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 .