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).
Formal definition
Let where
. Then
is called fully irreducible[1] 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.[2] 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.[3] For non-atoroidal automorphisms, Bestvina and Handel[2] 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.
Properties
- 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.[2]
- 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
.[2][4]
- Unlike for stretch factors of pseudo-Anosov surface homeomorphisms, it can happen that for a fully irreducible
one has
[5] and this behavior is believed to be generic. However, Handel and Mosher[6] 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
.[2]
- 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.[7]
- For a fully irreducible element
, its fixed points in
are projectivized
-trees
, where
, satisfying the property that
and
.[8]
- 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.[9][10]
- If
is fully irreducible, then the commensurator
is virtually cyclic.[11] 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
.[8]
- If
is fully irreducible and
, then either
is virtually cyclic or
contains a subgroup isomorphic to
.[8] [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
.[12]
- An element
acts as a loxodromic isometry on the free factor complex
if and only if
is fully irreducible.[13]
- 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
.[14]
- 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.[15] 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.[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.