Out(F_n)

In mathematics, Out(F_n) is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.

Outer space

Out(F_n) acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Teichmüller space for a bouquet of circles.

Definition

A point of the outer space is essentially an R-graph X homotopy equivalent to a bouquet of n circles together with a certain choice of a free homotopy class of a homotopy equivalence from X to the Bouquet of n circles. An $\mathbb {R}$ - Graph is just a weighted Graph 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 2.

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 $F_{n}$ in $n$ variables. Furthermore we may choose a maximal tree in $X$ and choose for each remaining edge a direction. We will now assign to each remaining edge $e$ a word in $F_{n}$ 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 a element in its fundamental group $F_{n}$ . 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 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 a orientation of the remaining edges.

The operation of $\mathrm {Out} (F_{n})$ on the outer space is defined as follows. Every Automorphism $g$ of $F_{n}$ 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 $l_{X}:F_{n}\rightarrow \mathbb {R}$ . A word in $F_{n}$ 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 $X\mapsto l_{X}$ defines an embedding of the outer space to some infinite dimensional projective space.

Simplicial structure on the outer space

In the second model a 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 a 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 $Out(F_{n})$ is simplicial and has finite isotropy groups.

Structure

The abelianization map F_n → Zⁿ induces a homomorphism Out(F_n) → GL(n,Z), the latter being the automorphism group of Zⁿ. This map is onto, making Out(F_n) a group extension

{\mbox{Tor}}(F_{n})\rightarrow {\mbox{Out}}(F_{n})\rightarrow {\mbox{GL}}(n,\mathbb {Z} ).

The kernel Tor(F_n) is the Torelli group of F_n.

In the case n = 2, the map Out(F₂) → GL(2,Z) is an isomorphism.

Analogy with mapping class groups

Because F_n is the fundamental group of a bouquet of circles, Out(F_n) can be thought of as the mapping class group of a bouquet of n circles. (The mapping class group of a surface is the outer automorphism group of the fundamental group of that surface.) By analogy, Out(F_n) can be described as the quotient G/H, where G is the group of all self-homotopy equivalences of the bouquet of circles, and H is the subgroup of G consisting of homotopy equivalences that are homotopic to the identity map.

References

Culler, Marc (1986). "Moduli of graphs and automorphisms of free groups" (PDF). Inventiones Mathematicae. 84 (1): 91–119. doi:10.1007/BF01388734. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
Vogtmann, Karen (2002). "Automorphisms of free groups and outer space" (PDF). Geometriae Dedicata. 94: 1–31. doi:10.1023/A:1020973910646.
Vogtmann, Karen (2008), "What Is . . . Outer Space?" (PDF), AMS Notices, 55 (07): 784–786.