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Ping-pong and outer space. (English) Zbl 1211.20027
An outer automorphism \(\varphi\) of a free group \(F_N\) is called reducible if there exists a free product decomposition \(F_N=C_1*\cdots*C_k*F'\), where \(k\geq 1\) and \(C_i\neq\{1\}\), such that \(\varphi\) permutes the conjugacy classes of subgroups \(C_1,\dots,C_k\) in \(F_N\), otherwise it is called irreducible.
An outer automorphism \(\varphi\in\text{Out}(F_N)\) is said to be irreducible with irreducible powers or an ‘iwip’ for short, if for every \(n\geq 1\) the power \(\varphi^n\) is irreducible. An outer automorphism \(\varphi\in\text{Out}(F_N)\) is hyperbolic or atoroidal if no positive power of \(\varphi\) fixes the conjugacy class of a nontrivial element of \(F_N\). An automorphism \(\Phi\in\operatorname{Aut}(F_N)\) is called hyperbolic or atoroidal if the outer automorphism \(\varphi\in\text{Out}(F_N)\) defined by \(\Phi\) is atoroidal.
In this paper the authors prove that, if \(\varphi,\psi\in\text{Out}(F_N)\) are hyperbolic iwips such that \(\langle\varphi,\psi\rangle\subseteq\text{Out}(F_N)\) is not virtually cyclic, then some high powers of \(\varphi\) and \(\psi\) generate a free subgroup of rank two for which all nontrivial elements are again hyperbolic iwips.
More precisely, their main result is: Let \(N\geq 3\) and let \(\varphi,\psi\in\text{Out}(F_N)\) be hyperbolic iwips such that the subgroup \(\langle\varphi,\psi\rangle\subseteq\text{Out}(F_N)\) is not virtually cyclic. Then there exist \(m,n\geq 1\) such that the subgroup \(G=\langle\varphi^m,\psi^n\rangle\subseteq\text{Out}(F_N)\) is free of rank two and such that every nontrivial element of \(G\) is again a hyperbolic iwip.
They end up with the important corollary: Let \(G\subseteq\text{Out}(F_N)\) be a subgroup which contains some hyperbolic iwip, and assume that \(G\) is not virtually cyclic. Then \(G\) contains a free subgroup of rank two where all nontrivial elements are hyperbolic iwips.
The reference list contains 49 articles.

MSC:
20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
20F28 Automorphism groups of groups
57M50 General geometric structures on low-dimensional manifolds
57M07 Topological methods in group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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