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 .