I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a paper of Bowditch) the following. Let $M^n$ be a negatively curved manifold, which is obtained as a quotient $X/\Gamma$, where $X$ is a pinched Hadamard manifold and $\Gamma \subseteq \text{Isom }X$ is discrete. Then the convex core of $M$ is defined as $C(M) := H/\Gamma$, where $H$ is the closed convex hull of the limit set of $\Gamma$.
Now, consider the "thick-thin" decomposition of $M$ as $M = M_{\text{thick}} \sqcup M_{\text{thin}}$. As mentioned by Bowditch (immediately following the definition), the intersection $C(M) \cap M_{\text{thick}}$ is compact. My question is, are the following statements correct?
- The complement of $C(M) \cap M_{\text{thick}}$ is a disjoint union of cusps and funnels, each of which is a warped product of the form $N^{n - 1} \times [1, \infty)$, having finite and infinite volumes respectively. "Most part" of each cusp (funnel) is inside $M_{\text{thin}}$ ($M_{\text{thick}}$ respectively). However, the cusps are contained inside the convex core, while the funnels are not.
- The fundamental group of each cusp and each funnel is amenable.
- The group $\Gamma$ must have parabolic elements for cusps, and loxodromic elements for funnels.
I have put together an attempt to understand the concepts around geometric finiteness in my own way. If some statements are wrong, please excuse me and please point them out. I am unable to find a nice enough reference which has all these in one place, so that a novice like me can understand. Thanks!
