map out


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


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

mbox{Tor}(F_n) rightarrow mbox{Out}(F_n) rightarrow mbox{GL}(n,mathbb{Z})

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

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

Analogy with mapping class groups

Because Fn is the fundamental group of a bouquet of circles, Out(Fn) 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(Fn) 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(Fn) acts geometrically on a cell complex known as outer space, which can be thought of as the Teichmüller space for a bouquet of circles.


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