In a solution to a recent post : [Fundamental group of a thick part of hyperbolic manifold](https://mathoverflow.net/questions/54401/fundamental-group-of-a-thick-part-of-hyperbolic-manifold), Igor Belegradek makes this claim that the thick part of a hyperbolic manifold is connected. To me it seems like the thick part of a hyperbolic annuli ($\mathbb{H}$ quotiented by the group of isometries generated by $z\mapsto r\_oz$, for some $r\_0\in \mathbb{R}$) itself will be disconnected for sufficiently small $r\_0$. Please forgive me if i am blatantly wrong. If not, then i would like to know under what extra conditions we can say that the thick part is connected. 



Connectedness of the thick part of a hyperbolic manifold?
In any dimension the thin part of a hyperbolic manifold $M$ is the union of regular neigborhoods of short geodesics and of cusps. This follows from the Margulis lemma. Since removing cusps is the same as removing collars of the boundary of a compact manifold, the only way for the thin part to disconnect $M$ is if the neighborhood of a geodesic (or collection of geodesics) separates $M$. This only happens when the geodesic is codimension one, that is, when M has dimension two.


