Cote n° 145 · pages 1–37
· 162 displayed formulas · Printemps 81 (vieille rédaction) : notes manuscrites (s.d.).
Inventory dating : 1981
Édition de démonstration
\[\pi \subset G\]
LaTeX source
\[ \pi \subset G \]
\[H = \Bigl\{ u \in G \Bigm| \exists\; \underbrace{\pi' \subset \pi}_{\substack{\text{ss-gr.} \\ \text{d'indice fini}}},\;
\underbrace{\pi'' \subset \pi}_{\text{id.}},\]
LaTeX source
\[
H = \Bigl\{ u \in G \Bigm| \exists\; \underbrace{\pi' \subset \pi}_{\substack{\text{ss-gr.} \\ \text{d'indice fini}}},\;
\underbrace{\pi'' \subset \pi}_{\text{id.}},
\]\[v_* : \pi' \longrightarrow \pi''\]
LaTeX source
\[ v_* : \pi' \longrightarrow \pi'' \]
\[\begin{array}{c}
\pi \\
\cup \quad \cup \\
\pi' \simeq \pi'' \\
\pi'_i \simeq \pi''_i
\end{array}\]
LaTeX source
\[
\begin{array}{c}
\pi \\
\cup \quad \cup \\
\pi' \simeq \pi'' \\
\pi'_i \simeq \pi''_i
\end{array}
\]\[p'' u = v p', \qquad \struck{p'' u \gamma =}, \qquad
p''(\gamma u) = (p''\gamma) u = p'' u = v p'\]
LaTeX source
\[
p'' u = v p', \qquad \struck{p'' u \gamma =}, \qquad
p''(\gamma u) = (p''\gamma) u = p'' u = v p'
\]\[\struck{\pi \subset H}\]
LaTeX source
\[
\struck{\pi \subset H}
\]\[\begin{aligned}
u(l_1) &= \mathrm{int}(g_1)\, l_1^{p} \\
u(l_\infty) &= \mathrm{int}(\underbrace{\sigma_0(g_1)}_{g_\infty})\, l_\infty^{p}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
u(l_1) &= \mathrm{int}(g_1)\, l_1^{p} \\
u(l_\infty) &= \mathrm{int}(\underbrace{\sigma_0(g_1)}_{g_\infty})\, l_\infty^{p}
\end{aligned}
\]\[g_1 \in \pi \bmod L_1, \qquad p \in \hat{\mathbb{Z}}^{*} \quad \text{satisfaisant}\]
LaTeX source
\[
g_1 \in \pi \bmod L_1, \qquad p \in \hat{\mathbb{Z}}^{*} \quad \text{satisfaisant}
\]\[(1) \qquad \boxed{l_\infty^{p}\,.\,\mathrm{int}(h_1)(l_1^{p})\,.\,
\mathrm{int}\bigl(h_1\, l_1^{-\gamma}\, \rho^{-1}(h_1)\bigr)(l_0^{p}) = 1}\]
LaTeX source
\[
(1) \qquad \boxed{l_\infty^{p}\,.\,\mathrm{int}(h_1)(l_1^{p})\,.\,
\mathrm{int}\bigl(h_1\, l_1^{-\gamma}\, \rho^{-1}(h_1)\bigr)(l_0^{p}) = 1}
\]\[\text{et} \quad (2) \qquad \boxed{\rho(h_1)\bigl(l_\infty^{-\gamma}\, h_1\, l_1^{-\gamma}\bigr)\rho^{-1}(h_1) = l_0^{\beta}}\]
LaTeX source
\[
\text{et} \quad (2) \qquad \boxed{\rho(h_1)\bigl(l_\infty^{-\gamma}\, h_1\, l_1^{-\gamma}\bigr)\rho^{-1}(h_1) = l_0^{\beta}}
\]\[h_1 = \sigma_0(g_1)^{-1}\, g_1\]
LaTeX source
\[
h_1 = \sigma_0(g_1)^{-1}\, g_1
\]\[h_1\, l_1^{-\gamma}\, \rho^{-1}(h_1) = \struck{l_\infty} l_\infty^{\gamma}\, \rho(h_1)^{-1}\, l_0^{\beta}\]
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\[
h_1\, l_1^{-\gamma}\, \rho^{-1}(h_1) = \struck{l_\infty} l_\infty^{\gamma}\, \rho(h_1)^{-1}\, l_0^{\beta}
\]\[(1') \qquad \boxed{l_\infty^{p}\,.\,\mathrm{int}(h_1)(l_1^{p})\,.\,
\mathrm{int}\bigl(l_\infty^{\gamma}\, \rho(h_1)^{-1}\bigr)(l_0^{p}) = 1}\]
LaTeX source
\[
(1') \qquad \boxed{l_\infty^{p}\,.\,\mathrm{int}(h_1)(l_1^{p})\,.\,
\mathrm{int}\bigl(l_\infty^{\gamma}\, \rho(h_1)^{-1}\bigr)(l_0^{p}) = 1}
\]\[h'_1 = l_\infty^{-\mu}\, h_1\, l_1^{\mu}\]
LaTeX source
\[
h'_1 = l_\infty^{-\mu}\, h_1\, l_1^{\mu}
\]\[l_\infty^{p}\,.\,\mathrm{int}(\uncertain{h'_1})(l_1^{p})\,.\,
\mathrm{int}\bigl(\struck{\ill{}}\; l_\infty^{\gamma - \mu}\, \rho^{-1}(h_1'^{-1})\bigr)(l_0^{p}) = 1\]
LaTeX source
\[
l_\infty^{p}\,.\,\mathrm{int}(\uncertain{h'_1})(l_1^{p})\,.\,
\mathrm{int}\bigl(\struck{\ill{}}\; l_\infty^{\gamma - \mu}\, \rho^{-1}(h_1'^{-1})\bigr)(l_0^{p}) = 1
\]\[\struck{u''\varphi} \qquad u'\varphi u^{-1} = \varphi\]
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\[
\struck{u''\varphi} \qquad u'\varphi u^{-1} = \varphi
\]\[u'\varphi = \varphi u \qquad u'\varphi = v'\]
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\[ u'\varphi = \varphi u \qquad u'\varphi = v' \]
\[v'\varphi = \varphi u\]
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\[ v'\varphi = \varphi u \]
\[u'\varphi u^{-1} = \mathrm{int}(g)\, \varphi u\]
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\[
u'\varphi u^{-1} = \mathrm{int}(g)\, \varphi u
\]\[u'\varphi = \struck{\varphi u}\; \mathrm{int}(g)\, \varphi(u)\]
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\[
u'\varphi = \struck{\varphi u}\; \mathrm{int}(g)\, \varphi(u)
\]\[\mathcal{E}_{\varphi} = \bigl\{ u \in \mathrm{Aut}(\pi) \bigm| \exists\, u' \in \mathrm{Aut}(\pi')\; \ldots\]
LaTeX source
\[
\mathcal{E}_{\varphi} = \bigl\{ u \in \mathrm{Aut}(\pi) \bigm| \exists\, u' \in \mathrm{Aut}(\pi')\; \ldots
\]\[l_0\, l_1\, l_\infty = 1\]
LaTeX source
\[ l_0\, l_1\, l_\infty = 1 \]
\[\rho(l_0) = l_1, \quad \rho(l_1) = l_\infty, \quad \rho(l_\infty) = l_0\]
LaTeX source
\[ \rho(l_0) = l_1, \quad \rho(l_1) = l_\infty, \quad \rho(l_\infty) = l_0 \]
\[\sigma_\infty(l_0) = l_1, \quad \sigma_\infty(l_1) = l_0, \quad
\sigma_\infty(l_\infty) = (l_1 l_0)^{-1} = l_0^{-1} l_1^{-1}
