Out(F_n)

Remarks

For $n = 1$, $F_1 = \Z$ has the two automorphisms $x \mapsto \pm x$ and no nontrivial inner automorphisms, so $\mathrm{Out}(F_1) \cong \Z/2\Z$. Every automorphism of $F_n$ induces an automorphism of the abelianisation $\Z^n$, and inner automorphisms induce the identity. This gives a homomorphism $\mathrm{Out}(F_n) \to \mathrm{GL}_n(\Z)$, into the general linear group over $\Z$, which is surjective, because elementary matrices and sign changes lift to automorphisms of $F_n$. For $n = 2$ this map is an isomorphism, so $\mathrm{Out}(F_2) \cong \mathrm{GL}_2(\Z)$. For $n \ge 3$ its kernel is infinite.

$\mathrm{Out}(F_n)$ is finitely presented. Culler and Vogtmann built a contractible space, Outer space, on which $\mathrm{Out}(F_n)$ acts properly; it plays the role that Teichmüller space plays for the mapping class group of a surface. $\mathrm{Out}(F_n)$ satisfies the Tits alternative.