In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group Fn is an element of Out(Fn) which has no periodic conjugacy classes of proper free factors in Fn (where n > 1). Fully irreducible automorphisms are also referred to as "irreducible with irreducible powers" or "iwip" automorphisms. The notion of being fully irreducible provides a key Out(Fn) counterpart of the notion of a pseudo-Anosov element of the mapping class group of a finite type surface. Fully irreducibles play an important role in the study of structural properties of individual elements and of subgroups of Out(Fn).
Formal definition Let φ ∈ Out ( F n ) {\displaystyle \varphi \in \operatorname {Out} (F_{n})} where n ≥ 2 {\displaystyle n\geq 2} . Then φ {\displaystyle \varphi } is called fully irreducible if there does not exist an integer p ≠ 0 {\displaystyle p\neq 0} and a proper free factor A {\displaystyle A} of F n {\displaystyle F_{n}} such that φ p ( [ A ] ) = [ A ] {\displaystyle \varphi ^{p}([A])=[A]} , where [ A ] {\displaystyle [A]} is the conjugacy class of A {\displaystyle A} in F n {\displaystyle F_{n}} . Equivalently, for all p > 0 {\displaystyle p>0} and any proper free factor A {\displaystyle A} of F n {\displaystyle F_{n}} , φ p ( [ A ] ) ≠ [ A ] {\displaystyle \varphi ^{p}([A])\neq [A]} . Here saying that A {\displaystyle A} is a proper free factor of F n {\displaystyle F_{n}} means that A ≠ 1 {\displaystyle A\neq 1} and there exists a subgroup B ≤ F n , B ≠ 1 {\displaystyle B\leq F_{n},B\neq 1} such that F n = A ∗ B {\displaystyle F_{n}=A\ast B} . Also, Φ ∈ Aut ( F n ) {\displaystyle \Phi \in \operatorname {Aut} (F_{n})} is called fully irreducible if the outer automorphism class φ ∈ Out ( F n ) {\displaystyle \varphi \in \operatorname {Out} (F_{n})} of Φ {\displaystyle \Phi } is fully irreducible. Two fully irreducibles φ , ψ ∈ Out ( F n ) {\displaystyle \varphi ,\psi \in \operatorname {Out} (F_{n})} are called independent if ⟨ φ ⟩ ∩ ⟨ ψ ⟩ = { 1 } {\displaystyle \langle \varphi \rangle \cap \langle \psi \rangle =\{1\}} .
Relationship to irreducible automorphisms The notion of being fully irreducible grew out of an older notion of an "irreducible" outer automorphism of F n {\displaystyle F_{n}} originally introduced in. An element φ ∈ Out ( F n ) {\displaystyle \varphi \in \operatorname {Out} (F_{n})} , where n ≥ 2 {\displaystyle n\geq 2} , is called irreducible if there does not exist a free product decomposition
F n = A 1 ∗ ⋯ ∗ A k ∗ C {\displaystyle F_{n}=A_{1}\ast \dots \ast A_{k}\ast C}
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