fully irreducible automorphism

Let $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ where $n\backslash ge\; 2$. Then $\backslash varphi$ is called fully irreducibleThierry Coulbois and Arnaud Hilion, Botany of irreducible automorphisms of free groups, Pacific Journal of Mathematics 256 (2012), 291–307 if there do not exist an integer $p\backslash ne\; 0$ and a proper free factor $A$ of $F\_n$ such that $\backslash varphi^p([A])=[A]$, 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 $A\backslash ne\; 1$ and there exists a subgroup $B\backslash le\; F\_n,\; B\backslash ne\; 1$ such that $F\_n=A\backslash ast\; B$.

Also, $\backslash Phi\backslash in\; \backslash operatorname\{Aut\}(F\_n)$ is called fully irreducible if the outer automorphism class $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ of $\backslash Phi$ is fully irreducible.

Two fully irreducibles $\backslash varphi,\backslash psi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ are called independent if $\backslash langle\; \backslash varphi\backslash rangle\; \backslash cap\; \backslash langle\; \backslash psi\; \backslash rangle\; =\; \backslash \{1\backslash \}$.

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 $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$, where $n\backslash ge\; 2$, is called irreducible if there does not exist a free product decomposition

:$F\_n=A\_1\backslash ast\backslash dots\; \backslash ast\; A\_k\; \backslash ast\; C$

with $k\backslash ge\; 1$, and with $A\_i\backslash ne\; 1,\; i=1,\backslash dots\; k$ being proper free factors of $F\_n$, such that $\backslash varphi$ permutes the conjugacy classes $[A\_1],\; \backslash dots,\; [A\_k]$.

Then $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ is fully irreducible in the sense of the definition above if and only if for every $p\backslash ne\; 0$ $\backslash varphi^p$ is irreducible.

It is known that for any atoroidal $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ (that is, without periodic conjugacy classes of nontrivial elements of $F\_n$), being irreducible is equivalent to being fully irreducible.Ilya Kapovich, Algorithmic detectability of iwip automorphisms. Bulletin of the London Mathematical Society 46 (2014), no. 2, 279–290. For non-atoroidal automorphisms, Bestvina and Handel produce an example of an irreducible but not fully irreducible element of $\backslash operatorname\{Out\}(F\_n)$, induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.

• If $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ and $p\backslash ne\; 0$ then $\backslash varphi$ is fully irreducible if and only if $\backslash varphi^p$ is fully irreducible.
• Every fully irreducible $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ can be represented by an expanding irreducible train track map.Mladen Bestvina, and Michael Handel, [https://www.jstor.org/pss/2946562 Train tracks and automorphisms of free groups.] Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51
• Every fully irreducible $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ has exponential growth in $F\_n$ given by a stretch factor $\backslash lambda=\backslash lambda(\backslash varphi)>1$. 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 $X\backslash in\; cv\_n$) and for every $1\backslash ne\; g\backslash in\; F\_n$ one has:

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

Moreover, $\backslash lambda=\backslash lambda(\backslash varphi)$ is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of $\backslash varphi$.Oleg Bogopolski. [https://books.google.com/books?id=jEw8MpP6DIgC&dq=Oleg+Bogopolski+Introduction+to+Group+Theory&pg=PT1 Introduction to group theory.] EMS Textbooks in Mathematics. European Mathematical Society, Zürich, 2008. {{ISBN|978-3-03719-041-8}}

• Unlike for stretch factors of pseudo-Anosov surface homeomorphisms, it can happen that for a fully irreducible $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ one has $\backslash lambda(\backslash varphi)\backslash ne\; \backslash lambda(\backslash varphi^\{-1\})$Michael Handel, and Lee Mosher, [https://www.ams.org/journals/tran/2007-359-07/S0002-9947-07-04065-2/S0002-9947-07-04065-2.pdf Parageometric outer automorphisms of free groups.] Transactions of the American Mathematical Society 359 (2007), no. 7, 3153–3183 and this behavior is believed to be generic. However, Handel and MosherMichael Handel, Lee Mosher, [https://www.ams.org/journals/tran/2007-359-07/S0002-9947-07-04066-4/S0002-9947-07-04066-4.pdf The expansion factors of an outer automorphism and its inverse.] Transactions of the American Mathematical Society 359 (2007), no. 7, 3185–3208 proved that for every $n\backslash ge\; 2$ there exists a finite constant such that for every fully irreducible $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$

:: $\backslash frac\{\backslash log\backslash lambda(\backslash varphi)\; \}\{\backslash log\; \backslash lambda(\backslash varphi^\{-1\})\}\; \backslash le\; C\_n.$

