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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 .
- 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.