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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 .
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 .
Finally, we are in a position to prove that a hyperbolic group has no subgroup isomorphic to .
Theorem 11. Let with . Then .
Proof. By Lemma 10, we can assume that is not conjugate to any element of length by replacing with a power of itself. Suppose . We need to bound .
Replacing with for some , we may assume that . We will be done if we can bound .
Suppose . By dividing into triangles, we see that any geodesic rectangle is -slim, in the same way that triangles are -slim.
Because the rectangle with vertices is -slim, there exists such that .
If , then , a contradiction. Similarly . So . Therefore, .
But . This is a contradiction since we assumed that is not conjugate to anything so short. Therefore . Thus .
An element of a group is torsion if its order is finite.
A group is torsion if every element is torsion.
A group is torsion-free if no nontrivial elements are torsion.
Corollary. Every non-trivial abelian subgroup of a hyperbolic group is virtually cyclic.
Lemma 11. Let be a torsion-free hyperbolic group. Whenever is not a proper power, then is malnormal.
Definition. A subgroup of a group is malnormal if for all , , then .
Remark. By Theorem 11, if is hyperbolic and torsion-free, centralizers are cyclic.
Proof of Lemma. Suppose .
Therefore for some , .
By Lemma 10, . Therefore . Thus . Therefore .
Exercise 17. Prove that if where is hyperbolic and torsion-free and and and , then . That is, is commutative transitive.
We now turn briefly to a fundamental open question about hyperbolic groups. This question is a theme of the course.
Question. Is every word-hyperbolic group residually finite?
The fundamental groups of hyperbolic manifolds are linear, so residually finite by Selberg’s Lemma.
What about for negative curved manifolds?
Theorem (Sela). Every torsion-free hyperbolic group is Hopfian.
Theorem (I. Kapovich-Wise). If every nontrivial hyperbolic group has a proper finite-index subgroup, then every hyperbolic group is residually finite.
Theorem (Agol-Groves-Manning). If every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable.