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).
φ ∈ Out (
is called fully irreducible[1] if there do not exist an integer
and a proper free factor
φ
is the conjugacy class of
is a proper free factor of
and there exists a subgroup
is called fully irreducible if the outer automorphism class
φ ∈ Out (
is fully irreducible.
Two fully irreducibles
φ , ψ ∈ Out (
are called independent if
⟨ φ ⟩ ∩ ⟨ ψ ⟩ = { 1 }
The notion of being fully irreducible grew out of an older notion of an "irreducible" outer automorphism of
originally introduced in.
, is called irreducible if there does not exist a free product decomposition with
being proper free factors of
permutes the conjugacy classes
is fully irreducible in the sense of the definition above if and only if for every
(that is, without periodic conjugacy classes of nontrivial elements of
), being irreducible is equivalent to being fully irreducible.
[3] For non-atoroidal automorphisms, Bestvina and Handel[2] produce an example of an irreducible but not fully irreducible element of
, induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.
λ = λ ( φ )
is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of