**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,

Also

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 .

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