In this package we provide methods to compute with automorphisms of the free group of rank \(2\). The details of the algorithms are available in [{Fer}26]. We use the following notation:
\[ \sigma=\begin{cases} x \mapsto y,\\ y \mapsto x, \end{cases}\quad \phi_1=\begin{cases} x \mapsto xy^{-1},\\ y \mapsto y, \end{cases}\quad \phi_2=\begin{cases} x \mapsto x,\\ y \mapsto yx, \end{cases}\quad \phi_3=\begin{cases} x \mapsto y^{-1}x,\\ y \mapsto y. \end{cases} \]
Then \(\mathrm{Aut}(F_2)\) has the following presentation.
\[ \begin{array}{r@{}l} \mathrm{Aut}(F_2) = \langle \sigma,\phi_1,\phi_2,\phi_3 \mid & \phi_1\phi_3=\phi_3\phi_1, \quad \phi_2\phi_3\phi_2 = \phi_3\phi_2\phi_3, \quad \phi_2\phi_1\phi_2 = \phi_1\phi_2\phi_1, \\ & (\phi_1\phi_2\phi_3)^4=1, \quad \sigma^2=1, \quad \phi_1^{\sigma} = \phi_2^{-1}, \\ & \phi_3\sigma = \sigma\phi_3^{-1}\phi_2^{-1}\phi_1^{-1}\phi_2\phi_3, \quad \phi_3^{-1}\sigma= \sigma\phi_3^{-1}\phi_2^{-1}\phi_1\phi_2\phi_3\rangle. \end{array} \]
Denote the commutator subgroup of a group \(G\) by \(G^{\prime}\). For \(\psi\in \mathrm{Aut}(F_2)\), we define the map \(\psi^\ast\) by \(\psi^\ast(x F_2^{\prime}) = \psi(x) F_2^{\prime}\). If \(\overline{\cdot}\) denotes the isomorphism between \(\mathrm{Aut}(F_2/F_2^\prime)\) and \(\mathrm{GL}_2(\mathbb{Z})\), then the map
\[ \begin{array}{r@{}l} \Psi \colon & \mathrm{Aut}(F_2) \to \mathrm{GL}_2(\mathbb{Z}) \\ & \psi \mapsto \overline{(\psi^\ast)} \end{array} \]
is a group homomorphism and the kernel of this map is \(\mathrm{Inn}(F_2)\). We denote by \(\mathrm{SA}_2\) the subgroup of \(\mathrm{Aut}(F_2)\) consisting of those automorphisms that are mapped into \(\mathrm{SL}_2(\mathbb{Z})\) under \(\Psi\). Let \(B_4\) denote the braid group on \(4\) strands. Then \(\mathrm{SA}_2 \cong B_4 / Z(B_4)\). Using this, we have that each automorphism can be written uniquely as a word of the form
\[ \alpha = \sigma^{\varepsilon_1}\Delta^{\varepsilon_2}\alpha_1\alpha_2\cdots\alpha_k \]
where \(\varepsilon_1,\varepsilon_2 \in \{0,1\}\), \(\Delta = \phi_1(\phi_2\phi_1)(\phi_3\phi_2\phi_1)\), and \(\alpha_1,\dots,\alpha_k\in \mathrm{SA}_2\) whose images under the isomorphism between \(\mathrm{SA}_2\) and \(B_4 / Z(B_4)\) are simple braids. We call this expression the left canonical form of \(\alpha\).
generated by GAPDoc2HTML