Fully irreducible automorphism
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  where . Then  is called fully irreducible if there do not exist an integer  and a proper free factor  of  such that , where  is the conjugacy class of  in . 
Here saying that   is a proper free factor of  means that  and there exists a subgroup  such that .

Also,  is called fully irreducible if the outer automorphism class  of  is fully irreducible.

Two fully irreducibles  are called independent if .

Relationship to irreducible automorphisms

The notion of being fully irreducible grew out of an older notion of an ``irreducible" outer automorphism of  originally introduced in. An element , where , is called irreducible if there does not exist a free product decomposition

with , and with  being proper free factors of ,  such that  permutes the conjugacy classes .

Then  is fully irreducible in the sense of the definition above  if and only if for every   is irreducible.

It is known that for any atoroidal  (that is, without periodic conjugacy classes of nontrivial elements of ), being irreducible is equivalent to being fully irreducible. For non-atoroidal automorphisms, Bestvina and Handel 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.

Properties

 If  and  then  is fully irreducible if and only if  is fully irreducible.
Every fully irreducible  can be represented by an expanding irreducible train track map.
Every fully irreducible  has exponential growth in  given by a stretch factor . This stretch factor has the property that for every free basis  of  (and, more generally, for every point of the Culler–Vogtmann Outer space ) and for every  one has:
 
Moreover,  is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of .
Unlike for stretch factors of pseudo-Anosov surface homeomorphisms, it can happen that for a fully irreducible  one has  and this behavior is believed to be generic. However, Handel and Mosher proved that for every  there exists a finite constant  such that for every fully irreducible 
 
A fully irreducible  is non-atoroidal, that is, has a periodic conjugacy class of a nontrivial element of , if and only if  is induced by a pseudo-Anosov homeomorphism of a compact connected surface with one boundary component and with the fundamental group isomorphic to .
A fully irreducible element  has exactly two fixed points in the Thurston compactification  of the projectivized Outer space , and  acts on  with ``North-South" dynamics.
For a fully irreducible element , its fixed points in  are projectivized -trees , where , satisfying the property that  and .
A fully irreducible element  acts on the space of projectivized geodesic currents  with either ``North-South" or ``generalized North-South" dynamics, depending on whether  is atoroidal or non-atoroidal.
 If  is fully irreducible, then the commensurator  is virtually cyclic. In particular, the centralizer and the normalizer of  in  are virtually cyclic.
If  are independent fully irreducibles, then  are four distinct points, and there exists  such that for every  the subgroup  is isomorphic to . 
 If  is fully irreducible and , then either  is virtually cyclic or  contains a subgroup isomorphic to .  [This statement provides a strong form of the Tits alternative for subgroups of  containing fully irreducibles.]
If  is an arbitrary subgroup, then either  contains a fully irreducible element, or there exist a finite index subgroup  and a proper free factor  of  such that .
 An element  acts as a loxodromic isometry on the free factor complex  if and only if  is fully irreducible.
It is known that ``random" (in the sense of random walks) elements of  are fully irreducible. More precisely, if  is a measure on  whose support generates a semigroup in   containing some two independent fully irreducibles. Then for the random walk of length  on  determined by , the probability that we obtain a fully irreducible element converges to 1 as .
A fully irreducible element  admits a (generally non-unique) periodic axis in the volume-one normalized Outer space , which is geodesic with respect to the asymmetric Lipschitz metric on  and possesses strong ``contraction"-type properties. A related object, defined for an atoroidal fully irreducible , is the axis bundle , which is a certain -invariant closed subset proper homotopy equivalent to a line.