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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 .
Theorem 12 (Gromov): Let be torsion-free -hyperbolic group. If such that , then for all sufficiently large , .
Remark: The torsion-free hypothesis is not necessary, but it allows us to avoid some technicalities. For instance, it is a non-obvious fact that an infinite hyperbolic group contains a copy of .
For the rest of this lecture will be a torsion-free -hyperbolic group, where are primitive (i.e. not proper powers).
Recall that for torsion-free -hyperbolic, primitive implies that .
If and do not commute we can show there is some point on arbitrarily far from .
Hence we have the following lemma.
If and do not commute there is some point on arbitrarily far from .
Proof: Suppose not. That means such that such that . So is in . But the Cayley graph is locally finite so has finitely many elements. By the Pigeonhole Principle such that for some . Then . But then . .
For a moment view and as the horizontal and vertical geodesics in . For two points on and on , we can argue that the geodesic between them curves toward the origin.
And so we have Lemma 14.
Lemma 14: There exists such that , .
Recall that by is a quasi-isometric embedding. So by Theorem 6, and
By Lemma 13 choose such that
. Choose such that . Now, must be -close to so for some point on the geodesic between and , . Then .
For a subgroup , one can choose a closest point projection which is -equivariant. (Write . Choose where and are close and declare to be -equivariant.) is typically not a group homomorphism.
We’re interested in and .
In , there is some such that either or .
Lemma 15: such that , or .
Let . WLOG, is -close to and since is the closest point to (in particular compared to ). So . .
Now we can prove the theorem.
Proof of Theorem 12:
The idea is to use the Ping-Pong Lemma on the Cayley graph.
Let and let , where is provided by Lemma 15. For all we have and likewise for all we have . In particular, .
Let . By -equivariance,
for any . In particular,
by the triangle inequality. Similarly,
for all and all . Because and are quasi-isometrically embedded, it follows that and for .
Therefore, by the Ping-Pong Lemma .