Hyperbolic turnovers

A hyperbolic turnover is the orbifold analog of the hyperbolic 3—holed sphere. It is a theorem that the presence of an immersed (but non-embedded) hyperbolic turnover T in a hyperbolic 3—orbifold Q places strong restrictions on the topology and geometry of a 3—dimensional hyperbolic suborbifold of Q that contains T (called the turnover core). We will explore ramifications of this, sketch the proof, and describe work in progress to classify the possibilities for immersed turnovers in orbifold quotients of 3—dimensional hyperbolic reflection groups.