\quad \bigl(= \mathrm{int}(l_0^{-1}).\, l_\infty\bigr)\]
LaTeX source
\[
\sigma_\infty(l_0) = l_1, \quad \sigma_\infty(l_1) = l_0, \quad
\sigma_\infty(l_\infty) = (l_1 l_0)^{-1} = l_0^{-1} l_1^{-1}
\quad \bigl(= \mathrm{int}(l_0^{-1}).\, l_\infty\bigr)
\]\[\rho\, \sigma_i\, \rho^{-1} = \sigma_{\rho(i)}\]
LaTeX source
\[
\rho\, \sigma_i\, \rho^{-1} = \sigma_{\rho(i)}
\]\[\sigma_0\sigma_1 \;\bigl(= \sigma_1\sigma_\infty = \sigma_\infty\sigma_0\bigr) = \rho\]
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\[ \sigma_0\sigma_1 \;\bigl(= \sigma_1\sigma_\infty = \sigma_\infty\sigma_0\bigr) = \rho \]
\[\struck{[\sigma_0, \sigma_1] \ill{}} = \struck{(\sigma_0\sigma_1)^2 = \rho^2}
\qquad \text{dans } \mathfrak{S}_3\]
LaTeX source
\[
\struck{[\sigma_0, \sigma_1] \ill{}} = \struck{(\sigma_0\sigma_1)^2 = \rho^2}
\qquad \text{dans } \mathfrak{S}_3
\]\[\begin{aligned}
\sigma_0\sigma_1(l_0) &= \sigma_0(l_\infty) = l_1 = \rho(l_0) \\
\sigma_0\sigma_1(l_1) &= \sigma_0(l_\infty^{-1} l_0^{-1}) = \sigma_0(l_\infty)^{-1}\sigma_0(l_0)^{-1}
= l_1^{-1}\,(l_1^{-1} l_\infty^{-1})^{-1} = l_1^{-1}\, l_\infty\, l_1 \\
\sigma_0\sigma_1(l_\infty) &= (l_1^{-1} l_\infty l_1)^{-1}\, l_1^{-1}
= l_1^{-1} l_\infty^{-1} \struck{l_1} = l_1^{-1}\,(l_\infty^{-1} l_1^{-1})\, l_1 = l_1^{-1}\, l_0\, l_1
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\sigma_0\sigma_1(l_0) &= \sigma_0(l_\infty) = l_1 = \rho(l_0) \\
\sigma_0\sigma_1(l_1) &= \sigma_0(l_\infty^{-1} l_0^{-1}) = \sigma_0(l_\infty)^{-1}\sigma_0(l_0)^{-1}
= l_1^{-1}\,(l_1^{-1} l_\infty^{-1})^{-1} = l_1^{-1}\, l_\infty\, l_1 \\
\sigma_0\sigma_1(l_\infty) &= (l_1^{-1} l_\infty l_1)^{-1}\, l_1^{-1}
= l_1^{-1} l_\infty^{-1} \struck{l_1} = l_1^{-1}\,(l_\infty^{-1} l_1^{-1})\, l_1 = l_1^{-1}\, l_0\, l_1
\end{aligned}
\]\[\sigma_0\sigma_1 = \mathrm{int}(l_1^{-1})\,\rho \qquad \text{dans } \mathrm{Aut}(\pi)\]
LaTeX source
\[
\sigma_0\sigma_1 = \mathrm{int}(l_1^{-1})\,\rho \qquad \text{dans } \mathrm{Aut}(\pi)
\]\[\sigma_1\sigma_\infty = \mathrm{int}(l_\infty^{-1})\,\rho, \qquad
\sigma_\infty\sigma_0 = \mathrm{int}(l_0^{-1})\,\rho\]
LaTeX source
\[
\sigma_1\sigma_\infty = \mathrm{int}(l_\infty^{-1})\,\rho, \qquad
\sigma_\infty\sigma_0 = \mathrm{int}(l_0^{-1})\,\rho
\]\[\tau(l_0) = l_0^{-1}, \qquad \tau(l_1) = l_1^{-1}, \qquad
\tau(l_\infty) = l_1\, l_0 = \mathrm{int}(l_1)\, l_\infty^{-1}\]
LaTeX source
\[
\tau(l_0) = l_0^{-1}, \qquad \tau(l_1) = l_1^{-1}, \qquad
\tau(l_\infty) = l_1\, l_0 = \mathrm{int}(l_1)\, l_\infty^{-1}
\]\[\tau^2 = 1\]
LaTeX source
\[ \tau^2 = 1 \]
\[(*) \qquad u(l_0) = l_0^{p}\]
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\[
(*) \qquad u(l_0) = l_0^{p}
\]\[(**) \qquad u\rho = \struck{\rho u}\; \mathrm{int}(g)\, \rho u ,\]
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\[
(**) \qquad u\rho = \struck{\rho u}\; \mathrm{int}(g)\, \rho u ,
\]\[\lambda\, g\, \rho(\lambda)^{-1} = \mathbf{l_0^{\alpha}\, g\, l_1^{-\alpha}}.\]
LaTeX source
\[
\lambda\, g\, \rho(\lambda)^{-1} = \mathbf{l_0^{\alpha}\, g\, l_1^{-\alpha}}.
\]\[\begin{cases}
u(l_0) = l_0^{p} \\
u(l_1) = u\rho(l_0) = \mathrm{int}(g)\,\rho\, \underbrace{u(l_0)}_{l_0^{p}} = \mathrm{int}(g)\, l_1^{p} \\
u(l_\infty) = u\rho(l_1) = \mathrm{int}(g)\, \rho\, \underbrace{u(l_1)} = \mathrm{int}(g)\, \underbrace{\rho\, \mathrm{int}(g)\, \rho^{-1}}_{\mathrm{int}(\rho(g))}\, \underbrace{(\rho\, l_1^{p})}_{l_\infty^{p}} = \struck{\mathrm{int}(g\rho(g))\ill{}}
\end{cases}\]
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\[
\begin{cases}
u(l_0) = l_0^{p} \\
u(l_1) = u\rho(l_0) = \mathrm{int}(g)\,\rho\, \underbrace{u(l_0)}_{l_0^{p}} = \mathrm{int}(g)\, l_1^{p} \\
u(l_\infty) = u\rho(l_1) = \mathrm{int}(g)\, \rho\, \underbrace{u(l_1)} = \mathrm{int}(g)\, \underbrace{\rho\, \mathrm{int}(g)\, \rho^{-1}}_{\mathrm{int}(\rho(g))}\, \underbrace{(\rho\, l_1^{p})}_{l_\infty^{p}} = \struck{\mathrm{int}(g\rho(g))\ill{}}
\end{cases}
\]\[\boxed{l_0^{p}\,.\,\mathrm{int}(g)\, l_1^{p}\,.\,\mathrm{int}\bigl(g\rho(g)\bigr)\, l_\infty^{p} = 1}\]
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\[
\boxed{l_0^{p}\,.\,\mathrm{int}(g)\, l_1^{p}\,.\,\mathrm{int}\bigl(g\rho(g)\bigr)\, l_\infty^{p} = 1}
\]\[l_0^{p}\,\bigl(g\, l_1^{p}\, g^{-1}\bigr)\bigl(g\rho(g)\, l_\infty^{p}\, \rho(g)^{-1} g^{-1}\bigr) = 1\]
LaTeX source
\[
l_0^{p}\,\bigl(g\, l_1^{p}\, g^{-1}\bigr)\bigl(g\rho(g)\, l_\infty^{p}\, \rho(g)^{-1} g^{-1}\bigr) = 1
\]\[l_0^{p}\, g\, l_1^{p}\, \rho(g)\, l_\infty^{p}\, \rho(g)^{-1}\, g^{-1} = 1\]
LaTeX source
\[
l_0^{p}\, g\, l_1^{p}\, \rho(g)\, l_\infty^{p}\, \rho(g)^{-1}\, g^{-1} = 1
\]\[\struck{l_0^{p}\,.\,\mathrm{int}(g)\bigl(l_1^{p}\,.\,\mathrm{int}(\rho(g))(l_\infty^{p})\bigr) = 1}\]
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\[
\struck{l_0^{p}\,.\,\mathrm{int}(g)\bigl(l_1^{p}\,.\,\mathrm{int}(\rho(g))(l_\infty^{p})\bigr) = 1}
