Fully irreducible automorphism

Short description: Concept in mathematics

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

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

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

Relationship to irreducible automorphisms

The notion of being fully irreducible grew out of an older notion of an ``irreducible" outer automorphism of [math]\displaystyle{ F_n }[/math] originally introduced in. An element [math]\displaystyle{ \varphi\in \operatorname{Out}(F_n) }[/math], where [math]\displaystyle{ n\ge 2 }[/math], is called irreducible if there does not exist a free product decomposition

[math]\displaystyle{ F_n=A_1\ast\dots \ast A_k \ast C }[/math]

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

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

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

Properties

• If [math]\displaystyle{ \varphi\in \operatorname{Out}(F_n) }[/math] and [math]\displaystyle{ p\ne 0 }[/math] then [math]\displaystyle{ \varphi }[/math] is fully irreducible if and only if [math]\displaystyle{ \varphi^p }[/math] is fully irreducible.
• Every fully irreducible [math]\displaystyle{ \varphi\in \operatorname{Out}(F_n) }[/math] can be represented by an expanding irreducible train track map.
• Every fully irreducible [math]\displaystyle{ \varphi\in \operatorname{Out}(F_n) }[/math] has exponential growth in [math]\displaystyle{ F_n }[/math] given by a stretch factor [math]\displaystyle{ \lambda=\lambda(\varphi)\gt 1 }[/math]. This stretch factor has the property that for every free basis [math]\displaystyle{ X }[/math] of [math]\displaystyle{ F_n }[/math] (and, more generally, for every point of the Culler–Vogtmann Outer space [math]\displaystyle{ X\in cv n }[/math]) and for every [math]\displaystyle{ 1\ne g\in F_n }[/math] one has:

[math]\displaystyle{ \lim_{k\to\infty}\sqrt[k]{\|\varphi^k(g)\|_X}=\lambda. }[/math]

Moreover, [math]\displaystyle{ \lambda=\lambda(\varphi) }[/math] is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of [math]\displaystyle{ \varphi }[/math].

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

[math]\displaystyle{ \frac{\log\lambda(\varphi) }{\log \lambda(\varphi^{-1})} \le C_n. }[/math]

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