Some intuition: Recall that if M is a closed hyperbolic manifold
then \pi_1(M) is word-hyperbolic. However, a lot of interesting hyperbolic manifolds are not closed.

Example: Let K\subset S^3 be the figure 8 knot.

figure 1

Then the complement M_{8}=S^{3} K admits a complete hyperbolic metric and is of finite volume.

So, here we have an example of a hyperbolic manifold which is not compact but is of finite volume. This is almost as which is almost as natural as being closed.

M_{8} is homotopy equivalent to M_{8}', the complement of a thickened K in S^{3}.

fig2

M_8' is a compact manifold with boundary and its interior admits a hyperbolic metric. The boundary of M_8' is homeomorphic to a 2-torus, so \partial M_8' \hookrightarrow M_8' induces a map \mathbb{Z}^2\hookrightarrow\pi_1M_8' . By Dehn’s lemma, the map is injective so \pi_1M_8' cannot be word hyperbolic. The point is that \pi_1M_8 acts nicely on \mathbb{H}^2 but no cocompactly so the Svarc=Milnor lemma does not apply.

The torus boundary component of M_8' corresponds to a cusp of M_8.

The point is that we can use cusped manifolds like M_8' to build a lot of manifolds and in particular a lot of hyperbolic manifolds.

Take M_8' and a solid Torus T .

Choose a homeomorphism \phi: \partial M_8' \hookrightarrow\partial T

Definition: The manifold M_{\phi}=M_{8}'\cup_{\phi}T is obtained from M_{8}' by Dehn filling .

We now want to understand what we have done to \pi_{1}M_{8}. The map \phi induces a map \phi_{*}:

lecture_4_17_09_xymatrix1

The surjectivity of \phi_{*} follows from the fact that \phi is a homeomorphism. The Seifert Van Kampen theorem implies that \pi_{1}M_{\phi}=\pi_{1}M_{8}\langle\langle \ker(\phi_{*})\rangle\rangle, where \langle\langle\ker(\phi_{*}) \rangle\rangle denotes the normal closure of \ker(\phi_{*}) .

Gromov-Thurston 2\pi theorem: Let M be any compact hyperbolic manifold and \partial_{0}M be a component of \partial M homeomorphic to a 2-torus for all but finitely many choices of

lecture_4_17_09_xymatrix2

the Dehn filling M_{\phi} is hyperbolic.

Note: by finitely many we mean finitely many maps up to homotopy.

This is a very fruitful way of building hyperbolic manifolds. The next question to ask is whether we can do the same thing for groups. So, now we will try to develop a group theoretic version of this picture.

Let \Gamma be a group theoretic graph with the induced length metric. Construct a new graph \mathcal{H}(\Gamma) called the combinatorial horoball on \Gamma as follows: Define the vertices V(\mathcal{H})=V(\Gamma)\times \mathbb{N}. There are two sorts of edges in {E}(\mathcal{H}). We say that (u,k) and (v,k) are joined by a (horizontal) edge if d_{\Gamma}(u,v) \leq 2^{k} and u\neq v. We say that (v,k) and (v,k+1) are joined by a (vertical) edge for all k.

fig6
For k large enough u' and v' will have distance one and L\leq 1 iff 2^{k} \ge d_{\Gamma}(u,v) iff k\leq \log_{2}d_{\Gamma}(u,v).

Exercise 27:
(A). For u,vin V(\Gamma), d_{\mathcal{H}}((u,0),(v,0))\approx \log_{2}d_{\Gamma}(u,v).

(B). For any connected \Gamma, \mathcal{H}(\Gamma) is Gromov hyperbolic .

fig7

Let G be a group and let \mathcal{P}=\{ P_{1},\ldots, P_{n} \} be a finite set of finitely generated subgroups of G. Choose a finite generating set S for G such that for each i, s_i=S \cap P_i generate P_i. Then \mathrm{Cay}(G,S) contains natural copies of \mathrm{Cay}(P_{i},S_{i}).

Construct the augmented Cayley graph X=X(G,\mathcal{P},S) by gluing on combinatorial horoballs equivariantly.

X(G,\mathcal{P},S) = \mathrm{Cay}(G,S) \cup \bigcup_{i} \lbrack \mathcal{H}(\mathrm{Cay}(P_{i},S_{i})) \times G/P_{i} \rbrack / \sim where for each i
and each gP_{i}\in G /P_{i}, \mathcal{H}(\mathrm{Cay}(P_{i},S_{i}) \times \{ gP_{i}\} is glued to g\mathrm{Cay}(P_{i},S_{i}) along \mathrm{Cay}(P_{i},S_ {i}) \times \{ 0 \} .

Definition: G is hyperbolic rel \mathcal{P} if and only if X(G,P,S) is Gromov hyperbolic for some (any) choice of S.

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