\]\[\mathrm{int}(g^{-1})\, l_0^{p}\,.\, l_1^{p}\,.\,\mathrm{int}(\rho(g))(l_\infty^{p}) = 1\]
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\[
\mathrm{int}(g^{-1})\, l_0^{p}\,.\, l_1^{p}\,.\,\mathrm{int}(\rho(g))(l_\infty^{p}) = 1
\]\[\tau\rho = \mathrm{int}(l_1)\, \rho\, \tau\]
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\[
\tau\rho = \mathrm{int}(l_1)\, \rho\, \tau
\]\[u\,\sigma_\infty = \mathrm{int}(h)\, \struck{\ill{}}\, \sigma_\infty\, u ,\]
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\[
u\,\sigma_\infty = \mathrm{int}(h)\, \struck{\ill{}}\, \sigma_\infty\, u ,
\]\[\underbrace{u\,\sigma_\infty(l_0)}_{\substack{\| \\ u(l_1) \\ \mathrm{int}(g)\, l_1^{p}}}
= \mathrm{int}(h)\, \underbrace{\sigma_\infty(l_0^{p})}_{l_1^{p}}
\qquad \text{i.e.} \qquad \mathrm{int}(g^{-1}h)\, l_1^{p} = l_1^{p}\]
LaTeX source
\[
\underbrace{u\,\sigma_\infty(l_0)}_{\substack{\| \\ u(l_1) \\ \mathrm{int}(g)\, l_1^{p}}}
= \mathrm{int}(h)\, \underbrace{\sigma_\infty(l_0^{p})}_{l_1^{p}}
\qquad \text{i.e.} \qquad \mathrm{int}(g^{-1}h)\, l_1^{p} = l_1^{p}
\]\[\text{i.e.} \qquad g^{-1}h \in L_1 \quad \text{i.e.} \quad h = g\, l_1^{q},\; q \in \hat{\mathbb{Z}}\]
LaTeX source
\[
\text{i.e.} \qquad g^{-1}h \in L_1 \quad \text{i.e.} \quad h = g\, l_1^{q},\; q \in \hat{\mathbb{Z}}
\]\[\underbrace{u\,\sigma_\infty(l_1)}_{l_0} = \mathrm{int}(h)\, \sigma_\infty\, \underbrace{u(l_1)}_{\mathrm{int}(g)\, l_1^{p}}
\qquad \text{i.e.}\]
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\[
\underbrace{u\,\sigma_\infty(l_1)}_{l_0} = \mathrm{int}(h)\, \sigma_\infty\, \underbrace{u(l_1)}_{\mathrm{int}(g)\, l_1^{p}}
\qquad \text{i.e.}
\]\[l_0^{p} = \mathrm{int}\bigl(h\, \sigma_\infty(g)\bigr)\, l_0^{p} \qquad \text{soit}\]
LaTeX source
\[
l_0^{p} = \mathrm{int}\bigl(h\, \sigma_\infty(g)\bigr)\, l_0^{p} \qquad \text{soit}
\]\[l_0^{p} = \mathrm{int}\bigl(g\, l_1^{q}\, \sigma_\infty(g)\bigr)\, l_0^{p}\]
LaTeX source
\[
l_0^{p} = \mathrm{int}\bigl(g\, l_1^{q}\, \sigma_\infty(g)\bigr)\, l_0^{p}
\]\[\struck{\mathrm{int}(g\, l_1^{q}\, \sigma_\infty g)\, L_1 = L_0}
\qquad \text{i.e. la condition}\]
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\[
\struck{\mathrm{int}(g\, l_1^{q}\, \sigma_\infty g)\, L_1 = L_0}
\qquad \text{i.e. la condition}
\]\[\exists\, q \quad \text{tel que} \quad g\, l_1^{q}\, \sigma_\infty(g) \in L_0\]
LaTeX source
\[
\exists\, q \quad \text{tel que} \quad g\, l_1^{q}\, \sigma_\infty(g) \in L_0
\]\[\exists\, q, r \in \hat{\mathbb{Z}}^{*} \quad \text{tels que} \quad
\boxed{g\, l_1^{q}\, \sigma_\infty(g) = l_0^{r}}\]
LaTeX source
\[
\exists\, q, r \in \hat{\mathbb{Z}}^{*} \quad \text{tels que} \quad
\boxed{g\, l_1^{q}\, \sigma_\infty(g) = l_0^{r}}
\]\[\sigma_\infty(g') = l_1^{\alpha}\, \sigma_\infty(g)\, l_0^{-\alpha},\]
LaTeX source
\[
\sigma_\infty(g') = l_1^{\alpha}\, \sigma_\infty(g)\, l_0^{-\alpha},
\]\[g'\, l_1^{q'}\, \sigma_\infty(g') = l_0^{r'}\]
LaTeX source
\[
g'\, l_1^{q'}\, \sigma_\infty(g') = l_0^{r'}
\]\[l_0^{\alpha}\, g\, l_1^{-\alpha}\, l_1^{q'}\, l_1^{\alpha}\, \sigma_\infty(g)\, l_0^{-\alpha} = l_0^{r'}\]
LaTeX source
\[
l_0^{\alpha}\, g\, l_1^{-\alpha}\, l_1^{q'}\, l_1^{\alpha}\, \sigma_\infty(g)\, l_0^{-\alpha} = l_0^{r'}
\]\[g\, l_1^{q'}\, \sigma_\infty(g) = l_0^{r'}\]
LaTeX source
\[
g\, l_1^{q'}\, \sigma_\infty(g) = l_0^{r'}
\]\[l_1\, l_1^{q}\, l_0 = l_0^{r} \qquad \text{donc} \qquad q = -1,\; r = 1 .\]
LaTeX source
\[
l_1\, l_1^{q}\, l_0 = l_0^{r} \qquad \text{donc} \qquad q = -1,\; r = 1 .
\]\[\begin{cases}
p = -1, \qquad g = l_0^{\alpha}\, l_1^{\beta} \quad (\alpha + \beta = 1), \quad \alpha, \beta \in \mathbb{Z}) \\
q = -1, \quad r = 1 \qquad \uncertain{au moins}, \quad \beta = -1
\end{cases}\]
LaTeX source
\[
\begin{cases}
p = -1, \qquad g = l_0^{\alpha}\, l_1^{\beta} \quad (\alpha + \beta = 1), \quad \alpha, \beta \in \mathbb{Z}) \\
q = -1, \quad r = 1 \qquad \uncertain{au moins}, \quad \beta = -1
\end{cases}
\]\[\mathrm{Autext}_{\mathrm{lac}}(\hat\pi_{0,3}, \mathfrak{S}_3) \overset{\text{déf}}{=} \Pi,\]
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\[
\mathrm{Autext}_{\mathrm{lac}}(\hat\pi_{0,3}, \mathfrak{S}_3) \overset{\text{déf}}{=} \Pi,
\]\[\varphi(\alpha).g = l_0^{\alpha}\, g\, l_1^{-\alpha}\]
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\[
\varphi(\alpha).g = l_0^{\alpha}\, g\, l_1^{-\alpha}
\]\[(\gamma, \chi) : \Pi \longrightarrow \pi/\hat{\mathbb{Z}} \times \hat{\mathbb{Z}}^{*},
\qquad u \longmapsto (\gamma(u), \chi(u))\]
LaTeX source
\[
(\gamma, \chi) : \Pi \longrightarrow \pi/\hat{\mathbb{Z}} \times \hat{\mathbb{Z}}^{*},
\qquad u \longmapsto (\gamma(u), \chi(u))
\]\[\sigma_\infty(g^{-1}) = l_0^{\beta}\, g\, l_1^{\beta}.\]
LaTeX source
\[
\sigma_\infty(g^{-1}) = l_0^{\beta}\, g\, l_1^{\beta}.
