View source for Out(Fn)

{{short description|Outer automorphism group of a free group on n generators}}
{{DISPLAYTITLE:Out(''F<sub>n</sub>'')}}
In [[mathematics]], '''Out(''F<sub>n</sub>'')''' is the [[outer automorphism group]] of a [[free group]] on ''n'' [[generating set of a group|generator]]s. These groups are at universal stage in [[geometric group theory]], as they act on the set of [[Presentation of a group|presentations]] with <math>n</math> generators of any [[finitely generated group]].<ref>{{Citation |last=Lubotzky |first=Alexander |title=Dynamics of Aut(Fn) Actions on Group Presentations and Representations |date=2011-12-15 |arxiv=1109.0155 }}</ref> Despite geometric analogies with general [[General linear group|linear groups]] and [[Mapping class group of a surface|mapping class groups]], their complexity is generally regarded as more challenging, which has fueled the development of new techniques in the field.

== Definition ==
Let <math>F_n</math> be the free nonabelian group of rank <math>n \ge 1</math>. The set of [[inner automorphism]]s of <math>F_n</math>, i.e. automorphisms obtained as conjugations by an element of <math>F_n</math>, is a [[normal subgroup]] <math>\mathrm{Inn}(F_n) \triangleleft \mathrm{Aut}(F_n)</math>. The outer automorphism group of <math>F_n</math> is the quotient<math display="block">\mathrm{Out}(F_n) := \mathrm{Aut}(F_n)/\mathrm{Inn}(F_n).</math>An element of <math>\mathrm{Out}(F_n)</math> is called an outer class.

==Relations to other groups==

=== Linear groups ===
The [[abelianization]] map <math>F_n \to \Z^n</math> induces a [[homomorphism]] from <math>\mathrm{Out}(F_n)</math> to the general linear group <math>\mathrm{GL}(n,\Z)</math>, the latter being the [[automorphism group]] of <math>\Z^n</math>.  This map is onto, making <math>\mathrm{Out}(F_n)</math> a [[group extension]],

:<math>1\to \mathrm{Tor}(F_n) \to \mathrm{Out}(F_n) \to \mathrm{GL}(n,\Z)\to 1</math>.

The kernel <math>\mathrm{Tor}(F_n)</math> is the [[Torelli group]] of <math>F_n</math>.

The map <math>\mathrm{Out}(F_2) \to \mathrm{GL}(2,\Z)</math> is an [[isomorphism]]. This no longer holds for higher ranks: the Torelli group of <math>F_3</math> contains the automorphism fixing two basis elements and multiplying the remaining one by the commutator of the two others.

=== Aut(F<sub>n</sub>) ===
By definition, <math>\mathrm{Aut}(F_n)</math> is an extension of the inner automorphism group <math>\mathrm{Inn}(F_n)</math> by <math>\mathrm{Out}(F_n)</math>. The inner automorphism group itself is the image of the [[Conjugacy class#Conjugacy as group action|action by conjugation]], which has kernel the [[Center (group theory)|center]] <math>Z(F_n)</math>. Since <math>Z(F_n)</math> is trivial for <math>n\ge 2</math>, this gives a short exact sequence<math display="block">1 \rightarrow F_n \rightarrow \mathrm{Aut}(F_n) \rightarrow \mathrm{Out}(F_n) \rightarrow 1.</math>For all <math>n \ge 2</math>, there are embeddings <math>\mathrm{Aut}(F_n) \longrightarrow \mathrm{Out}(F_{n+1})</math> obtained by taking the outer class of the extension of an automorphism of <math>F_n</math> fixing the additional generator. Therefore, when studying properties that are inherited by subgroups and quotients, the theories of <math>\mathrm{Aut}(F_n)</math> and <math>\mathrm{Out}(F_n)</math> are essentially the same.

=== Mapping class groups of surfaces ===
Because <math>F_n</math> is the [[fundamental group]] of a [[bouquet of circles|bouquet of ''n'' circles]], <math>\mathrm{Out}(F_n)</math> can be described topologically as the [[mapping class group]] of a bouquet of ''n'' circles (in the [[homotopy category]]), in analogy to the mapping class group of a closed [[surface (mathematics)|surface]] which is isomorphic to the outer automorphism group of the fundamental group of that surface.

