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. 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 $S^{3}$.  $M_8'$ 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 $\phi_{*}$: 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 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 $k$. 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 . 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 $S$.