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 $$\varphi\in \operatorname{Out}(F_n)$$ where $$n\ge 2$$. Then $$\varphi$$ is called fully irreducible if there do not exist an integer $$p\ne 0$$ and a proper free factor $$A$$ of $$F_n$$ such that $$\varphi^p([A])=[A]$$, where $$[A]$$ is the conjugacy class of $$A$$ in $$F_n$$. Here saying that $$A$$ is a proper free factor of $$F_n$$ means that $$A\ne 1$$ and there exists a subgroup $$B\le F_n, B\ne 1$$ such that $$F_n=A\ast B$$.

Also, $$\Phi\in \operatorname{Aut}(F_n)$$ is called fully irreducible if the outer automorphism class $$\varphi\in \operatorname{Out}(F_n)$$ of $$\Phi$$ is fully irreducible.

Two fully irreducibles $$\varphi,\psi\in \operatorname{Out}(F_n)$$ are called independent if $$\langle \varphi\rangle \cap \langle \psi \rangle = \{1\}$$.

Relationship to irreducible automorphisms
The notion of being fully irreducible grew out of an older notion of an "irreducible" outer automorphism of $$F_n$$ originally introduced in. An element $$\varphi\in \operatorname{Out}(F_n)$$, where $$n\ge 2$$, is called irreducible if there does not exist a free product decomposition
 * $$F_n=A_1\ast\dots \ast A_k \ast C$$

with $$k\ge 1$$, and with $$A_i\ne 1, i=1,\dots k$$ being proper free factors of $$F_n$$, such that $$\varphi$$ permutes the conjugacy classes $$[A_1], \dots, [A_k] $$.

Then $$\varphi\in \operatorname{Out}(F_n)$$ is fully irreducible in the sense of the definition above if and only if for every $$p\ne 0$$ $$\varphi^p$$ is irreducible.

It is known that for any atoroidal $$\varphi\in \operatorname{Out}(F_n)$$ (that is, without periodic conjugacy classes of nontrivial elements of $$F_n$$), 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 $$\operatorname{Out}(F_n)$$, induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.

Properties

 * If $$\varphi\in \operatorname{Out}(F_n)$$ and $$ p\ne 0$$ then $$\varphi$$ is fully irreducible if and only if $$\varphi^p$$ is fully irreducible.
 * Every fully irreducible $$\varphi\in \operatorname{Out}(F_n)$$ can be represented by an expanding irreducible train track map.
 * Every fully irreducible $$\varphi\in \operatorname{Out}(F_n)$$ has exponential growth in $$F_n$$ given by a stretch factor $$\lambda=\lambda(\varphi)>1$$. This stretch factor has the property that for every free basis $$X$$ of $$F_n$$ (and, more generally, for every point of the Culler–Vogtmann Outer space $X\in cv_n$) and for every $$1\ne g\in F_n$$ one has:
 * $$\lim_{k\to\infty}\sqrt[k]{\|\varphi^k(g)\|_X}=\lambda. $$

Moreover, $$\lambda=\lambda(\varphi)$$ is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of $$\varphi$$.
 * Unlike for stretch factors of pseudo-Anosov surface homeomorphisms, it can happen that for a fully irreducible $$\varphi\in \operatorname{Out}(F_n)$$ one has $$ \lambda(\varphi)\ne \lambda(\varphi^{-1})$$ and this behavior is believed to be generic. However, Handel and Mosher proved that for every $$n\ge 2$$ there exists a finite constant $$0< C_n <\infty$$ such that for every fully irreducible $$\varphi\in \operatorname{Out}(F_n)$$
 * $$\frac{\log\lambda(\varphi) }{\log \lambda(\varphi^{-1})} \le C_n. $$


