In a solution to a recent post : 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.
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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. 