• A fully irreducible $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ is non-atoroidal, that is, has a periodic conjugacy class of a nontrivial element of $F\_n$, if and only if $\backslash varphi$ 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 $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ has exactly two fixed points in the Thurston compactification $\backslash overline\{CV\}\_n$ of the projectivized Outer space $CV\_n$, and $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ acts on $\backslash overline\{CV\}\_n$ with "North-South" dynamics.{{citation

| last1 = Levitt | first1 = Gilbert

| last2 = Lustig | first2 = Martin

| title = Automorphisms of free groups have asymptotically periodic dynamics

| journal = Journal für die reine und angewandte Mathematik

| pages = 1–36

| volume = 2008

| year = 2008

| issue = 619

| doi=10.1515/CRELLE.2008.038| arxiv = math/0407437

| s2cid = 14724939

}}

• For a fully irreducible element $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$, its fixed points in $\backslash overline\{CV\}\_n$ are projectivized $\backslash mathbb\; R$-trees $[T\_+(\backslash varphi)],\; [T\_-(\backslash varphi)]$, where $T\_+(\backslash varphi),T\_-(\backslash varphi)\backslash in\; \backslash overline\{cv\}\_n$, satisfying the property that $T\_+(\backslash varphi)\backslash varphi=\backslash lambda(\backslash varphi)\; T\_+(\backslash varphi)$ and $T\_-(\backslash varphi)\backslash varphi^\{-1\}=\backslash lambda(\backslash varphi^\{-1\})\; T\_-(\backslash varphi)$.
• A fully irreducible element $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ acts on the space of projectivized geodesic currents $\backslash mathbb\; PCurr(F\_n)$ with either "North-South" or "generalized North-South" dynamics, depending on whether $\backslash varphi$ is atoroidal or non-atoroidal.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.
• If $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ is fully irreducible, then the commensurator $Comm(\backslash langle\; \backslash varphi\backslash rangle)\backslash le\; \backslash operatorname\{Out\}(F\_n)$ is virtually cyclic.Ilya Kapovich, and Martin Lustig, Stabilizers of $\backslash mathbb\{R\}$-trees with free isometric actions of F_N.

Journal of Group Theory 14 (2011), no. 5, 673–694. In particular, the centralizer and the normalizer of $\backslash langle\; \backslash varphi\backslash rangle$ in $\backslash operatorname\{Out\}(F\_n)$ are virtually cyclic.

• If $\backslash varphi,\backslash psi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ are independent fully irreducibles, then $[T\_\backslash pm(\backslash varphi)],\; [T\_\backslash pm(\backslash psi)]\backslash in\; \backslash overline\{CV\}\_n$ are four distinct points, and there exists $M\backslash ge\; 1$ such that for every $p,q\backslash ge\; M$ the subgroup $\backslash langle\; \backslash varphi^p,\; \backslash psi^q\backslash rangle\; \backslash le\; \backslash operatorname\{Out\}(F\_n)$ is isomorphic to $F\_2$.
• If $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ is fully irreducible and $\backslash varphi\backslash in\; H\backslash le\; \backslash operatorname\{Out\}(F\_n)$, then either $H$ is virtually cyclic or $H$ contains a subgroup isomorphic to $F\_2$.Mladen Bestvina, Mark Feighn and Michael Handel, Laminations, trees, and irreducible automorphisms of free groups. Geometric and Functional Analysis 7 (1997), 215–244. [This statement provides a strong form of the Tits alternative for subgroups of $\backslash operatorname\{Out\}(F\_n)$ containing fully irreducibles.]
• If $H\backslash le\; \backslash operatorname\{Out\}(F\_n)$ is an arbitrary subgroup, then either $H$ contains a fully irreducible element, or there exist a finite index subgroup $H\_0\backslash le\; H$ and a proper free factor $A$ of $F\_n$ such that $H\_0[A]=[A]$.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.
• An element $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ acts as a loxodromic isometry on the free factor complex $\backslash mathcal\{FF\}\_n$ if and only if $\backslash varphi$ is fully irreducible.Mladen Bestvina, and Mark Feighn, Hyperbolicity of the complex of free factors. Advances in Mathematics 256 (2014), 104–155.
• It is known that "random" (in the sense of random walks) elements of $\backslash operatorname\{Out\}(F\_n)$ are fully irreducible. More precisely, if $\backslash mu$ is a measure on $\backslash operatorname\{Out\}(F\_n)$ whose support generates a semigroup in $\backslash operatorname\{Out\}(F\_n)$ containing some two independent fully irreducibles. Then for the random walk of length $k$ on $\backslash operatorname\{Out\}(F\_n)$ determined by $\backslash mu$, the probability that we obtain a fully irreducible element converges to 1 as $k\backslash to\; \backslash infty$.Joseph Maher and Giulio Tiozzo, [https://www.degruyter.com/view/j/crll.ahead-of-print/crelle-2015-0076/crelle-2015-0076.xml Random walks on weakly hyperbolic groups], Journal für die reine und angewandte Mathematik, Ahead of print (Jan 2016); cf. Theorem 1.4
• A fully irreducible element $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$ 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.Yael Algom-Kfir,

Strongly contracting geodesics in outer space.

Geometry & Topology 15 (2011), no. 4, 2181–2233. A related object, defined for an atoroidal fully irreducible $\backslash varphi\backslash in\; \backslash operatorname\{Out\}(F\_n)$, is the axis bundle $A\_\backslash varphi\backslash subseteq\; X\_n$, which is a certain $\backslash varphi$-invariant closed subset proper homotopy equivalent to a line.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}}.

{{Reflist}}

• Thierry Coulbois and Arnaud Hilion, Botany of irreducible automorphisms of free groups, Pacific Journal of Mathematics 256 (2012), 291–307.
• Karen Vogtmann, [https://www.ams.org/journals/bull/2015-52-01/S0273-0979-2014-01466-1/S0273-0979-2014-01466-1.pdf On the geometry of outer space.] Bulletin of the American Mathematical Society 52 (2015), no. 1, 27–46.

Category:Geometric group theory

Category:Geometric topology