Classification of automorphisms of the free group of rank 2 by ranks of fixedpoint subgroups

Let Fn be a free group of rank n and let Aut(Fn) be the automorphism group of Fn. For any α ∈ Aut(Fn) let Fix(α) denote the fixed point subgroup of α: Fix(α) = {x ∈ Fn|α(x) = x}. Bestvina and Handel [BH] proved that rk(Fix(α)) ≤ n. Collins and Turner [CT] classified the automorphisms α with rk(Fix(α)) = n. Here we give a classification of automorphisms of the free group of rank 2 by ranks of fixed point subgroups, a classification of fixed point subgroups and a classification of stabilizers of elements. As a corollary we obtain an efficient algorithm solving the conjugacy problem in Aut(F2). We give an algorithm for finding a basis of Fix(α) for α ∈ Aut(F2). 1. Formulations of Theorems Let F2 be the free group with the base {a, b}. Define some automorphisms of F2: β: a 7 → a b 7 → ab, γ: a 7 → b−1a b 7 → b, ρ: a 7 → ab b 7 → a−1, τ: a 7 → b b 7 → a−1b−1, σ: a 7 → b−1 b 7 → a, σ1: a 7 → b b 7 → a, σ2: a 7 → a−1 b 7 → b−1, σ3: a 7 → a−1 b 7 → b. We use the following rule of composition of automorphisms: if ϕ, ψ ∈ Aut(F2) and x ∈ F2 then ϕψ(x) = ψ(ϕ(x)). For x ∈ F2 denote by St(x) the stabilizer of x in Aut(F2). For a subset X of a group denote by 〈X 〉 the subgroup generated by X. Denote [x, y] = x−1y−1xy, xy = y−1xy. For g ∈ F2 denote by g ̂ the automorphism induced by the conjugation by g: ĝ(x) = g−1xg, x ∈ F2. Let − : Aut(F2) → GL2(Z) be the homomorphism induced by the abelianization of F2. It is known that the group of inner automorphisms Inn(F2) is the kernel of this homomorphism and tha