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The recognition theorem for $$\text{Out}(F_n)$$. (English) Zbl 1239.20036
Let $$F_n$$ be the free group of rank $$n$$. For an automorphism $$\Phi$$ of $$F_n$$ let $$\varphi$$ denote the corresponding outer automorphism of $$F_n$$. Thus $$\Phi\in\operatorname{Aut}(F_n)$$ represents $$\varphi\in\text{Out}(F_n)$$.
Let $$R_n$$ be the rose with one vertex $$*$$ and $$n$$ petals. A marked graph $$G$$ is a graph of rank $$n$$ all of whose vertices have valence at least two, equipped with a homotopy equivalence $$m\colon R_n\to G$$. If $$b=m(*)$$, it is obtained an identification of $$F_n$$ with $$\pi_1(G,b)$$. A homotopy equivalence $$f\colon G\to G$$ and a path $$\sigma$$ from $$b$$ to $$f(b)$$ determines an automorphism of $$\pi_1(G,b)$$ and hence an automorphism $$\Phi$$ of $$F_n$$. This automorphism depends only on the homotopy class of $$\sigma$$. So, modulo the inner automorphisms of $$F_n$$, the $$f\colon G\to G$$ represents $$\varphi\in\text{Out}(F_n)$$.
Let $$G$$ be a marked graph with marking $$m\colon R_n\to G$$ and $$\Gamma$$ its universal cover. The set $$\mathcal E(\Gamma)$$ of ends of $$\Gamma$$ is naturally identified with the boundary $$\partial F_n$$ of $$F_n$$. If $$f\colon G\to G$$ represents $$\varphi\in\text{Out}(F_n)$$, then a path $$\sigma$$ from $$b$$ to $$f(b)$$ determines both an automorphism representing $$\varphi$$ and a lift $$\widetilde f\colon\Gamma\to\Gamma$$ of $$f\colon G\to G$$. This defines a bijection between the set of lifts $$\widetilde f$$ of $$f$$ and the set of automorphisms $$\Phi$$ of $$F_n$$ representing $$\varphi$$. Therefore $$\widetilde f$$ is determined by $$\Phi$$. Under the identification of $$\mathcal E(\Gamma)$$ with the $$\partial F_n$$ of $$F_n$$, a lift $$\widetilde f$$ determines a homeomorphism $$\widehat f$$ of $$\partial F_n$$. An automorphism $$\Phi$$ also determines a homeomorphism $$\widehat\Phi$$ of $$\partial F_n$$ and $$\widehat f=\widehat\Phi$$ if and only if $$\widetilde f$$ is determined by $$\Phi$$.
A point $$P\in\partial F_n$$ is an attractor for $$\widehat\Phi$$ if it has a neighborhood $$U$$ such that $$\widehat\Phi(U)\subset U$$ and $$\bigcap_{n=1}^\infty\widehat\Phi^n(U)=P$$. If $$Q$$ is an attractor for $$\widehat\Phi^{-1}$$, then it is a repeller for $$\widehat\Phi$$.
For the universal cover $$\Gamma$$ of a marked graph $$G$$ the space of lines in $$\Gamma$$ is endowed with a weak topology. Then the graph $$G$$ is equipped with the quotient topology. A closed set of lines in $$G$$ or a closed $$F_n$$-invariant set of lines in $$\Gamma$$ is called lamination. To each $$\varphi\in\text{Out}F_n)$$ is associated a finite $$\varphi$$-invariant set of laminations $$\mathcal L(\varphi)$$ called the set of attracting laminations for $$\varphi$$.
The relative train track maps have been introduced by M. Bestvina and M. Handel [in Ann. Math. (2) 135, No. 1, 1-51 (1992; Zbl 0757.57004)], in the present paper they play a central role and here the authors elaborate this notion and prove a relevant theorem (Theorem 2.19 in the paper).
Let $$f\colon G\to G$$ be a relative train track map representing $$\varphi\in\text{Out}(F_n)$$. For $$\Phi\in\operatorname{Aut}(F_n)$$ representing $$\varphi$$, let $$\text{Fix}_N(\widehat\Phi)\subset\text{Fix}(\widehat\Phi)$$ be the set of non-repelling fixed points of $$\widehat\Phi$$. $$\Phi$$ is a principal automorphism if either of the following hold.
$$\bullet$$ $$\text{Fix}_N(\widehat\Phi)$$ contains at least three points.
$$\bullet$$ $$\text{Fix}_N(\widehat\Phi)$$ is a two point set that is neither the set of endpoints of an axis $$A_c$$ nor the set of endpoints of a lift $$\widetilde\lambda$$ of $$\mathcal L(\varphi)$$. The corresponding lift $$\widetilde f\colon\Gamma\to\Gamma$$ of $$f$$ is a principal lift.
The set of principal automorphisms representing $$\varphi$$ is denoted $$P(\varphi)$$.
If $$\Phi$$ is a principal lift of $$\varphi$$, then $$\Phi^k$$ is a principal lift of $$\varphi^k$$ and $$\text{Fix}_N(\widehat\Phi)\subset\text{Fix}_N(\widehat\Phi^k)$$ for every $$k\geq 1$$. The set of all non-repelling periodic points in $$\text{Per}(\widehat\Phi)$$ is denoted $$\text{Per}_N(\widehat\Phi)$$. An outer automorphism $$\varphi$$ is forward rotationless if $$\text{Fix}_N(\widehat\Phi)=\text{Per}_N(\widehat\Phi)$$ for all $$\Phi\in P(\varphi)$$ and if for each $$k\geq 1$$ $$\Phi\mapsto\Phi^k$$ defines a bijection between $$P(\varphi)$$ and $$P(\varphi^k)$$.
Now we are in the position to state the main Theorem of this paper.
Theorem: (Recognition Theorem) Suppose that $$\varphi,\psi\in\text{Out}(F_n)$$ are forward rotationless and that
(1) $$PF_\Lambda(\varphi)=PF_\Lambda(\psi)$$, for all $$\Lambda\in\mathcal L(\varphi)=\mathcal(\psi)$$; and
(2) there is a bijection $$B\colon P(\varphi)\to P(\psi)$$ such that:
(i) (fixed sets preserved) $$\text{Fix}_N(\widehat\Phi)=\text{Fix}_N(\widehat{B(\Phi)})$$; and
(ii) (twist coordinates preserved) if $$w\in\text{Fix}(\Phi)$$ and $$\Phi,i_w\Phi\in P(\varphi)$$, then $$B(i_w\Phi)=i_wB(\Phi)$$.
Then $$\varphi=\psi$$.
Here $$i_w$$ denotes the inner automorphism of $$F_n$$ induced by $$w$$ and, for each $$\Lambda\in\mathcal L(\varphi)$$, $$PF_\Lambda\colon\text{Stab}(\Lambda)\to\mathbb Z$$ is a homomorphism such that $$\psi\in\text{Ker}(PF_\Lambda)$$ if and only if $$\Lambda\not\in\mathcal L(\psi)$$ and $$\Lambda\not\in\mathcal L(\psi^{-1})$$ (Lemma 2.14 in the paper).
The paper is based on previous articles, mainly on the previously mentioned one and on [M. Bestvina and the authors, Ann. Math. (2) 151, No. 2, 517-623 (2000; Zbl 0984.20025)] and on [D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, Duke Math. J. 93, No. 3, 425-452 (1998; Zbl 0946.20010)]. But here the authors go further in the study of some notions and develop new results.

##### MSC:
 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
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##### References:
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