Small hyperbolic polyhedra
  We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic turnover, we classify the triangle subgroups of the fundamental groups of orientable 3-dimensional hyperbolic tetrahedral orbifolds in the case when all of the vertices of the tetrahedra are non-finite, and we provide a conjectural classification of all the triangle subgroups of the fundamental groups of orientable 3-dimensional hyperbolic polyhedral orbifolds. Finally, we show that any triangle subgroup of a (non-orientable) 3-dimensional hyperbolic reflection group arises from a triangle reflection subgroup. 
Shawn Rafalski
math.GT