\]\[\struck{u(l_1) = \mathrm{int}(g)(l_1)}\]
LaTeX source
\[
\struck{u(l_1) = \mathrm{int}(g)(l_1)}
\]\[\varepsilon_0^2 = l_0, \quad \varepsilon_1^2 = l_1, \quad \varepsilon_\infty^2 = l_\infty\]
LaTeX source
\[ \varepsilon_0^2 = l_0, \quad \varepsilon_1^2 = l_1, \quad \varepsilon_\infty^2 = l_\infty \]
\[\struck{\varepsilon_\infty(l_\infty) = l_\infty, \quad \varepsilon_\infty(l_0) = }\]
LaTeX source
\[
\struck{\varepsilon_\infty(l_\infty) = l_\infty, \quad \varepsilon_\infty(l_0) = }
\]\[\varepsilon_\infty = \bigl(\mathrm{int}(l_0)\sigma_\infty\bigr)^{\mathrm{int}(l_\infty^{\alpha})}
\quad \text{i.e.} \quad
\varepsilon_\infty(l_0) = l_0 l_1 l_0^{-1}, \quad
\varepsilon_\infty(l_1) = l_0, \quad
\varepsilon_\infty(l_\infty) = l_\infty\]
LaTeX source
\[
\varepsilon_\infty = \bigl(\mathrm{int}(l_0)\sigma_\infty\bigr)^{\mathrm{int}(l_\infty^{\alpha})}
\quad \text{i.e.} \quad
\varepsilon_\infty(l_0) = l_0 l_1 l_0^{-1}, \quad
\varepsilon_\infty(l_1) = l_0, \quad
\varepsilon_\infty(l_\infty) = l_\infty
\]\[\struck{\varepsilon_0 = \mathrm{int}(l_1)\sigma_0, \qquad
\varepsilon_1 = \mathrm{int}(l_\infty)\sigma_1}\]
LaTeX source
\[
\struck{\varepsilon_0 = \mathrm{int}(l_1)\sigma_0, \qquad
\varepsilon_1 = \mathrm{int}(l_\infty)\sigma_1}
\]\[\varepsilon_\infty = l_0\, \sigma_\infty\, l_\infty^{\alpha}.\]
LaTeX source
\[
\varepsilon_\infty = l_0\, \sigma_\infty\, l_\infty^{\alpha}.
\]\[\struck{\varepsilon_\infty^2 = l_0\sigma_\infty l_0 \sigma_\infty
= l_0 \sigma_\infty l_0 \sigma_\infty^{-1} = l_0\, \sigma_\infty(l_0)}\]
LaTeX source
\[
\struck{\varepsilon_\infty^2 = l_0\sigma_\infty l_0 \sigma_\infty
= l_0 \sigma_\infty l_0 \sigma_\infty^{-1} = l_0\, \sigma_\infty(l_0)}
\]\[\begin{aligned}
\varepsilon_\infty^2 &= l_0\, \underbrace{\sigma_\infty\, l_\infty^{\alpha}\, l_0\, \sigma_\infty}\, l_\infty^{\alpha} \\
&= l_0\, \sigma_\infty(l_\infty^{\alpha} l_0)\, l_\infty^{\alpha}
= l_0\, (l_0^{-1} l_1^{-1})^{\alpha}\, l_1\, (l_1^{-1} l_0^{-1})^{\alpha} \\
&= \struck{l_0\, \sigma_\infty(l_\infty)\, \sigma_\infty(l_0)^{\alpha-1}\, \sigma_\infty(l_0)\, l_\infty^{\alpha}}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\varepsilon_\infty^2 &= l_0\, \underbrace{\sigma_\infty\, l_\infty^{\alpha}\, l_0\, \sigma_\infty}\, l_\infty^{\alpha} \\
&= l_0\, \sigma_\infty(l_\infty^{\alpha} l_0)\, l_\infty^{\alpha}
= l_0\, (l_0^{-1} l_1^{-1})^{\alpha}\, l_1\, (l_1^{-1} l_0^{-1})^{\alpha} \\
&= \struck{l_0\, \sigma_\infty(l_\infty)\, \sigma_\infty(l_0)^{\alpha-1}\, \sigma_\infty(l_0)\, l_\infty^{\alpha}}
\end{aligned}
\]\[\struck{l_0 l_\infty\, \sigma_\infty(l_\infty)^{\alpha-1}\, \sigma_\infty(l_0)\, l_\infty^{\alpha-1} = 1}\]
LaTeX source
\[
\struck{l_0 l_\infty\, \sigma_\infty(l_\infty)^{\alpha-1}\, \sigma_\infty(l_0)\, l_\infty^{\alpha-1} = 1}
\]\[\struck{\text{i.e.}\quad l_0^{-1} l_1\; l_0^{1-\alpha}\, l_1^{2-\alpha}\, l_\infty^{\alpha} = 1}\]
LaTeX source
\[
\struck{\text{i.e.}\quad l_0^{-1} l_1\; l_0^{1-\alpha}\, l_1^{2-\alpha}\, l_\infty^{\alpha} = 1}
\]\[l_0 l_1 l_0\, (l_0^{-1} l_1^{-1})^{\alpha}\, l_1\, (l_1^{-1} l_0^{-1})^{\alpha} = 1\]
LaTeX source
\[
l_0 l_1 l_0\, (l_0^{-1} l_1^{-1})^{\alpha}\, l_1\, (l_1^{-1} l_0^{-1})^{\alpha} = 1
\]\[\struck{l_0 l_1 l_0 l_0^{-1} l_1^{-1} l_0^{-1} l_1^{-1} l_1 l_1^{-1} l_0^{-1} l_1 l_0^{-1} = 1 \quad ? \quad \text{Non !}}\]
LaTeX source
\[
\struck{l_0 l_1 l_0 l_0^{-1} l_1^{-1} l_0^{-1} l_1^{-1} l_1 l_1^{-1} l_0^{-1} l_1 l_0^{-1} = 1 \quad ? \quad \text{Non !}}
\]\[\struck{l_0 l_1 l_0 l_0^{-1} l_1^{-1} l_1 l_1^{-1} l_0^{-1} = 1} \quad \text{ok}\]
LaTeX source
\[
\struck{l_0 l_1 l_0 l_0^{-1} l_1^{-1} l_1 l_1^{-1} l_0^{-1} = 1} \quad \text{ok}
\]\[\begin{cases}
\varepsilon_\infty = l_0\, \sigma_\infty\, l_\infty & \bigl(\text{ou encore } \mathrm{int}(l_0)\,\sigma_\infty\, \mathrm{int}(l_\infty)\bigr) \\
\varepsilon_0 = l_1\, \sigma_0\, l_0 \\
\varepsilon_1 = l_\infty\, \sigma_1\, l_1
\end{cases}\]
LaTeX source
\[
\begin{cases}
\varepsilon_\infty = l_0\, \sigma_\infty\, l_\infty & \bigl(\text{ou encore } \mathrm{int}(l_0)\,\sigma_\infty\, \mathrm{int}(l_\infty)\bigr) \\
\varepsilon_0 = l_1\, \sigma_0\, l_0 \\
\varepsilon_1 = l_\infty\, \sigma_1\, l_1
\end{cases}
\]\[\varepsilon_0^2 = l_0, \qquad \varepsilon_1^2 = l_1, \qquad \varepsilon_\infty^2 = l_\infty\]
LaTeX source
\[ \varepsilon_0^2 = l_0, \qquad \varepsilon_1^2 = l_1, \qquad \varepsilon_\infty^2 = l_\infty \]
\[\sigma_0 = \varepsilon_0\, \varepsilon_1^2, \qquad
