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Recall, that for any graph $\Gamma$ we built a combinatorial horoball $\mathcal{H}(\Gamma)$.  For a group $G$ and a collection of subgroups $\mathcal{P}=\{P_1,\ldots,P_n\}$and a generating set $S$, we built the augmented Cayley graph $X$ by gluing copies of $\mathcal{H}(\mathrm{Cay}(G))$.  $G$ is hyperbolic relative to $\mathcal{P}$ if and only if $X$ is Gromov hyperbolic.

Exercise 28: If $A$ and $B$ are finitely generated, then $A*B$ is hyperbolic relative $\{A,B\}$. (Hint: $X$ is a graph of spaces with underlying graph a tree and the combinatorial horoballs for vertex spaces.)

Example: Suppose $M$is a complete hyperbolic manifold of finite volume.  So, $\Gamma=\pi_1M$ acts on $\mathbb{H}^n$.  Let $\Lambda$ be a subset of $\partial\mathbb{H}^n$ consisting of points that are the unique fixed point of some element of $\Gamma$.  So $\Gamma$ acts on $\Lambda$, and there only finitely many orbits.  Let $P_1,\ldots,P_n$ be stabilizers of representatives from these orbits and let $\mathcal{P}=\{P_1,\ldots,P_n\}$.  Then, $\Gamma$ is hyperbolic relative to $\mathcal{P}$.

Example: Let $G$ be a torsion-free word-hyperbolic group.  Then, $G$ is clearly hyperbolic relative to $\{1\}$.  A collection of subgroups $P_1,ldots,P_n$ is malnormal if for any $g\in G$, $P_i\cap gP_jg^{-1}\neq1$ implies that $i=j$ and $g\in P_i$.  $G$ is hyperbolic relative to $\mathcal{P}=\{P_1,\ldots,P_n\}$ if and only if $\mathcal{P}$ is malnormal.

The collection of subgroups $\mathcal{P}$ is the collection of peripheral subgroups.

Lemma 31: If $G$ is torsion-free and hyperbolic relative to a set of quasiconvex subgroups $\mathcal{P}$, then $\mathcal{P}$ is malnormal.

Sketch of Proof: Suppose that $P_1\cap gP_2g^{-1}$ is infinite.  Consider the following rectangles:  Note that if $k=l(g)$, then $gP_2g^{-1}$ is contained in a $k$-neighborhood of $gP_2$.  Now, there exists infinite sequences $p_i\in P_1$ and $q_i\in P_2$ such that $d(p_i,gq_i)\leq k$.  Look at the rectangles with vertices $1, g, gp_i, p_i$.  The geodesics in $X$ between 1 and $p_i$ and $g$ and $gq_i$ go arbitrarily deep into the combinatorial horoballs.  Therefore, they are arbitrarily far apart.  It follows that these rectangles cannot be uniformly slim.

Let $\mathcal{N}=\{N_1,\ldots,N_n\}$ where each $N_i\lhd P_i$.  Write $G/\langle\langle\bigcup_iN_i\rangle\rangle=G/\mathcal{N}$.  Call this the Dehn filling of $G$.

Note: If $G$ is hyperbolic relative to $\mathcal{P}$, then $G$ is hyperbolic.

Theorem 21: (Groves-Manning-Osin). Suppose $G$ is hyperbolic relative to $\mathcal{P}$.  Then, there exists a finite set $A$ contained in $G\smallsetminus 1$ such that whenever $(\bigcup_i N_i)\cap A\neq\emptyset$ we have

1. $P_i/N_i\to G/\mathcal{N}$ is injective for all $i$, and
2. $G/\mathcal{N}$ is hyperbolic relative to the collection $\{P_i/N_i\}$;

In particular, if $P_i/N_i$ are all hyperbolic, then so is $G/\mathcal{N}$.

One application of this theorem is a simple proof of a theorem of Gromov, Olshanskii, and Delzant:

Theorem 22: Let $G$ be hyperbolic and suppose $\{\langle g_1\rangle,\ldots,\langle g_n\rangle\}$ is malnormal, with each $\langle g_i\rangle$ infinite.  Then, there is constant $K$ such that for all positive integers $l_1,\ldots,l_n$ there is an epimorphism to a hyperbolic group $\phi:G\to G'$ such that $o(\phi(g_i))=Kl_i$ for each $i$.

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 $S$.