I've been reading Sullivan's paper "Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups", and I was hoping I could get some clarification regardig concepts that come up in section 7, where he sets out to show Kleinian groups satisfying certain properties are convex cocompact (i.e. geometrically finite and without parabolics). I would be happy with a reference to somewhere where this material is presented in more detail, since I suspect I'm missing background knowledge.
Let $\Gamma \subset \mathrm{PSL}(2,\mathbb{C})$ be a finitely generated non-elementary Kleinian group, and $\Lambda(\Gamma) \subset \partial \mathbb{H}^3, \Omega(\Gamma) = \partial \mathbb{H}^3 – \Lambda(\Gamma) \subset \partial \mathbb{H}^3$ be the limit and ordinary sets for $\Gamma$, respectively. Then, $\Gamma$ acts properly discontinuously by Moebius transformations on $\Omega$, so the quotient $\Omega/\Gamma$ is a hyperbolic Riemann surface, which is of finite type by Ahlfors' finiteness theorem. That is, $\Omega/\Gamma = S= \bigcup_{i=1}^r S_{g_i,n_i}$, where $g_i$ and $n_i$ are the genus and number of punctures of $S_{g_i,n_i}$. I'll refer to S as the ideal boundary at infinity of $M = \mathbb{H}^3/\Gamma$.
On the other hand, by the Scott core theorem, there exists a compact submanifold $C \subset M$ (the compact core of $M$) such that the inclusion $C \to M$ is a homotopy equivalence. Let $E = \bigcup_{j=1}^n E_j$ be the closure of the connected components of $M – C$ (where the $E_j$ are the connected components of $E$). Then, we have that $\partial E_j$ is a compact surface, for each $j = 1, \dots, n$. Using excision, you can show that the homology groups of $E_j$ coincide with those of $\partial E_j$, so all the $\partial E_j$ are connected surfaces. I'll refer to the $E_j$ as the ends of $M$.
So, the overarching question is (I'll number them for convenience):
- How do the surfaces $\partial E_j$ relate to the $S_{g_i,n_i}$?
For example, I know that if $M$ has rank two cusps, then you cannot "see" these in the ideal boundary (the intersection of a lift of the cusp with a fundamental domain only intersects $\partial \mathbb{H}^3$ at a limit point), but you should be able to "see" it in the ends of $M$, i.e. some of the $\partial E_j$ should be tori corresponding to these rank two cusps.
More specifically:
-
How do the ideal boundary components $S_{g_i,n_i}$ where $n_i \neq 0$ look in terms of the ends of $M$? Since all the $\partial E_j$ are compact surfaces, none of them can be homeomorphic to $S_{g_i,n_i}$, but it seems to me that they should show up in some way.
-
Do all ideal boundary components of the form $S_{g_i,0}$ have a unique component $\partial E_{j(i)}$ corresponding to them?
Thanks in advance!
Best Answer
Here's a rough idea of what is going on. And I'm going to restrict my comments to the torsion free case which simplifies the language (and, I think, is implicit in your question anyway).
Let $\mathcal H$ be the convex hull of $\Lambda(\Gamma)$, and consider the quotient $\mathcal H / \Gamma$ which is embedded in $M$. Outside of some exceptional situations (e.g. Fuchsian groups), $\mathcal H / \Gamma$ is homeomorphic to $M$, and the difference between them, by which I mean the closure $\overline{M - \mathcal H / \Gamma}$, is homeomorphic to $\partial M \times [0,1]$. So, in particular, we have a rather natural homeomorphism $S \approx \partial \mathcal H / \Gamma$. (One can even get around the exceptions by replacing $\mathcal H$ with its $\epsilon$-neighborhood, as is explained in work of Thurston that one can find developed in work of Epstein and Marden; but I will stick with this simpler notation.)
Of course, under the assumption that $S$ is not compact, i.e. the number of punctures $n_i$ is nonzero for some $i$, it follows that $\mathcal H / \Gamma$ is not compact. So $\mathcal H / \Gamma$ is the "convex core" of $M$, but not yet its compact core.
To get the actual compact core of $M$, you now look at the "cusps" of $\mathcal H / \Gamma$. The compact core itself is obtained by cutting off each cusp and adding a new boundary portion associated to that cusp. Cusps are of two types: $\mathbb Z \oplus \mathbb Z$ cusps; and $\mathbb Z$ cusps. When you cut off a $\mathbb Z \oplus \mathbb Z$ cusp of $\mathcal H / \Gamma$, you just add a new boundary component homeomorphic to the torus. But when you cut of a $\mathbb Z$ cusp of $\mathcal H / \Gamma$, what you add is a cylinder that connects up two circle cusps of the Riemann surface $S = \partial \mathcal H / \Gamma$.
So to summarize, via the inclusions $C \subset \mathcal H / \Gamma \approx M$, one can think of $S$ as embedded in $\partial C$, in such a way that the complement $\partial C - S$ has components in one-to-one correspondence with the cusps of $\mathcal H / \Gamma$: a torus component for each $\mathbb Z \oplus \mathbb Z$ cusp; and an annulus component for each $\mathcal Z$ cusp.