Fully irreducible automorphism
In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group Fn is an element of Out(Fn) which has no periodic conjugacy classes of proper free factors in Fn (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(Fn) 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(Fn).
Let where . Then is called fully irreducible if there do not exist an integer and a proper free factor of such that , where is the conjugacy class of in . Here saying that is a proper free factor of means that and there exists a subgroup such that .
Also, is called fully irreducible if the outer automorphism class of is fully irreducible.
Two fully irreducibles are called independent if .
Relationship to irreducible automorphisms
The notion of being fully irreducible grew out of an older notion of an ``irreducible" outer automorphism of originally introduced in. An element , where , is called irreducible if there does not exist a free product decomposition
with , and with being proper free factors of , such that permutes the conjugacy classes .
Then is fully irreducible in the sense of the definition above if and only if for every is irreducible.
It is known that for any atoroidal (that is, without periodic conjugacy classes of nontrivial elements of ), 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 , induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.
- If and then is fully irreducible if and only if is fully irreducible.
- Every fully irreducible can be represented by an expanding irreducible train track map.
- Every fully irreducible has exponential growth in given by a stretch factor . This stretch factor has the property that for every free basis of (and, more generally, for every point of the Culler–Vogtmann Outer space ) and for every one has:
Moreover, is equal to the Perron–Frobenius eigenvalue of the transition matrix of any train track representative of .
- Unlike for stretch factors of pseudo-Anosov surface homeomorphisms, it can happen that for a fully irreducible one has and this behavior is believed to be generic. However, Handel and Mosher proved that for every there exists a finite constant such that for every fully irreducible
- A fully irreducible is non-atoroidal, that is, has a periodic conjugacy class of a nontrivial element of , if and only if is induced by a pseudo-Anosov homeomorphism of a compact connected surface with one boundary component and with the fundamental group isomorphic to .
- A fully irreducible element has exactly two fixed points in the Thurston compactification of the projectivized Outer space , and acts on with ``North-South" dynamics.
- For a fully irreducible element , its fixed points in are projectivized -trees , where , satisfying the property that and .
- A fully irreducible element acts on the space of projectivized geodesic currents with either ``North-South" or ``generalized North-South" dynamics, depending on whether is atoroidal or non-atoroidal.
- If is fully irreducible, then the commensurator is virtually cyclic. In particular, the centralizer and the normalizer of in are virtually cyclic.
- If are independent fully irreducibles, then are four distinct points, and there exists such that for every the subgroup is isomorphic to .
- If is fully irreducible and , then either is virtually cyclic or contains a subgroup isomorphic to . [This statement provides a strong form of the Tits alternative for subgroups of containing fully irreducibles.]
- If is an arbitrary subgroup, then either contains a fully irreducible element, or there exist a finite index subgroup and a proper free factor of such that .
- An element acts as a loxodromic isometry on the free factor complex if and only if is fully irreducible.
- It is known that ``random" (in the sense of random walks) elements of are fully irreducible. More precisely, if is a measure on whose support generates a semigroup in containing some two independent fully irreducibles. Then for the random walk of length on determined by , the probability that we obtain a fully irreducible element converges to 1 as .
- A fully irreducible element admits a (generally non-unique) periodic axis in the volume-one normalized Outer space , which is geodesic with respect to the asymmetric Lipschitz metric on and possesses strong ``contraction"-type properties. A related object, defined for an atoroidal fully irreducible , is the axis bundle , which is a certain -invariant closed subset proper homotopy equivalent to a line.
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