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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.
## Structure

The abelianization map F_{n} → Z^{n} induces a homomorphism Out(F_{n}) → GL(n,Z), the latter being the automorphism group of Z^{n}. This map is onto, making Out(F_{n}) a group extension## 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 surfaces is the outer automorphism group of the fundamental group of that surface.) In particular, 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 isotopic to the identity map.
## Outer space

Out(F_{n}) acts geometrically on a cell complex known as outer space, which can be thought of as the Teichmüller space for a bouquet of circles.
## References

- $mbox\{Tor\}(F\_n)\; rightarrow\; mbox\{Out\}(F\_n)\; rightarrow\; mbox\{GL\}(n,mathbb\{Z\})$

The kernel Tor(F_{n}) is the Torelli group of F_{n}.

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

- Culler, Marc; Vogtmann, Karen "Moduli of graphs and automorphisms of free groups".
*Inventiones Mathematicae*84 (1): 91–119. - Vogtmann, Karen "Automorphisms of free groups and outer space".
*Geometriae Dedicata*94 1–31.

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Last updated on Monday June 09, 2008 at 02:58:15 PDT (GMT -0700)

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Last updated on Monday June 09, 2008 at 02:58:15 PDT (GMT -0700)

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