neumann, in an effort to understand the Banach-Tarski paradox. Now, 75 years later, the concept remains mysterious in many ways. For example, it can be quite difficult to decide whether or not a given group is amenable. In this talk I will give a survey of amenability, leading up to some open questions.
The Organizers
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Have any questions? Please contact either
Philip Hotchkiss
or
John Meier.
*The organizers of the workshop have no idea how this sentence has appeared on the webpage, and they should in no way be held responsible for any claims implicit or explicitly stated here.