Could somebody teach me about the relationship, if any, between hyperbolicity in groups (in Gromov sense) and hyperbolicity in 3-dimensional orbifolds? To be more specific, let Q be a 3-dimensional orbifold. If the orbifold fundamental group of Q is a hyperbolic group (in Gromov sense), then can we say that Q is a hyperbolic orbifold? 



Relationship between hyperbolicity in group theory and hyperbolicity in geometry
[See Peter Scott's Bulletin article for more information.] Typically, we say an orbifold $Q$ is hyperbolic if it comes to us as a quotient of hyperbolic space $H^n$ by the action of a discrete group $G$ of isometries. If the action $G$ is cocompact then $G$ will be a Gromov hyperbolic group. This is the "easy direction".


On the other hand, if $Q$ is an orbifold with enough topological hypotheses (for example, dimension three, irreducible, "good" as Agol says, perhaps more...) then, if the orbifold fundamental group of $Q$ is Gromov hyperbolic it follows from the geometrization theorem (Perelman and so on) that $Q$ is orbifold homeomorphic to a quotient as in the first paragraph. 


So, roughly, the two notions are equivalent. However one direction is easy -- it follows from basic definitions in the field of coarse geometry -- and the other direction is one of the most famous recent results in mathematics. 