\sigma_1 = \varepsilon_1\, \varepsilon_\infty^2, \qquad
\sigma_\infty = \varepsilon_\infty\, \varepsilon_0^2\]
LaTeX source
\[ \sigma_0 = \varepsilon_0\, \varepsilon_1^2, \qquad \sigma_1 = \varepsilon_1\, \varepsilon_\infty^2, \qquad \sigma_\infty = \varepsilon_\infty\, \varepsilon_0^2 \]
\[\begin{cases}
\varepsilon_0^2\, \varepsilon_1^2\, \varepsilon_\infty^2 = 1 \\
(\varepsilon_0\, \varepsilon_1^2)^2 = 1 \\
(\varepsilon_1\, \varepsilon_\infty^2)^2 = 1 \\
(\varepsilon_\infty\, \varepsilon_0^2)^2 = 1
\end{cases}\]
LaTeX source
\[
\begin{cases}
\varepsilon_0^2\, \varepsilon_1^2\, \varepsilon_\infty^2 = 1 \\
(\varepsilon_0\, \varepsilon_1^2)^2 = 1 \\
(\varepsilon_1\, \varepsilon_\infty^2)^2 = 1 \\
(\varepsilon_\infty\, \varepsilon_0^2)^2 = 1
\end{cases}
\]\[\sigma_0 \sigma_1 = l_1^{-1} \rho \qquad \text{donc} \qquad
\rho = l_1 \sigma_0 \sigma_1
= \varepsilon_1^2\, \varepsilon_0\, \varepsilon_1^2\, \varepsilon_1\, \varepsilon_\infty^2
= \varepsilon_1^2\, \varepsilon_0\, \varepsilon_1\, \underbrace{\varepsilon_1^2\, \varepsilon_\infty^2}_{\varepsilon_0^{-2}}\]
LaTeX source
\[
\sigma_0 \sigma_1 = l_1^{-1} \rho \qquad \text{donc} \qquad
\rho = l_1 \sigma_0 \sigma_1
= \varepsilon_1^2\, \varepsilon_0\, \varepsilon_1^2\, \varepsilon_1\, \varepsilon_\infty^2
= \varepsilon_1^2\, \varepsilon_0\, \varepsilon_1\, \underbrace{\varepsilon_1^2\, \varepsilon_\infty^2}_{\varepsilon_0^{-2}}
\]\[\struck{\rho = \sigma_0\sigma_1 = \sigma_1\sigma_\infty = \sigma_\infty\sigma_0}
\qquad\qquad
\struck{\varepsilon_0\, \varepsilon_1^3\, \varepsilon_{\ill{}}}\]
LaTeX source
\[
\struck{\rho = \sigma_0\sigma_1 = \sigma_1\sigma_\infty = \sigma_\infty\sigma_0}
\qquad\qquad
\struck{\varepsilon_0\, \varepsilon_1^3\, \varepsilon_{\ill{}}}
\]\[\begin{aligned}
\rho &= \varepsilon_1^2\, \varepsilon_0\, \varepsilon_1\, \varepsilon_0^{-2} \\
&= \varepsilon_\infty^2\, \varepsilon_1\, \varepsilon_\infty\, \varepsilon_1^{-2} \\
&= \varepsilon_0^2\, \varepsilon_\infty\, \varepsilon_0\, \varepsilon_\infty^{-2}.
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\rho &= \varepsilon_1^2\, \varepsilon_0\, \varepsilon_1\, \varepsilon_0^{-2} \\
&= \varepsilon_\infty^2\, \varepsilon_1\, \varepsilon_\infty\, \varepsilon_1^{-2} \\
&= \varepsilon_0^2\, \varepsilon_\infty\, \varepsilon_0\, \varepsilon_\infty^{-2}.
\end{aligned}
\]\[(\varepsilon_0\, \varepsilon_1\, \varepsilon_\infty)^2 = 1\]
LaTeX source
\[ (\varepsilon_0\, \varepsilon_1\, \varepsilon_\infty)^2 = 1 \]
\[\varepsilon_0\, \varepsilon_1\, \varepsilon_\infty = 1\]
LaTeX source
\[ \varepsilon_0\, \varepsilon_1\, \varepsilon_\infty = 1 \]
\[l_1 \underbrace{\sigma_0\, l_0}\, l_\infty\, \underbrace{\sigma_1\, l_1}\, l_0\, \underbrace{\sigma_\infty\, l_\infty} = 1
\qquad \text{i.e.} \qquad l_\infty^{-1}\, l_0\, l_1^{-1} = 1 \quad \text{ok}\]
LaTeX source
\[
l_1 \underbrace{\sigma_0\, l_0}\, l_\infty\, \underbrace{\sigma_1\, l_1}\, l_0\, \underbrace{\sigma_\infty\, l_\infty} = 1
\qquad \text{i.e.} \qquad l_\infty^{-1}\, l_0\, l_1^{-1} = 1 \quad \text{ok}
\]\[\boxed{\varepsilon_0\, \varepsilon_1\, \varepsilon_\infty = 1}\]
LaTeX source
\[
\boxed{\varepsilon_0\, \varepsilon_1\, \varepsilon_\infty = 1}
\]\[\sigma_0 = \varepsilon_\infty^{-1} \varepsilon_1, \qquad
\sigma_1 = \varepsilon_0^{-1} \varepsilon_\infty, \qquad
\sigma_\infty = \varepsilon_1^{-1} \varepsilon_0\]
LaTeX source
\[
\sigma_0 = \varepsilon_\infty^{-1} \varepsilon_1, \qquad
\sigma_1 = \varepsilon_0^{-1} \varepsilon_\infty, \qquad
\sigma_\infty = \varepsilon_1^{-1} \varepsilon_0
\]\[\sigma_\infty\, \sigma_1\, \sigma_0 = 1\]
LaTeX source
\[ \sigma_\infty\, \sigma_1\, \sigma_0 = 1 \]
\[\text{i.e.} \qquad
\varepsilon_0 = \varepsilon_1\, \varepsilon_\infty\, \varepsilon_1^{-1}, \qquad
\varepsilon_1 = \varepsilon_\infty\, \varepsilon_0\, \varepsilon_\infty^{-1}, \qquad
\varepsilon_\infty = \varepsilon_0\, \varepsilon_1\, \varepsilon_0^{-1}\]
LaTeX source
\[
\text{i.e.} \qquad
\varepsilon_0 = \varepsilon_1\, \varepsilon_\infty\, \varepsilon_1^{-1}, \qquad
\varepsilon_1 = \varepsilon_\infty\, \varepsilon_0\, \varepsilon_\infty^{-1}, \qquad
\varepsilon_\infty = \varepsilon_0\, \varepsilon_1\, \varepsilon_0^{-1}
\]\[\Pi = \pi_1\bigl(\overbrace{\mathbb{P}^1(\mathbb{C}) - \{0, 1, \infty\}}^{U = U_{0,3}},\, P\bigr)\]
LaTeX source
\[
\Pi = \pi_1\bigl(\overbrace{\mathbb{P}^1(\mathbb{C}) - \{0, 1, \infty\}}^{U = U_{0,3}},\, P\bigr)
