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Out(F_n)

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Outer automorphism group of a free group on n generators

Inmathematics,Out(F_n) is theouter automorphism group of afree group onngenerators. These groups are at universal stage ingeometric group theory, as they act on the set ofpresentations with $n {\displaystyle n}$ generators of anyfinitely generated group.^[1] Despite geometric analogies with generallinear groups andmapping class groups, their complexity is generally regarded as more challenging, which has fueled the development of new techniques in the field.

Definition

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Let $F_{n}$ be the free nonabelian group of rank $n\geq 1$ . The set ofinner automorphisms of $F_{n}$ , i.e. automorphisms obtained as conjugations by an element of $F_{n}$ , is anormal subgroup $\mathrm {Inn} (F_{n})\triangleleft \mathrm {Aut} (F_{n})$ . The outer automorphism group of $F_{n}$ is the quotient $\mathrm {Out} (F_{n}):=\mathrm {Aut} (F_{n})/\mathrm {Inn} (F_{n}).$ An element of $\mathrm {Out} (F_{n})$ is called an outer class.

Relations to other groups

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Linear groups

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Theabelianization map $F_{n}\to \mathbb {Z} ^{n}$ induces ahomomorphism from $\mathrm {Out} (F_{n})$ to the general linear group $\mathrm {GL} (n,\mathbb {Z} )$ , the latter being theautomorphism group of $\mathbb {Z} ^{n}$ . This map is onto, making $\mathrm {Out} (F_{n})$ agroup extension,

1\to \mathrm {Tor} (F_{n})\to \mathrm {Out} (F_{n})\to \mathrm {GL} (n,\mathbb {Z} )\to 1

The kernel $\mathrm {Tor} (F_{n})$ is theTorelli group of $F_{n}$ .

The map $\mathrm {Out} (F_{2})\to \mathrm {GL} (2,\mathbb {Z} )$ is anisomorphism. This no longer holds for higher ranks: the Torelli group of $F_{3}$ contains the automorphism fixing two basis elements and multiplying the remaining one by the commutator of the two others.

Aut(F_n)

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By definition, $\mathrm {Aut} (F_{n})$ is an extension of the inner automorphism group $\mathrm {Inn} (F_{n})$ by $\mathrm {Out} (F_{n})$ . The inner automorphism group itself is the image of theaction by conjugation, which has kernel thecenter $Z(F_{n})$ . Since $Z(F_{n})$ is trivial for $n\geq 2$ , this gives a short exact sequence $1\rightarrow F_{n}\rightarrow \mathrm {Aut} (F_{n})\rightarrow \mathrm {Out} (F_{n})\rightarrow 1.$ For all $n\geq 2$ , there are embeddings $\mathrm {Aut} (F_{n})\longrightarrow \mathrm {Out} (F_{n+1})$ obtained by taking the outer class of the extension of an automorphism of $F_{n}$ fixing the additional generator. Therefore, when studying properties that are inherited by subgroups and quotients, the theories of $\mathrm {Aut} (F_{n})$ and $\mathrm {Out} (F_{n})$ are essentially the same.

Mapping class groups of surfaces

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Because $F_{n}$ is thefundamental group of abouquet ofn circles, $\mathrm {Out} (F_{n})$ can be described topologically as themapping class group of a bouquet ofn circles (in thehomotopy category), in analogy to the mapping class group of a closedsurface which is isomorphic to the outer automorphism group of the fundamental group of that surface.

Given any finite graph with fundamental group $F_{n}$ , the graph can be "thickened" to a surface $S {\displaystyle S}$ with one boundary component that retracts onto the graph. TheBirman exact sequence yields a map from the mapping class group $\mathrm {MCG} (S)\longrightarrow \mathrm {Out} (F_{n})$ . The elements of $\mathrm {Out} (F_{n})$ 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,^[2] providing geometric realization for outer classes fixing a conjugacy class.

