Fully irreducible automorphism

Template:Short description 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 $\phi \in \mathrm{Out}(F_n)$ where $n\ge 2$. Then $\phi$ is called fully irreducible if there does not exist an integer $p\ne 0$ and a proper free factor $A$ of $F_n$ such that ${\phi }^p([A])=[A]$, where $[A]$ is the conjugacy class of $A$ in $F_n$. Equivalently, for all $p>0$ and any proper free factor $A$ of $F_n$, ${\phi }^p([A])\ne [A]$. 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, $\mathrm{\Phi }\in \mathrm{Aut}(F_n)$ is called fully irreducible if the outer automorphism class $\phi \in \mathrm{Out}(F_n)$ of $\mathrm{\Phi }$ is fully irreducible.

Two fully irreducibles $\phi ,\psi \in \mathrm{Out}(F_n)$ are called independent if $⟨\phi ⟩\cap ⟨\psi ⟩=\{1\}$.

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 $\phi \in \mathrm{Out}(F_n)$, 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 $\phi$ permutes the conjugacy classes $[A_1],\dots ,[A_k]$.

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

It is known that for any atoroidal $\phi \in \mathrm{Out}(F_n)$ (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 $\mathrm{Out}(F_n)$, induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.

Properties

• If $\phi \in \mathrm{Out}(F_n)$ and $p\ne 0$ then $\phi$ is fully irreducible if and only if ${\phi }^p$ is fully irreducible.
• Every fully irreducible $\phi \in \mathrm{Out}(F_n)$ can be represented by an expanding irreducible train track map.
• Every fully irreducible $\phi \in \mathrm{Out}(F_n)$ has exponential growth in $F_n$ given by a stretch factor $\lambda =\lambda (\phi )>1$. 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 $X\in c{v}_n$) and for every $1\ne g\in F_n$ one has:

$\underset{k\to \mathrm{\infty }}{lim}\sqrt[k]{\Vert {\phi }^k(g){\Vert }_X}=\lambda .$

Moreover, $\lambda =\lambda (\phi )$ is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of $\phi$.

• Unlike for stretch factors of pseudo-Anosov surface homeomorphisms, it can happen that for a fully irreducible $\phi \in \mathrm{Out}(F_n)$ one has $\lambda (\phi )\ne \lambda ({\phi }^{-1})$ and this behavior is believed to be generic. However, Handel and Mosher proved that for every $n\ge 2$ there exists a finite constant $0<C_n<\mathrm{\infty }$ such that for every fully irreducible $\phi \in \mathrm{Out}(F_n)$

$\frac{\log \lambda (\phi )}{\log \lambda ({\phi }^{-1})}\le C_n.$

• A fully irreducible $\phi \in \mathrm{Out}(F_n)$ is non-atoroidal, that is, has a periodic conjugacy class of a nontrivial element of $F_n$, if and only if $\phi$ 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 $\phi \in \mathrm{Out}(F_n)$ has exactly two fixed points in the Thurston compactification ${\overline{CV}}_n$ of the projectivized Outer space $C{V}_n$, and $\phi \in \mathrm{Out}(F_n)$ acts on ${\overline{CV}}_n$ with "North-South" dynamics.
• For a fully irreducible element $\phi \in \mathrm{Out}(F_n)$, its fixed points in ${\overline{CV}}_n$ are projectivized $ℝ$-trees $[T_{+}(\phi )],[T_{-}(\phi )]$, where $T_{+}(\phi ),T_{-}(\phi )\in {\overline{cv}}_n$, satisfying the property that $T_{+}(\phi )\phi =\lambda (\phi )T_{+}(\phi )$ and $T_{-}(\phi ){\phi }^{-1}=\lambda ({\phi }^{-1})T_{-}(\phi )$.
• A fully irreducible element $\phi \in \mathrm{Out}(F_n)$ 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 $\phi$ is atoroidal or non-atoroidal.
• If $\phi \in \mathrm{Out}(F_n)$ is fully irreducible, then the commensurator $Comm(⟨\phi ⟩)\le \mathrm{Out}(F_n)$ is virtually cyclic. In particular, the centralizer and the normalizer of $⟨\phi ⟩$ in $\mathrm{Out}(F_n)$ are virtually cyclic.
• If $\phi ,\psi \in \mathrm{Out}(F_n)$ are independent fully irreducibles, then $[T_{\pm }(\phi )],[T_{\pm }(\psi )]\in {\overline{CV}}_n$ are four distinct points, and there exists $M\ge 1$ such that for every $p,q\ge M$ the subgroup $⟨{\phi }^p,{\psi }^q⟩\le \mathrm{Out}(F_n)$ is isomorphic to $F_2$.
• If $\phi \in \mathrm{Out}(F_n)$ is fully irreducible and $\phi \in H\le \mathrm{Out}(F_n)$, 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 $\mathrm{Out}(F_n)$ containing fully irreducibles.]
• If $H\le \mathrm{Out}(F_n)$ 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 $\phi \in \mathrm{Out}(F_n)$ acts as a loxodromic isometry on the free factor complex ${ℱℱ}_n$ if and only if $\phi$ is fully irreducible.
• It is known that "random" (in the sense of random walks) elements of $\mathrm{Out}(F_n)$ are fully irreducible. More precisely, if $\mu$ is a measure on $\mathrm{Out}(F_n)$ whose support generates a semigroup in $\mathrm{Out}(F_n)$ containing some two independent fully irreducibles. Then for the random walk of length $k$ on $\mathrm{Out}(F_n)$ determined by $\mu$, the probability that we obtain a fully irreducible element converges to 1 as $k\to \mathrm{\infty }$.
• A fully irreducible element $\phi \in \mathrm{Out}(F_n)$ 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 $\phi \in \mathrm{Out}(F_n)$, is the axis bundle $A_{\phi }\subseteq X_n$, which is a certain $\phi$-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.