Recall, that for any graph we built a combinatorial horoball . For a group and a collection of subgroups and a generating set , we built the *augmented Cayley graph* by gluing copies of . is hyperbolic relative to if and only if is Gromov hyperbolic.

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

**Example:** Suppose is a complete hyperbolic manifold of finite volume. So, acts on . Let be a subset of consisting of points that are the unique fixed point of some element of . So acts on , and there only finitely many orbits. Let be stabilizers of representatives from these orbits and let . Then, is hyperbolic relative to .

**Example:** Let be a torsion-free word-hyperbolic group. Then, is clearly hyperbolic relative to . A collection of subgroups is *malnormal* if for any , implies that and . is hyperbolic relative to if and only if is malnormal.

The collection of subgroups is the collection of *peripheral subgroups*.

**Lemma 31:** If is torsion-free and hyperbolic relative to a set of quasiconvex subgroups , then is malnormal.

**Sketch of Proof:** Suppose that is infinite. Consider the following rectangles: Note that if , then is contained in a -neighborhood of . Now, there exists infinite sequences and such that . Look at the rectangles with vertices . The geodesics in between 1 and and and go arbitrarily deep into the combinatorial horoballs. Therefore, they are arbitrarily far apart. It follows that these rectangles cannot be uniformly slim.

Let where each . Write . Call this the *Dehn filling* of .

**Note:** If is hyperbolic relative to , then is hyperbolic.

**Theorem 21:** (Groves-Manning-Osin). Suppose is hyperbolic relative to . Then, there exists a finite set contained in such that whenever we have

- is injective for all , and
- is hyperbolic relative to the collection ;

In particular, if are all hyperbolic, then so is .

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

**Theorem 22:** Let be hyperbolic and suppose is malnormal, with each infinite. Then, there is constant such that for all positive integers there is an epimorphism to a hyperbolic group such that for each .

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