Affine Deligne-Lusztig varieties and the geometry of Euclidean reflection groups

Let $G$ be a reductive group such as $SL_n$ over the field $k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the affine Weyl group of $G$.  The associated affine Deligne-Lusztig varieties $X_x(b)$ were introduced by Rapoport.  These are indexed by elements $x$ in $G$ and $b$ in $W$, and are related to many important concepts in algebraic geometry over fields of positive characteristic.  Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension.  We use techniques inspired by geometric group theory and representation theory to address these questions in the case that $b$ is a translation.  Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns.  Since we work only in the standard apartment of the building for $G$, which is just the tessellation of Euclidean space induced by the action of the reflection group $W$, our results also hold over the p-adics.  This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).