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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 .

Today we will see some methods of constructing groups.

**Definition.** Let be groups and let and be injective homomorphisms. If the diagram below is a pushout then we say write and we say that is the *amalgamated (free) product of and over .*

**Example.** If , we write and say is the *free product of and .*

As usual, we need to prove existence.

**Recall. **If is a group, then the *Eilenberg-MacLane Space* satisfies the following properties:

- is connected;
- ;
- for .

**Facts.**

- exists;
- The construction of is functorial;
- is unique, up to homotopy equivalence.

For as above, let and realize as a map and as a map . Now, let , where . By the Seifert-Van Kampen theorem, . Suppose that , and . Then,

.

In particular, if is finitely generated, then so is , and if are finitely presented and is finitely generated, then is finitely presented.

**Example.** Let be a connected surface and let be a separating, simple closed curve. Let . Then,

But, what if is non-separating (but still 2-sided)? Then, there are two natural maps representing , where . Associated to , we have a map , , which maps a curve to its signed (algebraic) intersection number with .

Let be a covering map corresponding to . Then,

This has a shift-automorphism . We can now recover :

**Defintion.** If are injective homomorphisms, then let

Let be the shift automorphism on . Now, is called the *HNN (*Higman, Neumann, Neumann) *Extension of over .* We often realize as , where and . It is easy to write down a presentation:. is called a *stable letter*.

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