\]\[\mathcal{E} = \pi_1\bigl(\underset{\substack{\| \\ U_{0,3}}}{U},\;
\underbrace{\mathfrak{S}_3 \times \mathbb{Z}/2}_{\Gamma};\; P\bigr)\]
LaTeX source
\[
\mathcal{E} = \pi_1\bigl(\underset{\substack{\| \\ U_{0,3}}}{U},\;
\underbrace{\mathfrak{S}_3 \times \mathbb{Z}/2}_{\Gamma};\; P\bigr)
\]\[(\gamma, l)(\gamma', l') = \bigl(\gamma\gamma',\; \gamma'^{-1}(l) \circ l'\bigr)\]
LaTeX source
\[
(\gamma, l)(\gamma', l') = \bigl(\gamma\gamma',\; \gamma'^{-1}(l) \circ l'\bigr)
\]\[\dot\sigma_0, \dot\sigma_1, \dot\sigma_\infty \quad \text{sur} \quad
\sigma_0, \sigma_1, \sigma_\infty\]
LaTeX source
\[
\dot\sigma_0, \dot\sigma_1, \dot\sigma_\infty \quad \text{sur} \quad
\sigma_0, \sigma_1, \sigma_\infty
\]\[(1) \qquad l_\infty\, l_1\, l_0 = 1\]
LaTeX source
\[ (1) \qquad l_\infty\, l_1\, l_0 = 1 \]
\[(2) \qquad
\begin{cases}
\sigma_0^2 = \sigma_1^2 = \sigma_\infty^2 = 1 \\
\sigma_0\sigma_1 = \rho\, l_0, \qquad \sigma_1\sigma_\infty = \rho\, l_1, \qquad \sigma_\infty\sigma_0 = \rho\, l_\infty
\end{cases}\]
LaTeX source
\[
(2) \qquad
\begin{cases}
\sigma_0^2 = \sigma_1^2 = \sigma_\infty^2 = 1 \\
\sigma_0\sigma_1 = \rho\, l_0, \qquad \sigma_1\sigma_\infty = \rho\, l_1, \qquad \sigma_\infty\sigma_0 = \rho\, l_\infty
\end{cases}
\]\[\begin{cases}
\rho\sigma_0\rho^{-1} = \sigma_1 \\
\rho\sigma_1\rho^{-1} = \sigma_\infty \\
\rho\sigma_\infty\rho^{-1} = \sigma_0
\end{cases}
\qquad \text{i.e.} \qquad
\begin{cases}
\rho\sigma_0 = \sigma_1\rho \\
\rho\sigma_1 = \sigma_\infty\rho \\
\rho\sigma_\infty = \sigma_0\rho
\end{cases}\]
LaTeX source
\[
\begin{cases}
\rho\sigma_0\rho^{-1} = \sigma_1 \\
\rho\sigma_1\rho^{-1} = \sigma_\infty \\
\rho\sigma_\infty\rho^{-1} = \sigma_0
\end{cases}
\qquad \text{i.e.} \qquad
\begin{cases}
\rho\sigma_0 = \sigma_1\rho \\
\rho\sigma_1 = \sigma_\infty\rho \\
\rho\sigma_\infty = \sigma_0\rho
\end{cases}
\]\[\struck{\rho = l_0 l_1 l_\infty}\]
LaTeX source
\[
\struck{\rho = l_0 l_1 l_\infty}
\]\[\begin{cases}
\dot\sigma_0^2 = \dot\sigma_1^2 = \dot\sigma_\infty^2 = 1 \\
\dot\sigma_0\dot\sigma_1 = \dot\sigma_1\dot\sigma_\infty = \dot\sigma_\infty\dot\sigma_0 = \dot\rho \\
\dot\rho^3 = 1
\end{cases}\]
LaTeX source
\[
\begin{cases}
\dot\sigma_0^2 = \dot\sigma_1^2 = \dot\sigma_\infty^2 = 1 \\
\dot\sigma_0\dot\sigma_1 = \dot\sigma_1\dot\sigma_\infty = \dot\sigma_\infty\dot\sigma_0 = \dot\rho \\
\dot\rho^3 = 1
\end{cases}
\]\[\rho^3 = 1\]
LaTeX source
\[ \rho^3 = 1 \]
\[(8) \qquad
\begin{cases}
\varepsilon_0 = l_0^{\ill{}}\, \sigma_0\, l_1^{\ill{}} \\
\varepsilon_1 = l_1^{\ill{}}\, \sigma_1\, l_\infty^{\ill{}} \\
\varepsilon_\infty = l_\infty\, \sigma_\infty\, l_0^{\ill{}}
\end{cases}
\qquad\qquad
\text{NB} \quad \rho\,\varepsilon_i\,\rho^{-1} = \varepsilon_{\rho(i)}
\quad \text{i.e.} \quad
\begin{cases}
\rho\varepsilon_0\rho^{-1} = \varepsilon_1 \\
\rho\varepsilon_1\rho^{-1} = \varepsilon_\infty \\
\rho\varepsilon_\infty\rho^{-1} = \varepsilon_0
\end{cases}\]
LaTeX source
\[
(8) \qquad
\begin{cases}
\varepsilon_0 = l_0^{\ill{}}\, \sigma_0\, l_1^{\ill{}} \\
\varepsilon_1 = l_1^{\ill{}}\, \sigma_1\, l_\infty^{\ill{}} \\
\varepsilon_\infty = l_\infty\, \sigma_\infty\, l_0^{\ill{}}
\end{cases}
\qquad\qquad
\text{NB} \quad \rho\,\varepsilon_i\,\rho^{-1} = \varepsilon_{\rho(i)}
\quad \text{i.e.} \quad
\begin{cases}
\rho\varepsilon_0\rho^{-1} = \varepsilon_1 \\
\rho\varepsilon_1\rho^{-1} = \varepsilon_\infty \\
\rho\varepsilon_\infty\rho^{-1} = \varepsilon_0
\end{cases}
\]\[\varepsilon_0^2 = l_0\, \underbrace{\sigma_0\, l_1\, l_0\, \sigma_0}\, l_1
= l_0\, l_\infty\, l_\infty^{-1}\, l_1^{-1}\, l_1 = l_0 \qquad \text{ok}\]
LaTeX source
\[
\varepsilon_0^2 = l_0\, \underbrace{\sigma_0\, l_1\, l_0\, \sigma_0}\, l_1
= l_0\, l_\infty\, l_\infty^{-1}\, l_1^{-1}\, l_1 = l_0 \qquad \text{ok}
\]\[\sigma_0(l_1 l_0) = \sigma_0(l_1)\, \sigma_0(l_0) = l_\infty\, (l_1 l_\infty)^{-1} = l_1^{-1}\]
LaTeX source
\[
\sigma_0(l_1 l_0) = \sigma_0(l_1)\, \sigma_0(l_0) = l_\infty\, (l_1 l_\infty)^{-1} = l_1^{-1}
\]\[\begin{cases}
l_\infty\, l_1\, l_0 = 1 \\
\varepsilon_0^2 = l_0, \quad \varepsilon_1^2 = l_1, \quad \varepsilon_\infty^2 = l_\infty \\
\varepsilon_0\varepsilon_1 = \rho(\ldots), \quad \varepsilon_1\varepsilon_\infty = \ldots, \quad \varepsilon_\infty\varepsilon_0 = \ldots, \quad \rho^3 = 1
\end{cases}\]
LaTeX source
\[
\begin{cases}
l_\infty\, l_1\, l_0 = 1 \\
\varepsilon_0^2 = l_0, \quad \varepsilon_1^2 = l_1, \quad \varepsilon_\infty^2 = l_\infty \\
