Automorphisms of free groups

The groups Aut(F_n) and Out(F_n): fixed points and fixed subgroups of automorphisms, word problem, automaticity, linearity of IA-automorphisms, embeddings of free products into Aut(F_n).

• Automaticity of automorphism groups of free groups
• Endomorphisms of free and free metabelian groups
• Finite-index subgroups of automorphism groups of free groups
• Fixed subgroups of free group automorphisms
• Free products in automorphism groups of free groups
• Linearity of IA-automorphism groups
• Word problem for automorphism groups of free groups

AbbreviationOuter automorphism group of a free group#✓fully proved

For a natural number \(n\), let \(F_n\) be the free group on \(n\) generators. Its outer automorphism group \(\mathrm{Out}(F_n)\) is the quotient of the automorphism group \(\mathrm{Aut}(F_n)\) by the normal subgroup generated by the inner automorphisms. Since the inner automorphisms already form a normal subgroup \(\mathrm{Inn}(F_n)\), this is \(\mathrm{Aut}(F_n) / \mathrm{Inn}(F_n)\).

abbrev OutFreeGroup (n : ℕ) : Type :=
  MulAut (FreeGroup (Fin n)) ⧸ Subgroup.normalClosure
    (Set.range (MulAut.conj : FreeGroup (Fin n) →* MulAut (FreeGroup (Fin n))))

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