 * A fully irreducible $$\varphi\in \operatorname{Out}(F_n)$$ is non-atoroidal, that is, has a periodic conjugacy class of a nontrivial element of $$F_n$$, if and only if $$\varphi$$ is induced by a pseudo-Anosov homeomorphism of a compact connected surface with one boundary component and with the fundamental group isomorphic to $$F_n$$.
 * A fully irreducible element $$\varphi\in \operatorname{Out}(F_n)$$ has exactly two fixed points in the Thurston compactification $$\overline{CV}_n$$ of the projectivized Outer space $$CV_n$$, and $$\varphi\in \operatorname{Out}(F_n)$$ acts on $$\overline{CV}_n$$ with "North-South" dynamics.
 * For a fully irreducible element $$\varphi\in \operatorname{Out}(F_n)$$, its fixed points in $$\overline{CV}_n$$ are projectivized $$\mathbb R$$-trees $$[T_+(\varphi)], [T_-(\varphi)]$$, where $$T_+(\varphi),T_-(\varphi)\in \overline{cv}_n$$, satisfying the property that $$T_+(\varphi)\varphi=\lambda(\varphi) T_+(\varphi) $$ and $$T_-(\varphi)\varphi^{-1}=\lambda(\varphi^{-1}) T_-(\varphi) $$.
 * A fully irreducible element $$\varphi\in \operatorname{Out}(F_n)$$ acts on the space of projectivized geodesic currents $$\mathbb PCurr(F_n)$$ with either "North-South" or "generalized North-South" dynamics, depending on whether $$\varphi$$ is atoroidal or non-atoroidal.
 * If $$\varphi\in \operatorname{Out}(F_n)$$ is fully irreducible, then the commensurator $$ Comm(\langle \varphi\rangle)\le \operatorname{Out}(F_n)$$ is virtually cyclic. In particular, the centralizer and the normalizer of $$\langle \varphi\rangle$$ in $$\operatorname{Out}(F_n)$$ are virtually cyclic.
 * If $$\varphi,\psi\in \operatorname{Out}(F_n)$$ are independent fully irreducibles, then $$[T_\pm(\varphi)], [T_\pm(\psi)]\in \overline{CV}_n$$ are four distinct points, and there exists $$M\ge 1$$ such that for every $$p,q\ge M $$ the subgroup $$\langle \varphi^p, \psi^q\rangle \le \operatorname{Out}(F_n)$$ is isomorphic to $$F_2$$.
 * If $$\varphi\in \operatorname{Out}(F_n)$$ is fully irreducible and $$\varphi\in H\le \operatorname{Out}(F_n)$$, then either $$H$$ is virtually cyclic or $$H$$ contains a subgroup isomorphic to $$F_2$$. [This statement provides a strong form of the Tits alternative for subgroups of $$\operatorname{Out}(F_n)$$ containing fully irreducibles.]
 * If $$H\le \operatorname{Out}(F_n)$$ is an arbitrary subgroup, then either $$H$$ contains a fully irreducible element, or there exist a finite index subgroup $$H_0\le H$$ and a proper free factor $$A$$ of $$F_n$$ such that $$H_0[A]=[A]$$.
 * An element $$\varphi\in \operatorname{Out}(F_n)$$ acts as a loxodromic isometry on the free factor complex $$\mathcal{FF}_n$$ if and only if $$\varphi$$ is fully irreducible.
 * It is known that "random" (in the sense of random walks) elements of $$\operatorname{Out}(F_n)$$ are fully irreducible. More precisely, if $$\mu$$ is a measure on $$\operatorname{Out}(F_n)$$ whose support generates a semigroup in $$\operatorname{Out}(F_n)$$ containing some two independent fully irreducibles. Then for the random walk of length $$k$$ on $$\operatorname{Out}(F_n)$$ determined by $$\mu$$, the probability that we obtain a fully irreducible element converges to 1 as $$k\to \infty$$.
 * A fully irreducible element $$\varphi\in \operatorname{Out}(F_n)$$ admits a (generally non-unique) periodic axis in the volume-one normalized Outer space $$X_n$$, which is geodesic with respect to the asymmetric Lipschitz metric on $$X_n$$ and possesses strong "contraction"-type properties. A related object, defined for an atoroidal fully irreducible $$\varphi\in \operatorname{Out}(F_n)$$, is the axis bundle $$A_\varphi\subseteq X_n$$, which is a certain $$\varphi$$-invariant closed subset proper homotopy equivalent to a line.