Out(Fn)
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- The correct title of this article is Out(Fn). It features superscript or subscript characters that are substituted or omitted because of technical limitations.
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.
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[edit] Structure
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
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.
[edit] 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.
[edit] 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.
[edit] References
- Culler, Marc; Vogtmann, Karen (1986). "Moduli of graphs and automorphisms of free groups". Inventiones Mathematicae 84 (1): 91–119. doi:.
- Vogtmann, Karen (2002). "Automorphisms of free groups and outer space". Geometriae Dedicata 94: 1–31. doi:.
