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

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

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

Properties

• If ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ and ${\displaystyle p\neq 0}$ then ${\displaystyle \varphi }$ is fully irreducible if and only if ${\displaystyle \varphi ^{p}}$ is fully irreducible.
• Every fully irreducible ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ can be represented by an expanding irreducible train track map.^[2]
• Every fully irreducible ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ has exponential growth in ${\displaystyle F_{n}}$ given by a stretch factor ${\displaystyle \lambda =\lambda (\varphi )>1}$. This stretch factor has the property that for every free basis ${\displaystyle X}$ of ${\displaystyle F_{n}}$ (and, more generally, for every point of the Culler–Vogtmann Outer space ${\displaystyle X\in cv_{n}}$) and for every ${\displaystyle 1\neq g\in F_{n}}$ one has:

${\displaystyle \lim _{k\to \infty }{\sqrt[{k}]{\|\varphi ^{k}(g)\|_{X}}}=\lambda .}$

Moreover, ${\displaystyle \lambda =\lambda (\varphi )}$ is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of ${\displaystyle \varphi }$.^[2][4]

• Unlike for stretch factors of pseudo-Anosov surface homeomorphisms, it can happen that for a fully irreducible ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ one has ${\displaystyle \lambda (\varphi )\neq \lambda (\varphi ^{-1})}$^[5] and this behavior is believed to be generic. However, Handel and Mosher^[6] proved that for every ${\displaystyle n\geq 2}$ there exists a finite constant ${\displaystyle 0<C_{n}<\infty }$ such that for every fully irreducible ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$

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

• A fully irreducible ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ is non-atoroidal, that is, has a periodic conjugacy class of a nontrivial element of ${\displaystyle F_{n}}$, if and only if ${\displaystyle \varphi }$ is induced by a pseudo-Anosov homeomorphism of a compact connected surface with one boundary component and with the fundamental group isomorphic to ${\displaystyle F_{n}}$.^[2]
• A fully irreducible element ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ has exactly two fixed points in the Thurston compactification ${\displaystyle {\overline {CV}}_{n}}$ of the projectivized Outer space ${\displaystyle CV_{n}}$, and ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ acts on ${\displaystyle {\overline {CV}}_{n}}$ with ``North-South" dynamics.<sup>[7]</sup>
• For a fully irreducible element ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$, its fixed points in ${\displaystyle {\overline {CV}}_{n}}$ are projectivized ${\displaystyle \mathbb {R} }$-trees ${\displaystyle [T_{+}(\varphi )],[T_{-}(\varphi )]}$, where ${\displaystyle T_{+}(\varphi ),T_{-}(\varphi )\in {\overline {cv}}_{n}}$, satisfying the property that ${\displaystyle T_{+}(\varphi )\varphi =\lambda (\varphi )T_{+}(\varphi )}$ and ${\displaystyle T_{-}(\varphi )\varphi ^{-1}=\lambda (\varphi ^{-1})T_{-}(\varphi )}$.<sup>[8]</sup>
• A fully irreducible element ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ acts on the space of projectivized geodesic currents ${\displaystyle \mathbb {P} Curr(F_{n})}$ with either ``North-South" or ``generalized North-South" dynamics, depending on whether ${\displaystyle \varphi }$ is atoroidal or non-atoroidal.<sup>[9][10]</sup>
• If ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ is fully irreducible, then the commensurator ${\displaystyle Comm(\langle \varphi \rangle )\leq \operatorname {Out} (F_{n})}$ is virtually cyclic.<sup>[11]</sup> In particular, the centralizer and the normalizer of ${\displaystyle \langle \varphi \rangle }$ in ${\displaystyle \operatorname {Out} (F_{n})}$ are virtually cyclic.
• If ${\displaystyle \varphi ,\psi \in \operatorname {Out} (F_{n})}$ are independent fully irreducibles, then ${\displaystyle [T_{\pm }(\varphi )],[T_{\pm }(\psi )]\in {\overline {CV}}_{n}}$ are four distinct points, and there exists ${\displaystyle M\geq 1}$ such that for every ${\displaystyle p,q\geq M}$ the subgroup ${\displaystyle \langle \varphi ^{p},\psi ^{q}\rangle \leq \operatorname {Out} (F_{n})}$ is isomorphic to ${\displaystyle F_{2}}$.<sup>[8]</sup>
• If ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ is fully irreducible and ${\displaystyle \varphi \in H\leq \operatorname {Out} (F_{n})}$, then either ${\displaystyle H}$ is virtually cyclic or ${\displaystyle H}$ contains a subgroup isomorphic to ${\displaystyle F_{2}}$.<sup>[8]</sup> [This statement provides a strong form of the Tits alternative for subgroups of ${\displaystyle \operatorname {Out} (F_{n})}$ containing fully irreducibles.]
• If ${\displaystyle H\leq \operatorname {Out} (F_{n})}$ is an arbitrary subgroup, then either ${\displaystyle H}$ contains a fully irreducible element, or there exist a finite index subgroup ${\displaystyle H_{0}\leq H}$ and a proper free factor ${\displaystyle A}$ of ${\displaystyle F_{n}}$ such that ${\displaystyle H_{0}[A]=[A]}$.<sup>[12]</sup>
• An element ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ acts as a loxodromic isometry on the free factor complex ${\displaystyle {\mathcal {FF}}_{n}}$ if and only if ${\displaystyle \varphi }$ is fully irreducible.<sup>[13]</sup>
• It is known that ``random" (in the sense of random walks) elements of ${\displaystyle \operatorname {Out} (F_{n})}$ are fully irreducible. More precisely, if ${\displaystyle \mu }$ is a measure on ${\displaystyle \operatorname {Out} (F_{n})}$ whose support generates a semigroup in ${\displaystyle \operatorname {Out} (F_{n})}$ containing some two independent fully irreducibles. Then for the random walk of length ${\displaystyle k}$ on ${\displaystyle \operatorname {Out} (F_{n})}$ determined by ${\displaystyle \mu }$, the probability that we obtain a fully irreducible element converges to 1 as ${\displaystyle k\to \infty }$.^[14]
• A fully irreducible element ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$ admits a (generally non-unique) periodic axis in the volume-one normalized Outer space ${\displaystyle X_{n}}$, which is geodesic with respect to the asymmetric Lipschitz metric on ${\displaystyle X_{n}}$ and possesses strong ``contraction"-type properties.^[15] A related object, defined for an atoroidal fully irreducible ${\displaystyle \varphi \in \operatorname {Out} (F_{n})}$, is the axis bundle ${\displaystyle A_{\varphi }\subseteq X_{n}}$, which is a certain ${\displaystyle \varphi }$-invariant closed subset proper homotopy equivalent to a line.^[16]

References

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.