Given any finite graph with fundamental group <math>F_n</math>, the graph can be "thickened" to a surface <math>S</math> with one boundary component that retracts onto the graph. The [[Mapping class group of a surface#The Birman exact sequence|Birman exact sequence]] yields a map from the mapping class group <math>\mathrm{MCG}(S) \longrightarrow \mathrm{Out}(F_n)</math>. The elements of <math>\mathrm{Out}(F_n)</math> that are in the image of such a map are called geometric. Such outer classes must leave invariant the cyclic word corresponding to the boundary, hence there are many non-geometric outer classes. A converse is true under some irreducibility assumptions,<ref>{{Cite journal |last1=Bestvina |first1=Mladen |last2=Handel |first2=Michael |date=1992 |title=Train Tracks and Automorphisms of Free Groups |url=https://www.jstor.org/stable/2946562 |journal=Annals of Mathematics |volume=135 |issue=1 |pages=1–51 |doi=10.2307/2946562 |jstor=2946562 |issn=0003-486X}}</ref> providing geometric realization for outer classes fixing a conjugacy class.

== Known results ==

* For <math display="inline">n\ge 4</math>, <math display="inline">\mathrm{Out}(F_n)</math> is not linear, i.e. it has no [[faithful representation]] by matrices over a field (Formanek, Procesi, 1992);<ref>{{Cite journal |last1=Formanek |first1=Edward |last2=Procesi |first2=Claudio |date=1992-07-01 |title=The automorphism group of a free group is not linear |url=https://dx.doi.org/10.1016/0021-8693%2892%2990029-L |journal=Journal of Algebra |volume=149 |issue=2 |pages=494–499 |doi=10.1016/0021-8693(92)90029-L |issn=0021-8693}}</ref>
* For <math>n\ge 3</math>, the [[Dehn function|isoperimetric function]] of <math>\mathrm{Out}(F_n)</math> is exponential (Hatcher, Vogtmann, 1996);<ref>{{Cite journal |last1=Hatcher |first1=Allen |last2=Vogtmann |first2=Karen |date=1996-04-01 |title=Isoperimetric inequalities for automorphism groups of free groups |url=https://msp.org/pjm/1996/173-2/p09.xhtml |journal=Pacific Journal of Mathematics |volume=173 |issue=2 |pages=425–441 |issn=0030-8730}}</ref>
* The [[Tits alternative|Tits Alternative]] holds in <math>\mathrm{Out}(F_n)</math>: each subgroup is either [[virtually]] [[Solvable group|solvable]] or else it contains a free group of rank 2 (Bestvina, Feighn, Handel, 2000);<ref>{{Cite journal |last1=Bestvina |first1=Mladen |last2=Feighn |first2=Mark |last3=Handel |first3=Michael |date=2000 |title=The Tits Alternative for out(Fn) I: Dynamics of Exponentially-Growing Automorphisms |url=https://www.jstor.org/stable/121043 |journal=Annals of Mathematics |volume=151 |issue=2 |pages=517–623 |doi=10.2307/121043 |jstor=121043 |issn=0003-486X|arxiv=math/9712217 }}</ref>
* For <math display="inline">n\ge 3</math>, <math display="inline">\mathrm{Out}(\mathrm{Out}(F_n)) = {1}</math> (Bridson and Vogtmann, 2000);<ref>{{Cite journal |last=Bridson |first=Martin R. |last2=Vogtmann |first2=Karen |date=2000-07-15 |title=Automorphisms of Automorphism Groups of Free Groups |url=https://www.sciencedirect.com/science/article/pii/S0021869300983273?via=ihub |journal=Journal of Algebra |volume=229 |issue=2 |pages=785–792 |doi=10.1006/jabr.2000.8327 |issn=0021-8693}}</ref>
* Every solvable subgroup of <math>\mathrm{Out}(F_n)</math> has a finitely generated [[Free abelian group|free abelian]] subgroup of finite index (Bestvina, Feighn, Handel, 2004);<ref>{{Cite journal |last1=Bestvina |first1=Mladen |last2=Feighn |first2=Mark |last3=Handel |first3=Michael |date=2004-03-01 |title=Solvable Subgroups of Out(Fn) are Virtually Abelian |url=https://link.springer.com/article/10.1023/B:GEOM.0000022864.30278.34 |journal=Geometriae Dedicata |language=en |volume=104 |issue=1 |pages=71–96 |doi=10.1023/B:GEOM.0000022864.30278.34 |issn=1572-9168|arxiv=math/9712219 }}</ref>
