Geometric Structures on 3-manifolds

I will first describe a simple graph coloring problem and survey some examples of graphs for which the coloring problem has or has no solution. I will then give a quick introduction to Bestvina-Brady Morse theory. Finally, I will describe the relationship between the coloring problem and some amusing virtual algebraic fibering consequences for geometric group theory and hyperbolic manifolds. This is joint work with Kasia Jankiewicz and Sergey Norin.