Algebra Seminar talk
2025-10-10
Christopher Cashen
Asymptotic cones of snowflake groups
Abstract:
This will be a gentle introduction to Geometric Group Theory. I will give a quick sketch of Gromov’s Polynomial Growth Theorem saying that a finitely generated group with polynomial growth is virtually nilpotent. This is a prototype “geometric hypotheses with algebraic conclusions” theorem. Asymptotic cones will appear in the proof. Then I will talk about complexity of the Word Problem, and introduce the family of "snowflakes groups" that show that the Dehn function spectrum is dense. Finally, I’ll talk about join work with Hoda and Woodhouse where we exhibit an infinite family of snowflake groups G such that every asymptotic cone of G is simply connected, even though G has neither polynomial growth nor quadratic Dehn function. Furthermore, there is an asymptotic cone of G that contains an isometrically embedded circle. These are the first examples of groups that have “metrically nontrivial” loops even though they have no topologically nontrivial ones.