10/15: Right-Angled Coxeter Groups — Adrienne Nolt
Abstract:
10/1: Growth Rates in $\mathrm{CAT}(0)$ Cube Complexes — Lydia Ahlstrom
Abstract: A $\mathrm{CAT}(0)$ cube complex (CCC) is a collection of Euclidean unit cubes of various dimensions that are glued together along their faces so that the resulting space is $\mathrm{CAT}(0)$ (a generalization of non-positive curvature) and simply connected. Once we have a clear understanding of what CCC’s are, we will learn about hyperplanes – a useful tool in the study of such spaces. We will then compare the exponential growth rate of the volume of balls in a CCC to the exponential growth rate of the hyperplane count.
9/24: The Ping-Pong Lemma — Eleanor Rhoads
Abstract: The core question of geometric group theory is: what can a group acting on a space tell us about the group, or the space? The Ping-Pong Lemma is a classic example of this, as it lets us recognize whether a group is free by how it acts. We’ll prove the Ping-Pong Lemma (for 2 players), talk about why one might care about free groups, see why it’s called a lemma, and then discuss some other examples of classic “groups acting on spaces” results. Note: This talk assumes no background beyond group theory that appears in undergraduate algebra.
9/17: Heavenly Elliptic Curves over Cubic Fields — Suzanne O’Hara
Abstract: The study of heavenly elliptic curves is motivated by a question of Ihara which compares Galois extensions generated by arithmetic and geometric methods. Knowing an elliptic curve $E/K$ is heavenly an odd prime $\ell$ lets us deduce extra information about the representation of the absolute Galois group $G_K$ on the $\ell$-torsion of the curve. Using this, we deduce a formula for mod $\ell$ trace of $E$ at primes $p$ distinct from $\ell$. Previous work of Rasmussen and Tamagawa has shown that for a fixed number field $K$ of degree at most $3$, there are on finitely many $K$-isomorphism classes of heavenly elliptic curves when we let $\ell$ vary through all primes. Recently, McLeman and Rasmussen have studied the set of heavenly elliptic curves where both the quadratic field of definition and the prime $\ell$ are allowed to vary. In this talk, we discuss analogues of their results to the cubic field case and show a further classification of heavenly elliptic curves which have a “CM-like” trace.