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

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

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

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.^[2] An element ${\displaystyle \phi \in Out(F_{n})}$, where $n\geq 2$, is called irreducible if there does not exist a free product decomposition

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

with $k\geq 1$, and with ${\displaystyle A_{i}\neq 1,i=1,\dots k}$ being proper free factors of $F_{n}$, such that $\phi$ permutes the conjugacy classes ${\displaystyle [A_{1}],\dots ,[A_{k}]}$.

Then ${\displaystyle \phi \in Out(F_{n})}$ is fully irreducible in the sense of the definition above if and only if for every $p\neq 0$ $\phi ^{p}$ is irreducible.

It is known that for any atoroidal ${\displaystyle \phi \in Out(F_{n})}$ (that is, without periodic conjugacy classes of nontrivial elements of $F_{n}$), being irreducible is equivalent to being fully irreducible.^[3] For non-atoroidal automorphisms, Bestvina and Handel^[2] produce an example of an irreducible but not fully irreducible element of ${\displaystyle Out(F_{n})}$, induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.

Properties

• If ${\displaystyle \phi \in Out(F_{n})}$ and ${\displaystyle p\neq 0}$ then $\phi$ is fully irreducible if and only if $\phi ^{p}$ is fully irreducible.
• Every fully irreducible ${\displaystyle \phi \in Out(F_{n})}$ can be represented by an expanding irreducible train track map.^[2]
• Every fully irreducible ${\displaystyle \phi \in Out(F_{n})}$ has exponential growth in $F_{n}$ given by a stretch factor ${\displaystyle \lambda =\lambda (\phi )>1}$. 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 ${\displaystyle X\in cv_{n}}$) and for every ${\displaystyle 1\neq g\in F_{n}}$ one has:

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

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

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

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

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

References

1. Thierry Coulbois and Arnaud Hilion, Botany of irreducible automorphisms of free groups, Pacific Journal of Mathematics 256 (2012), 291–307
2. Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51
3. Ilya Kapovich, Algorithmic detectability of iwip automorphisms. Bulletin of the London Mathematical Society 46 (2014), no. 2, 279-290.
4. Oleg Bogopolski. Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society, Zürich, 2008. ISBN 978-3-03719-041-8
5. Michael Handel, and Lee Mosher, Parageometric outer automorphisms of free groups. Transactions of the American Mathematical Society 359 (2007), no. 7, 3153–3183
6. Michael Handel, Lee Mosher, The expansion factors of an outer automorphism and its inverse. Transactions of the American Mathematical Society 359 (2007), no. 7, 3185–3208
7. Gilbert Levitt, and Martin Lustig, Automorphisms of free groups have asymptotically periodic dynamics. Crelle's journal, vol. 619 (2008), pp. 1–36
8. Mladen Bestvina, Mark Feighn and Michael Handel, Laminations, trees, and irreducible automorphisms of free groups. Geometric and Functional Analysis (GAFA) 7 (1997), 215–244.
9. Caglar Uyanik, Dynamics of hyperbolic iwips. Conformal Geometry and Dynamics 18 (2014), 192–216.
10. Caglar Uyanik, Generalized north-south dynamics on the space of geodesic currents. Geometriae Dedicata 177 (2015), 129–148.
11. Ilya Kapovich, and Martin Lustig, Stabilizers of ℝ-trees with free isometric actions of F_N. Journal of Group Theory 14 (2011), no. 5, 673–694.
12. Camille Horbez, A short proof of Handel and Mosher's alternative for subgroups of Out(F_N). Groups, Geometry, and Dynamics 10 (2016), no. 2, 709–721.
13. Mladen Bestvina, and Mark Feighn, Hyperbolicity of the complex of free factors. Advances in Mathematics 256 (2014), 104–155.
14. Josef Maher and Guilio Tiozzo, Random walks on weakly hyperbolic groups, Journal für die reine und angewandte Mathematik, Ahead of print (Jan 2016); c.f. Theorem 1.4
15. Yael Algom-Kfir, Strongly contracting geodesics in outer space. Geometry & Topology 15 (2011), no. 4, 2181–2233.
16. Michael Handel, and Lee Mosher, Axes in outer space. Memoirs of the American Mathematical Society 213 (2011), no. 1004; ISBN 978-0-8218-6927-7.

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.