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 $\phi \in Out(F_{n})$ 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 $\phi ^{p}([A])=[A]$ , 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 $A\neq 1$ and there exists a subgroup $B\leq F_{n},B\neq 1$ such that $F_{n}=A\ast B$ .

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

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

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 $\phi \in Out(F_{n})$ , 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 $\phi$ permutes the conjugacy classes $[A_{1}],\dots ,[A_{k}]$ .

Then $\phi \in Out(F_{n})$ 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 $\phi \in Out(F_{n})$ (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 $Out(F_{n})$ , induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.

Properties

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

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

Moreover, $\lambda =\lambda (\phi )$ 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 $\phi \in Out(F_{n})$ one has $\lambda (\phi )\neq (\lambda (\phi ^{-1})$ ^[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 $0<C_{n}<\infty$ such that for every fully irreducible $\phi \in Out(F_{n})$

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

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

References

↑ Thierry Coulbois and Arnaud Hilion, Botany of irreducible automorphisms of free groups, Pacific Journal of Mathematics 256 (2012), 291–307
1 2 3 4 5 Mladen Bestvina, and Michael Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51
↑ Ilya Kapovich, Algorithmic detectability of iwip automorphisms. Bulletin of the London Mathematical Society 46 (2014), no. 2, 279-290.
↑ Oleg Bogopolski. Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society, Zürich, 2008. ISBN 978-3-03719-041-8
↑ Michael Handel, and Lee Mosher, Parageometric outer automorphisms of free groups. Transactions of the American Mathematical Society 359 (2007), no. 7, 3153–3183
↑ 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
↑ Gilbert Levitt, and Martin Lustig, Automorphisms of free groups have asymptotically periodic dynamics. Crelle's journal, vol. 619 (2008), pp. 1–36
1 2 3 Mladen Bestvina, Mark Feighn and Michael Handel, Laminations, trees, and irreducible automorphisms of free groups. Geometric and Functional Analysis (GAFA) 7 (1997), 215–244.
↑ Caglar Uyanik, Dynamics of hyperbolic iwips. Conformal Geometry and Dynamics 18 (2014), 192–216.
↑ Caglar Uyanik, Generalized north-south dynamics on the space of geodesic currents. Geometriae Dedicata 177 (2015), 129–148.
↑ 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.
↑ 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.
↑ Mladen Bestvina, and Mark Feighn, Hyperbolicity of the complex of free factors. Advances in Mathematics 256 (2014), 104–155.
↑ 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
↑ Yael Algom-Kfir, Strongly contracting geodesics in outer space. Geometry & Topology 15 (2011), no. 4, 2181–2233.
↑ 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.

Fully irreducible automorphism

Formal definition

Relationship to irreducible automorphisms

Properties

References

Further reading