* For <math>i>0</math>, all but finitely many of the [[Group cohomology#Group homology|<math>i</math><sup>th</sup>-degree homology]] morphisms induced by the sequence<math display="block">\ldots \rightarrow \mathrm{Out}(F_{n-1}) \rightarrow \mathrm{Out}(F_n) \rightarrow \mathrm{Out}(F_{n+1}) \rightarrow \ldots</math> are isomorphisms (Hatcher and Vogtmann, 2004);<ref>{{Cite journal |last1=Hatcher |first1=Allen |last2=Vogtmann |first2=Karen |date=2004-12-24 |title=Homology stability for outer automorphism groups of free groups |url=https://msp.org/agt/2004/4-2/p25.xhtml |journal=Algebraic & Geometric Topology |volume=4 |issue=2 |pages=1253–1272 |doi=10.2140/agt.2004.4.1253 |issn=1472-2739|arxiv=math/0406377 }}</ref>
* For <math display="inline">n\ge 2</math>, the reduced [[C*-algebra|<math display="inline">C^*</math>-algebra]] of <math display="inline">\mathrm{Out}(F_n)</math> (i.e. the closure of its image under the [[regular representation]]) is simple;<ref>{{Cite journal |last=Bridson |first=Martin R. |last2=de la Harpe |first2=Pierre |date=2004-07-01 |title=Mapping class groups and outer automorphism groups of free groups are C∗-simple |url=https://www.sciencedirect.com/science/article/pii/S0022123603002167?via=ihub |journal=Journal of Functional Analysis |volume=212 |issue=1 |pages=195–205 |doi=10.1016/S0022-1236(03)00216-7 |issn=0022-1236|arxiv=math/0306204 }}</ref>
* For <math display="inline">n\ge 4</math>, if <math display="inline">\Gamma</math> is a finite index subgroup of <math display="inline">\mathrm{Out}(F_n)</math>, then any subgroup of <math display="inline">\mathrm{Out}(F_n)</math> isomorphic to <math display="inline">\Gamma</math> is a conjugate of <math display="inline">\Gamma</math> (Farb and Handel, 2007);<ref>{{Cite journal |last=Farb |first=Benson |last2=Handel |first2=Michael |date=2007-06-01 |title=Commensurations of Out(Fn) |url=https://link.springer.com/article/10.1007/s10240-007-0007-7 |journal=Publications mathématiques |language=en |volume=105 |issue=1 |pages=1–48 |doi=10.1007/s10240-007-0007-7 |issn=1618-1913}}</ref>
* For <math display="inline">n\ge 5</math>, <math display="inline">\mathrm{Out}(F_n)</math> has [[Kazhdan's property (T)]] (Kaluba, Nowak, Ozawa, 2019 for <math display="inline">n=5</math>; Kaluba, Kielak, Nowak, 2021 for <math>n\ge 6</math>);<ref>{{Citation |last1=Kaluba |first1=Marek |title=On property (T) for $\operatorname{Aut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$ |date=2021-01-20 |url=https://arxiv.org/abs/1812.03456 |access-date=2024-10-13 |arxiv=1812.03456 |last2=Kielak |first2=Dawid |last3=Nowak |first3=Piotr W.}}</ref>
* Actions on [[Hyperbolic metric space|hyperbolic]] complexes satisfying [[Acylindrically hyperbolic group|acylindricity]] conditions were constructed, in analogy with complexes like the [[Curve complex|complex of curves]] for mapping class groups;<ref>{{Cite web |last=Bestvina |first=Mladen |date=2023-12-15 |title=Groups acting on hyperbolic spaces—a survey |url=https://ems.press/books/standalone/274/5413 |access-date=2024-10-13 |website=ems.press |language=en}}</ref>
* For <math display="inline">n\ge 3</math>, <math display="inline">\mathrm{Out}(F_n)</math> is rigid with respect to [[Measured group theory|measure equivalence]] (Guirardel and Horbez, 2021 preprint).<ref>{{Citation |last=Guirardel |first=Vincent |title=Measure equivalence rigidity of $\mathrm{Out}(F_N)$ |date=2021-03-29 |url=https://arxiv.org/abs/2103.03696 |access-date=2024-12-03 |doi=10.48550/arXiv.2103.03696 |last2=Horbez |first2=Camille}}</ref>

==Outer space==
{{main article|Outer space (mathematics)}}
Out(''F<sub>n</sub>'') acts [[geometric group action|geometrically]] on a [[cell complex]] known as [[Marc Culler|Culler]]–[[Karen Vogtmann|Vogtmann]] Outer space, which can be thought of as the [[Teichmüller space|Fricke-Teichmüller space]] for a [[bouquet of circles]].