\varepsilon_0\varepsilon_1 = \rho(\ldots), \quad \varepsilon_1\varepsilon_\infty = \ldots, \quad \varepsilon_\infty\varepsilon_0 = \ldots, \quad \rho^3 = 1
\end{cases}
\]\[\varepsilon_0\varepsilon_1 = l_0\sigma_0 l_1\, l_1\sigma_1 l_\infty
= l_0\, \underbrace{\sigma_0\, l_1^2\, \sigma_0^{-1}}_{l_\infty^2}\,
\underbrace{\sigma_0\sigma_1}_{\rho\, l_0}\, l_\infty
= l_0\, l_\infty^2\, \rho\, l_0\, l_\infty\]
LaTeX source
\[
\varepsilon_0\varepsilon_1 = l_0\sigma_0 l_1\, l_1\sigma_1 l_\infty
= l_0\, \underbrace{\sigma_0\, l_1^2\, \sigma_0^{-1}}_{l_\infty^2}\,
\underbrace{\sigma_0\sigma_1}_{\rho\, l_0}\, l_\infty
= l_0\, l_\infty^2\, \rho\, l_0\, l_\infty
\]\[(9) \qquad
\begin{cases}
\varepsilon_0\varepsilon_1 = l_0\, l_\infty^2\, \rho\, l_0\, l_\infty \\
\varepsilon_1\varepsilon_\infty = l_1\, l_0^2\, \rho\, l_1\, l_0 \\
\varepsilon_\infty\varepsilon_0 = l_\infty\, l_1^2\, \rho\, l_\infty\, l_1
\end{cases}\]
LaTeX source
\[
(9) \qquad
\begin{cases}
\varepsilon_0\varepsilon_1 = l_0\, l_\infty^2\, \rho\, l_0\, l_\infty \\
\varepsilon_1\varepsilon_\infty = l_1\, l_0^2\, \rho\, l_1\, l_0 \\
\varepsilon_\infty\varepsilon_0 = l_\infty\, l_1^2\, \rho\, l_\infty\, l_1
\end{cases}
\]\[\begin{aligned}
&(4) \qquad
\overbrace{(\rho^{-1}\sigma_\infty\sigma_0 \struck{\ill{}})}^{l_\infty}\,
\overbrace{(\rho^{-1}\sigma_1\sigma_\infty \struck{\ill{}})}^{l_1}\,
\overbrace{(\rho^{-1}\sigma_0\sigma_1 \struck{\ill{}})}^{l_0} = 1 \\
&(5) \qquad \sigma_0^2 = \sigma_1^2 = \sigma_\infty^2 = 1 \\
&(6) \qquad \rho^3 = 1.
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
&(4) \qquad
\overbrace{(\rho^{-1}\sigma_\infty\sigma_0 \struck{\ill{}})}^{l_\infty}\,
\overbrace{(\rho^{-1}\sigma_1\sigma_\infty \struck{\ill{}})}^{l_1}\,
\overbrace{(\rho^{-1}\sigma_0\sigma_1 \struck{\ill{}})}^{l_0} = 1 \\
&(5) \qquad \sigma_0^2 = \sigma_1^2 = \sigma_\infty^2 = 1 \\
&(6) \qquad \rho^3 = 1.
\end{aligned}
\]\[(7) \qquad
\begin{cases}
\mathrm{int}(\sigma_i) : \text{échange } l_j \text{ et } l_k \quad (i, j, k \text{ distincts}) \\
\mathrm{int}(\rho)\, l_i = l_{\rho(i)}
\end{cases}\]
LaTeX source
\[
(7) \qquad
\begin{cases}
\mathrm{int}(\sigma_i) : \text{échange } l_j \text{ et } l_k \quad (i, j, k \text{ distincts}) \\
\mathrm{int}(\rho)\, l_i = l_{\rho(i)}
\end{cases}
\]\[\text{avec} \qquad \varepsilon_i^2 = l_i \quad (i = 0, 1, \infty),\]
LaTeX source
\[
\text{avec} \qquad \varepsilon_i^2 = l_i \quad (i = 0, 1, \infty),
\]\[\struck{(8) \quad
\begin{cases}
\varepsilon_0 = l_0^{\ill{}}\, \sigma_1\, l_{\ill{}}^{\ill{}} \\
\varepsilon_1 = l_\infty^{-1}\, \sigma_\infty\, l_1^{-1} \\
\varepsilon_\infty = l_0^{-1}\, \sigma_0\, l_\infty
\end{cases}}\]
LaTeX source
\[
\struck{(8) \quad
\begin{cases}
\varepsilon_0 = l_0^{\ill{}}\, \sigma_1\, l_{\ill{}}^{\ill{}} \\
\varepsilon_1 = l_\infty^{-1}\, \sigma_\infty\, l_1^{-1} \\
\varepsilon_\infty = l_0^{-1}\, \sigma_0\, l_\infty
\end{cases}}
\]\[(13) \qquad
\begin{cases}
\tilde\sigma_0^2 = \tilde\sigma_1^2 = \tilde\sigma_\infty^2 = 1 \\
\tilde\sigma_0\tilde\sigma_1 = \tilde\sigma_1\tilde\sigma_\infty = \tilde\sigma_\infty\tilde\sigma_0 = \rho\, \struck{(\ill{})} \\
(\rho^3 = 1 \text{ est conséquence des précédentes})
\end{cases}\]
LaTeX source
\[
(13) \qquad
\begin{cases}
\tilde\sigma_0^2 = \tilde\sigma_1^2 = \tilde\sigma_\infty^2 = 1 \\
\tilde\sigma_0\tilde\sigma_1 = \tilde\sigma_1\tilde\sigma_\infty = \tilde\sigma_\infty\tilde\sigma_0 = \rho\, \struck{(\ill{})} \\
(\rho^3 = 1 \text{ est conséquence des précédentes})
\end{cases}
\]\[(14) \qquad
\begin{cases}
\tilde\sigma_0\, l_0\, \tilde\sigma_0^{-1} = l_0^{-1} \qquad
\tilde\sigma_0\, l_1\, \tilde\sigma_0^{-1} = l_\infty^{-1} \qquad
\tilde\sigma_0\, l_\infty\, \tilde\sigma_0^{-1} = l_1^{-1} \\
\tilde\sigma_1\, l_1\, \tilde\sigma_1^{-1} = l_1^{-1} \qquad \ldots \\
\ldots
\end{cases}\]
LaTeX source
\[
(14) \qquad
\begin{cases}
\tilde\sigma_0\, l_0\, \tilde\sigma_0^{-1} = l_0^{-1} \qquad
\tilde\sigma_0\, l_1\, \tilde\sigma_0^{-1} = l_\infty^{-1} \qquad
\tilde\sigma_0\, l_\infty\, \tilde\sigma_0^{-1} = l_1^{-1} \\
\tilde\sigma_1\, l_1\, \tilde\sigma_1^{-1} = l_1^{-1} \qquad \ldots \\
\ldots
\end{cases}
\]\[(15) \qquad
\begin{cases}
\tau_0 = \sigma_0\tilde\sigma_0 = \tilde\sigma_0\sigma_0 \\
\tau_1 = \sigma_1\tilde\sigma_1 = \tilde\sigma_1\sigma_1 \\
\tau_\infty = \sigma_\infty\tilde\sigma_\infty = \tilde\sigma_\infty\sigma_\infty
\end{cases}
\qquad \text{donc} \qquad
(16) \qquad
\begin{cases}
\tau_0^2 = 1 \\
