Automorphisms of free groups have finitely generated fixed point sets

The Recognition Theorem for $\mathrm{Out}(F_n)$

“…/ is completely split for each such . After k applications of Lemma 4.27 with j D r (see in particular item (7) of that lemma) we have that f jG s is completely split. This completes the induction step and so also the proof of the theorem.…”

“…The symmetric argument using a relative train track map for  1 proves that P is not a repeller so Lemma 2.3 completes the proof. Choose K greater than the number of edges with height s 0 in any indivisible Nielsen path for h. By [7] there is a positive constant C so that ifˇ1 ˇ2 are finite subpaths in G then g # .ˇ2/ G 0 contains the subpath of g # .ˇ1/ obtained by removing the initial and terminal segments of edge length C . Since generic leaves of ƒ are birecurrent and since the realization of ƒ in G 0 can not be contained in G Proof.…”

Fractal representation of the attractive lamination of an automorphism of the free group

“…As explained earlier, every inner automorphism extends naturally to ∂F N . More generally, it is proved in [20] that such an extension exists for every automorphism ϕ ∈ Aut(F N ); we denote by ∂ϕ : ∂F N → ∂F N the homeomorphism extending ϕ. The proof relies on the fact that an automorphism ϕ of F N is a quasi-isometry for the word metric associated with some given basis.…”

Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups

“…If it has only one orbit of periodic points, then this orbit has order two. (2) Suppose X ∈ ∂F n is periodic of period q under ∂α. Then q ≤ M n , where M n depends only on n and log M n ∼ √ n log n as n → ∞.…”

“…In particular, the natural action of Fix α on the set of regular fixed points of ∂α is free. This action has finitely many orbits [2], indeed it follows from [5] that the number of orbits is at most 4n. It is not clear to us whether there is a bound depending only on G when G is an arbitrary hyperbolic group.…”