Learning about Geometric Groups Summary
Monday - Friday, July 9 - 13, 2012
Our working group has continued grappling with the problems of Geometric Group Theory. In the lectures this week, we formalized the idea of a group presentation, proved that a group is a free group if and only if it acts freely on a tree, and began to explore property FA, a property of some groups that act on trees so that at least one element of the tree is globally fixed. Now the class is focusing more on infinite groups, from simple examples like the infinite dihedral group to the exceedingly strange Baumslag-Solitar groups (which we learned can serve as discrete models of hyperbolic space). In class, we encountered the same troublesome group introduced by Martin Bridson in his Clay lecture, a group which seems at first glance to be related to the Baumslag-Solitar groups and to share their intricate structure, but upon close examination is revealed to be simply the trivial group.
This week saw a more intensive and formal presentation of the material, with the proof of one theorem taking up all of one class period and a good portion of another. We are starting to deal with metrics, including the word metric, and are running into some of difficulties that trouble researchers in the field, including how to decide if a word is really the identity ("the word problem") or whether two words in a group are actually equivalent ("the equality problem"). For some groups, these problems are solvable, but for others, they are not. Even groups with a finite amount of generators and relations can have unsolvable word problems, which seems counter-intuitive. Up until now, even though their has been an emphasis on trees and Cayley graphs, we have been focused mostly on groups as algebraic structures, but with the introduction of the notion of a metric, we can now start to explore their geometry.
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.