A group is a collection of objects with a binary operation that satisfies certain natural conditions: associativity, identity, inverses. Groups arise naturally as a way to encode symmetries of geometric objects such as manifolds, graphs, trees, cell complexes (which are higher dimensional versions of trees and graphs). Conversely, given a set of rules that define the operation of a group, one can construct a geometric object (the Cayley graph) whose symmetries form the group. Geometric group theory expoits this connection between groups and geometry to study groups.
The goal of this particular workshop is to compile an annotated list of problems in the field. Prior to the workshop, twenty different sub-areas were designated and people assigned to write drafts of problem lists for each area. Most of the drafts were ready in time for the workshop, and copies made for the participants. The activities at the workshop involved analyzing these drafts, prioritizing questions, filling in history and background details, and making sure there were no important omissions.
A fundamental result of the field that gives a flavor for the types of questions asked and the methods employed is the relationship between a certain class of groups introduced by Michael Gromov (called "Gromov hyperbolic groups") and a famous question of Max Dehn known as the "Word Problem". Gromov hyperbolic groups are a generalization of groups acting on hyperbolic space---they are certain groups described by specifying generators and a set of rules that the operation must satisfy. For example, if one forms a "random" example by declaring some set of generators and some single product of generators to be the identity element of the group, then the resulting group is almost always a Gromov hyperbolic group.
Given any group described by generators and multiplication rules, the Word Problem asks if there is an algorithm to decide if a given product of generators is the identity or not. It turns out that Gromov hyperbolic groups are precisely the groups for which the Word Problem has a solution algorithm that has linear-time complexity.
The Geometric Group Theory workshop was unusual for AIM in that its primary focus was not to produce solutions to problems, but instead to catalogue existing mathematics and to chart new directions for the future. It should lead to important new initiatives. There will be a program at the Mathematical Sciences Research Institute in Berkeley in the Fall of 2007, which will, in part, build on the work done here in Palo Alto.
One of the important byproducts of the Geometric Group Theory workshop is a "wiki" page for problems in the subject. Modeled after the OnLine encyclopedia called Wikipedia, this will be a Web page---available at first only to workshop participants---that can be developed and embellished in real time by OnLine users. Such an innovation will set a new standard for mathematical communication.