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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 .
Some intuition: Recall that if is a closed hyperbolic manifold
then is word-hyperbolic. However, a lot of interesting hyperbolic manifolds are not closed.
Example: Let be the figure 8 knot.
Then the complement admits a complete hyperbolic metric and is of finite volume.
So, here we have an example of a hyperbolic manifold which is not compact but is of finite volume. This is almost as which is almost as natural as being closed.
is homotopy equivalent to , the complement of a thickened in .
is a compact manifold with boundary and its interior admits a hyperbolic metric. The boundary of is homeomorphic to a 2-torus, so induces a map . By Dehn’s lemma, the map is injective so cannot be word hyperbolic. The point is that acts nicely on but no cocompactly so the Svarc=Milnor lemma does not apply.
The point is that we can use cusped manifolds like to build a lot of manifolds and in particular a lot of hyperbolic manifolds.
Take and a solid Torus .
Choose a homeomorphism
Definition: The manifold is obtained from by Dehn filling .
We now want to understand what we have done to . The map induces a map :
The surjectivity of follows from the fact that is a homeomorphism. The Seifert Van Kampen theorem implies that , where denotes the normal closure of .
Gromov-Thurston theorem: Let M be any compact hyperbolic manifold and be a component of homeomorphic to a 2-torus for all but finitely many choices of
the Dehn filling is hyperbolic.
Note: by finitely many we mean finitely many maps up to homotopy.
This is a very fruitful way of building hyperbolic manifolds. The next question to ask is whether we can do the same thing for groups. So, now we will try to develop a group theoretic version of this picture.
Let be a group theoretic graph with the induced length metric. Construct a new graph called the combinatorial horoball on as follows: Define the vertices . There are two sorts of edges in . We say that and are joined by a (horizontal) edge if and . We say that and are joined by a (vertical) edge for all .
For large enough and will have distance one and iff iff .
(A). For , .
(B). For any connected , is Gromov hyperbolic .
Let be a group and let be a finite set of finitely generated subgroups of . Choose a finite generating set for such that for each , generate . Then contains natural copies of .
Construct the augmented Cayley graph by gluing on combinatorial horoballs equivariantly.
where for each
and each /, is glued to along .
Definition: G is hyperbolic rel if and only if is Gromov hyperbolic for some (any) choice of .