Learning about Geometric Groups Summary
Monday - Friday, July 16 - 20, 2012
In keeping with the theme of learning which questions to ask about groups, we began our final week of studying Geometric Group Theory by looking at three important problems posed by Dehn, one of the pioneers of the field: the word problem, the conjugacy problem, and the isomorphism problem. These are complicated questions of what type of information we can learn about a given group -- whether we can determine the length of an arbitrary word, or determine whether two group elements are conjugates, or determine if one group is isomorphic to another group. Although these seem like basic questions, they are in fact highly complex, and whether or not they can be solved is dependent on what group we are considering -- there is no successful algorithmic approach! We looked at some related questions that implied information about these important questions, for example the property of almost convexity (AC) and minimal almost convexity (MAC), conditions which implied a solvable word problem.
However, we learned of even more subtleties of these problems as we explored the infinite Lamplighter group. Imagine an infinitely long street, with lampposts at each integer point and a lamplighter who walks down the street, illuminating or extinguishing bulbs. Each element of this group consists of a series of illuminated bulbs, plus the position of the lamplighter. Though this group is not minimally almost convex, its word problem is solvable. We teachers even wrote a computer program that calculates the word length of any given element. We explored the notions of efficient paths and geodesics in this group, based on two different possible generating sets. Despite the somewhat exotic nature of the group, we learned that it is actually representable by 2 x 2 matrices! As opposed to previous approaches, we first looked at this group through algebra (including a discussion of semi-direct products and wreath products) before moving on to the geometric representation, which turned out to be a pair of trees with associated height functions called a Diestel-Leader graph.
It was interesting to see that not every graph has an associated finitely-generated group, a result which was definitely not obvious. Certain Diestel-Leader graphs (not those associated with Lamplighter groups) do not have finitely-generated groups that act on them, a theorem which was proven by a mathematician present at PCMI! It was a great experience to work on these problems and to see some of the important results of Geometric Group Theory. The material we learned in class strongly connected with the content of the Clay lectures and of the SSTP morning problem sets, and the experience of being a student again helped push everyone to some in-depth reflection on our own teaching practices. Go Groups Group!
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.