Special topics course on Geometric Group Theory, offered in Spring 2021 by Dan Margalit.
Text: Groups, Graphs, and Trees, John Meier
Supplementary text: Office Hours with a Geometric Group Theorist, Matt Clay and Dan Margalit
Geometric group theory is the study of group actions on geometric spaces. A basic theorem in the subject is: if a group acts freely on a tree then the group is free. Geometric group theory is an exciting and active area of research that is very accessible to undergraduates.
The course will roughly follow Meier's textbook: background on group theory and group actions, Cayley graphs, groups generated by reflections, groups acting on trees, Baumslag-Solitar groups, the word problem, lamplighter groups, Thompson groups, hyperbolic groups.
In our algebra sequence, the students don't see a lot of interesting infinite groups. Geometric group theory is a study of this wide, beautiful, and wonderful world.