===Definition===
A point of the outer space is essentially an <math>\R</math>-graph ''X'' homotopy equivalent to a bouquet of ''n'' circles together with a certain choice of a free [[homotopy]] class of a [[Homotopy#Homotopy equivalence and null-homotopy|homotopy equivalence]] from ''X'' to the bouquet of ''n'' circles. An <math>\R</math>-graph is just a weighted [[Graph (discrete mathematics)|graph]] with weights in <math>\R</math>. The sum of all weights should be 1 and all weights  should be positive. To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least&nbsp;3.

A more descriptive view avoiding the homotopy equivalence ''f'' is the following. We may fix an identification of the [[fundamental group]] of the bouquet of ''n'' circles with the [[free group]] <math>F_n</math> in ''n'' variables. Furthermore, we may choose a [[Spanning tree (mathematics)|maximal tree]] in ''X'' and choose for each remaining edge a direction. We will now assign to each remaining edge ''e'' a word in <math>F_n</math> in the following way. Consider the closed path starting with ''e'' and then going back to the origin of ''e'' in the maximal tree. Composing this path with ''f'' we get a closed path in a bouquet of ''n'' circles and hence an element in its fundamental group <math>F_n</math>. This element is not well defined; if we change ''f'' by a free homotopy we obtain another element. It turns out, that those two elements are conjugate to each other, and hence we can choose the unique [[Free group#Construction|cyclically reduced]] element in this conjugacy class. It is possible to reconstruct the free homotopy type of ''f'' from these data. This view has the advantage, that it avoids the extra choice of ''f'' and has the disadvantage that additional ambiguity arises, because one has to choose a maximal tree and an orientation of the remaining edges.

The operation of Out(''F<sub>n</sub>'') on the outer space is defined as follows. Every automorphism ''g'' of <math>F_n</math> induces a self homotopy equivalence ''g′'' of the bouquet of ''n'' circles. Composing ''f'' with ''g′'' gives the desired action. And in the other model it is just application of ''g'' and making the resulting word cyclically reduced.

===Connection to length functions===
Every point in the outer space determines a unique length function <math>l_X \colon F_n \to \R</math>. A word in <math>F_n</math> determines via the chosen homotopy  equivalence a closed path in ''X''. The length of the word is then the minimal length of a path in the free homotopy class of that closed path. Such a length function is constant on each conjugacy class. The assignment <math>X \mapsto l_X</math> defines an embedding of the outer space to some infinite dimensional projective space.

===Simplicial structure on the outer space===
In the second model an open simplex is given by all those <math>\R</math>-graphs, which have combinatorically the same underlying graph and the same edges are labeled with the same words (only the length of the edges may differ). The boundary simplices of such a simplex consists of all graphs, that arise from this graph by collapsing an edge. If that edge is a loop it cannot be collapsed without changing the homotopy type of the graph. Hence there is no boundary simplex. So one can think about the outer space as a simplicial complex with some simplices removed. It is easy to verify, that the action of <math>\mathrm{Out}(F_n)</math> is simplicial and has finite isotropy groups.

==See also==
*[[Train track map]]
*[[Automorphism group of a free group]]
*[[Outer space (mathematics)|Outer space]]

==References==
{{Reflist}}
*{{cite journal | author1-link=Marc Culler|last1=Culler | first1 = Marc |author2-link=Karen Vogtmann|first2=Karen|last2= Vogtmann  | title = Moduli of graphs and automorphisms of free groups | journal = [[Inventiones Mathematicae]] | volume = 84 | issue = 1 | pages = 91–119 | year=1986 | url=http://www.math.cornell.edu/~vogtmann/ScannedPapers/1986.0084.pdf | doi = 10.1007/BF01388734|mr=0830040}}
*{{cite journal | last = Vogtmann | first=Karen | title = Automorphisms of free groups and outer space | journal = [[Geometriae Dedicata]] | volume = 94 | pages=1–31 | year=2002 | url=http://www.math.cornell.edu/~vogtmann/papers/Autosurvey/autosurvey.pdf | doi = 10.1023/A:1020973910646|mr=1950871}}
*{{citation | first = Karen | last = Vogtmann | title = What is … outer space? | journal = [[Notices of the American Mathematical Society]]| volume = 55 | issue = 7 | pages = 784–786 | year = 2008
    | url = https://www.ams.org/notices/200807/tx080700784p.pdf|mr=2436509 }}

[[Category:Geometric group theory]]