You can cut a cake in half, a pizza into slices, but can you cut an infinite group? I'll tell you about my sensei's secret cutting technique and demonstrate a couple of examples. There might be some spectral goodies at the end too. 

We will describe aspects of the geometry of non-minuscule rigid analytic period domains and their covering spaces, and pose some questions about p-adic period mappings and period images by analogy with the complex analytic theory.