Theorem 8: Let be a -hyperbolic group with respect to . If are conjugate then there exists such that
where depends only on .
Proof: We work in . Let be such that . Let be such that . We want to find a bound on .
Let . By Lemma 9,
So . Thus . Suppose that . By the Pigeonhole Principle there exist integers such that . It follows that one can find a shorter conjugating element by cutting out the section of between and .
Recall, for , is the centralizer of .
Theorem 9: If is -hyperbolic with respect to and , then is quasi-convex in .
Proof: Again we work in . Let , . We need to prove that is in a bounded neighborhood .
Just as in the proof of Theorem 8,
Well, and are conjugate. By Theorem 8 there exists such that
But so that and .
Exercise 15: Prove that
is not hyperbolic for any Anosov .