Known results

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For ${\textstyle n\geq 4}$ , ${\textstyle \mathrm {Out} (F_{n})}$ is not linear, i.e. it has nofaithful representation by matrices over a field (Formanek, Procesi, 1992);^[3]
For $n\geq 3$ , theisoperimetric function of $\mathrm {Out} (F_{n})$ is exponential (Hatcher, Vogtmann, 1996);^[4]
TheTits Alternative holds in $\mathrm {Out} (F_{n})$ : each subgroup is eithervirtually solvable or else it contains a free group of rank 2 (Bestvina, Feighn, Handel, 2000);^[5]
For ${\textstyle n\geq 3}$ , ${\textstyle \mathrm {Out} (\mathrm {Out} (F_{n}))={1}}$ (Bridson and Vogtmann, 2000);^[6]
Every solvable subgroup of $\mathrm {Out} (F_{n})$ has a finitely generatedfree abelian subgroup of finite index (Bestvina, Feighn, Handel, 2004);^[7]
For $i>0$ , all but finitely many of the $i {\displaystyle i}$ ^th-degree homology morphisms induced by the sequence $\ldots \rightarrow \mathrm {Out} (F_{n-1})\rightarrow \mathrm {Out} (F_{n})\rightarrow \mathrm {Out} (F_{n+1})\rightarrow \ldots$ are isomorphisms (Hatcher and Vogtmann, 2004);^[8]
For ${\textstyle n\geq 2}$ , the reduced ${\textstyle C^{*}}$ -algebra of ${\textstyle \mathrm {Out} (F_{n})}$ (i.e. the closure of its image under theregular representation) is simple;^[9]
For ${\textstyle n\geq 4}$ , if ${\textstyle \Gamma }$ is a finite index subgroup of ${\textstyle \mathrm {Out} (F_{n})}$ , then any subgroup of ${\textstyle \mathrm {Out} (F_{n})}$ isomorphic to ${\textstyle \Gamma }$ is a conjugate of ${\textstyle \Gamma }$ (Farb and Handel, 2007);^[10]
For ${\textstyle n\geq 5}$ , ${\textstyle \mathrm {Out} (F_{n})}$ hasKazhdan's property (T) (Kaluba, Nowak, Ozawa, 2019 for ${\textstyle n=5}$ ; Kaluba, Kielak, Nowak, 2021 for $n\geq 6$ );^[11]
Actions onhyperbolic complexes satisfyingacylindricity conditions were constructed, in analogy with complexes like thecomplex of curves for mapping class groups;^[12]
For ${\textstyle n\geq 3}$ , ${\textstyle \mathrm {Out} (F_{n})}$ is rigid with respect tomeasure equivalence (Guirardel and Horbez, 2021 preprint).^[13]

Outer space

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Main article:Outer space (mathematics)

Out(F_n) actsgeometrically on acell complex known asCuller–Vogtmann Outer space, which can be thought of as theFricke-Teichmüller space for abouquet of circles.

Definition

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A point of the outer space is essentially an $\mathbb {R}$ -graphX homotopy equivalent to a bouquet ofn circles together with a certain choice of a freehomotopy class of ahomotopy equivalence fromX to the bouquet ofn circles. An $\mathbb {R}$ -graph is just a weightedgraph with weights in $\mathbb {R}$ . 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 3.

A more descriptive view avoiding the homotopy equivalencef is the following. We may fix an identification of thefundamental group of the bouquet ofn circles with thefree group $F_{n}$ inn variables. Furthermore, we may choose amaximal tree inX and choose for each remaining edge a direction. We will now assign to each remaining edgee a word in $F_{n}$ in the following way. Consider the closed path starting withe and then going back to the origin ofe in the maximal tree. Composing this path withf we get a closed path in a bouquet ofn circles and hence an element in its fundamental group $F_{n}$ . This element is not well defined; if we changef 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 uniquecyclically reduced element in this conjugacy class. It is possible to reconstruct the free homotopy type off from these data. This view has the advantage, that it avoids the extra choice off 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_n) on the outer space is defined as follows. Every automorphismg of $F_{n}$ induces a self homotopy equivalenceg′ of the bouquet ofn circles. Composingf withg′ gives the desired action. And in the other model it is just application ofg and making the resulting word cyclically reduced.

Connection to length functions

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Every point in the outer space determines a unique length function $l_{X}\colon F_{n}\to \mathbb {R}$ . A word in $F_{n}$ determines via the chosen homotopy equivalence a closed path inX. 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 $X\mapsto l_{X}$ defines an embedding of the outer space to some infinite dimensional projective space.

Simplicial structure on the outer space

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In the second model an open simplex is given by all those $\mathbb {R}$ -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 $\mathrm {Out} (F_{n})$ is simplicial and has finite isotropy groups.