MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
The claim is for dimensions $>2$. Sorry for the confusion. – Igor Belegradek Feb 5 '11 at 18:24
up vote 7 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.