** "Geometry and Analysis on Groups" Research Seminar **

**Title:**
Reflection dynamics on Euclidean Coxeter groups.

**Speaker:**
Jon McCammond (University of California, Santa Barbara)

**Abstract:**
The collection of irreducible Euclidean Coxeter groups (also know as affine Coxeter groups) are indexed by the extended Dynkin diagrams and they act nicely on a metric simplicial complex isometric to some Euclidean space. Despite being a core example of a group acting on a space in both geometric group theory and geometric combinatorics, there are still mysteries to unravel. In this talk I will describe a relatively new geometric way to compute the reflection length of an element in one of these Coxeter groups (i.e. combinatorial distance in the Cayley graphs generated by the infinite set of all of its reflections). This is joint work with Joel Lewis, Kyle Petersen, and Petra Schwer. Time permitting, I will also highlight the way in the which the combinatorial notion of a parking function can be given a purely dynamical characterization (which is joint work with Hugh Thomas and Nathan Williams).