Fully irreducible automorphism

In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group F_n is an element of Out(F_n) which has no periodic conjugacy classes of proper free factors in F_n (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(F_n) 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(F_n).

Formal definition

Let $\varphi \in \operatorname {Out} (F_{n})$ where $n\geq 2$. Then $\varphi$ is called fully irreducible if there do not exist an integer $p\neq 0$ 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 $A\neq 1$ and there exists a subgroup $B\leq F_{n},B\neq 1$ 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 $n\geq 2$, is called irreducible if there does not exist a free product decomposition

$F_{n}=A_{1}\ast \dots \ast A_{k}\ast C$

with $k\geq 1$, and with $A_{i}\neq 1,i=1,\dots k$ 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 $p\neq 0$ $\varphi ^{p}$ 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 $p\neq 0$ 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 $1\neq g\in F_{n}$ 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 $\lambda (\varphi )\neq \lambda (\varphi ^{-1})$ and this behavior is believed to be generic. However, Handel and Mosher proved that for every $n\geq 2$ there exists a finite constant $0<C_{n}<\infty$ such that for every fully irreducible $\varphi \in \operatorname {Out} (F_{n})$

${\frac {\log \lambda (\varphi )}{\log \lambda (\varphi ^{-1})}}\leq C_{n}.$

• 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 $\mathbb {P} Curr(F_{n})$ 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 $Comm(\langle \varphi \rangle )\leq \operatorname {Out} (F_{n})$ 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 $M\geq 1$ such that for every $p,q\geq M$ the subgroup $\langle \varphi ^{p},\psi ^{q}\rangle \leq \operatorname {Out} (F_{n})$ is isomorphic to $F_{2}$.
• If $\varphi \in \operatorname {Out} (F_{n})$ is fully irreducible and $\varphi \in H\leq \operatorname {Out} (F_{n})$, 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 $H\leq \operatorname {Out} (F_{n})$ is an arbitrary subgroup, then either $H$ contains a fully irreducible element, or there exist a finite index subgroup $H_{0}\leq H$ 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.

Further reading

• Thierry Coulbois and Arnaud Hilion, Botany of irreducible automorphisms of free groups, Pacific Journal of Mathematics 256 (2012), 291–307.
• Karen Vogtmann, On the geometry of outer space. Bulletin of the American Mathematical Society 52 (2015), no. 1, 27–46.