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 $i$.