Learning about Geometric Groups Summary

Monday - Friday, July 2 - 6, 2012

The purpose of the Geometric Group Theory Working Group ("Groups group") is to give teachers the opportunity to learn more higher-level mathematics in a traditional undergraduate course setting. Each day, the group attends a one-hour undergraduate lecture given by Jen Taback of Bowdoin College, then spends an additional hour discussing the lecture, debriefing, and working on problem sets under the guidance of Brian Hopkins.

One of Jen's main goals is for students to develop an idea of what types of questions should be asked about groups, so that when we encounter an unfamiliar group, we have both algebraic and geometric tools we can use to help develop an intuition about the group and its properties. The first lecture developed the idea of what distinguishes geometry from topology, then explored some of the important theorems of group theory, in particular Cayley's Theorem. We constructed Cayley graphs of the integers based on different generating sets, then proceeded to construct graphs for other groups, including permutation groups, dihedral groups, alternating groups, and free groups. We proved some non-intuitive results, including that the free group F3 is a subset of the free group F2. We also used the Ping-Pong Lemma to prove that F2, despite seeming like a completely arbitrary bit of abstract algebra, actually is a subset of SL2(Z), a much more familiar group that acts on the same coordinate plane studied by high school students everywhere.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.