\tau_1^2 = 1 \\
\tau_\infty^2 = 1
\end{cases}\]
LaTeX source
\[
(15) \qquad
\begin{cases}
\tau_0 = \sigma_0\tilde\sigma_0 = \tilde\sigma_0\sigma_0 \\
\tau_1 = \sigma_1\tilde\sigma_1 = \tilde\sigma_1\sigma_1 \\
\tau_\infty = \sigma_\infty\tilde\sigma_\infty = \tilde\sigma_\infty\sigma_\infty
\end{cases}
\qquad \text{donc} \qquad
(16) \qquad
\begin{cases}
\tau_0^2 = 1 \\
\tau_1^2 = 1 \\
\tau_\infty^2 = 1
\end{cases}
\]\[(10)\quad
\left\{
\begin{aligned}
&\rho^{3} = 1\\
&\varepsilon_\infty^{2}\ \varepsilon_1^{2}\ \varepsilon_0^{2} = 1\\
&\varepsilon_1 = \varepsilon_0\,\varepsilon_\infty^{\uncertain{4}}\,\rho\,\varepsilon_{\uncertain{0}}^{2}\ \varepsilon_\infty^{2}\\
&\varepsilon_\infty = \varepsilon_1\,\varepsilon_0^{\uncertain{4}}\,\rho\,\varepsilon_{\uncertain{\infty}}^{2}\ \varepsilon_0^{2}\\
&\varepsilon_0 = \varepsilon_\infty\,\varepsilon_1^{\uncertain{4}}\,\rho\,\varepsilon_\infty^{2}\ \varepsilon_1^{2}
\end{aligned}
\right.\]
LaTeX source
\[
(10)\quad
\left\{
\begin{aligned}
&\rho^{3} = 1\\
&\varepsilon_\infty^{2}\ \varepsilon_1^{2}\ \varepsilon_0^{2} = 1\\
&\varepsilon_1 = \varepsilon_0\,\varepsilon_\infty^{\uncertain{4}}\,\rho\,\varepsilon_{\uncertain{0}}^{2}\ \varepsilon_\infty^{2}\\
&\varepsilon_\infty = \varepsilon_1\,\varepsilon_0^{\uncertain{4}}\,\rho\,\varepsilon_{\uncertain{\infty}}^{2}\ \varepsilon_0^{2}\\
&\varepsilon_0 = \varepsilon_\infty\,\varepsilon_1^{\uncertain{4}}\,\rho\,\varepsilon_\infty^{2}\ \varepsilon_1^{2}
\end{aligned}
\right.
\]\[\dot\sigma^{*} = (\dot\sigma, \dot\tau^{\operatorname{sg}(\dot\sigma)}) \in \Gamma\]
LaTeX source
\[
\dot\sigma^{*} = (\dot\sigma, \dot\tau^{\operatorname{sg}(\dot\sigma)}) \in \Gamma
\]\[(11)\quad
\varphi : \mathfrak{S}_3 \longrightarrow \Gamma = \mathfrak{S}_3 \times \mathbb{Z}/2,
\qquad \dot\sigma \longmapsto (\dot\sigma, \operatorname{sg}(\dot\sigma))\]
LaTeX source
\[
(11)\quad
\varphi : \mathfrak{S}_3 \longrightarrow \Gamma = \mathfrak{S}_3 \times \mathbb{Z}/2,
\qquad \dot\sigma \longmapsto (\dot\sigma, \operatorname{sg}(\dot\sigma))
\]\[(12)\quad \sigma \longmapsto \tilde\sigma : \mathfrak{S}_3 \longrightarrow \mathcal{E} .\]
LaTeX source
\[
(12)\quad \sigma \longmapsto \tilde\sigma : \mathfrak{S}_3 \longrightarrow \mathcal{E} .
\]\[(17)\quad
\left\{
\begin{aligned}
\struck{\tau_1\tau_\infty =}\ l_0 &= \tau_\infty\tau_1\\
\struck{\tau_\infty\tau_0 =}\ l_1 &= \tau_0\tau_\infty\\
l_\infty &= \tau_1\tau_0
\end{aligned}
\right.\]
LaTeX source
\[
(17)\quad
\left\{
\begin{aligned}
\struck{\tau_1\tau_\infty =}\ l_0 &= \tau_\infty\tau_1\\
\struck{\tau_\infty\tau_0 =}\ l_1 &= \tau_0\tau_\infty\\
l_\infty &= \tau_1\tau_0
\end{aligned}
\right.
\]\[(18)\quad 1 \longrightarrow \hat\pi \longrightarrow \hat{\hat{\mathcal{E}}}
\longrightarrow \hat{\hat\Gamma} \longrightarrow 1,
\qquad \hat{\hat\Gamma} \subset \operatorname{Autext}_{\mathrm{lac}}(\hat\pi)\]
LaTeX source
\[
(18)\quad 1 \longrightarrow \hat\pi \longrightarrow \hat{\hat{\mathcal{E}}}
\longrightarrow \hat{\hat\Gamma} \longrightarrow 1,
\qquad \hat{\hat\Gamma} \subset \operatorname{Autext}_{\mathrm{lac}}(\hat\pi)
\]\[\left\{
\begin{aligned}
&\Gamma \cap \boldsymbol{\Gamma} = \{1, \tau\}\\
&\boldsymbol{\Gamma} \cap \mathfrak{S}_3 = \{1\}
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
&\Gamma \cap \boldsymbol{\Gamma} = \{1, \tau\}\\
&\boldsymbol{\Gamma} \cap \mathfrak{S}_3 = \{1\}
\end{aligned}
\right.
\]\[(19)\quad p = \chi(u) \in \hat{\mathbb{Z}}^{*}\]
LaTeX source
\[
(19)\quad p = \chi(u) \in \hat{\mathbb{Z}}^{*}
\]\[(20)\quad g_i \in \hat\pi \qquad i = 0, 1, \infty\]
LaTeX source
\[ (20)\quad g_i \in \hat\pi \qquad i = 0, 1, \infty \]
\[(21)\quad u(l_i) = \operatorname{int}(g_i^{-1})\, l_i^{\,p}\]
LaTeX source
\[
(21)\quad u(l_i) = \operatorname{int}(g_i^{-1})\, l_i^{\,p}
\]\[(22)\quad (\operatorname{int}(g_\infty^{-1})\, l_\infty^{\,p})\,
(\operatorname{int}(g_1^{-1})\, l_1^{\,p})\,
(\operatorname{int}(g_0^{-1})\, l_0^{\,p}) = 1,\]
LaTeX source
\[
(22)\quad (\operatorname{int}(g_\infty^{-1})\, l_\infty^{\,p})\,
(\operatorname{int}(g_1^{-1})\, l_1^{\,p})\,
(\operatorname{int}(g_0^{-1})\, l_0^{\,p}) = 1,
\]\[vu(l_i) = v(\operatorname{int}(g_i^{-1})\, l_i^{\,p})
= \operatorname{int}(v(g_i^{-1}))\, v(l_i)^{p}\]
LaTeX source
\[
vu(l_i) = v(\operatorname{int}(g_i^{-1})\, l_i^{\,p})
= \operatorname{int}(v(g_i^{-1}))\, v(l_i)^{p}
\]\[v(l_i) = \operatorname{int}(h_i^{-1})\, l_i^{\,q} \quad \text{donc} \quad
v(l_i)^{p} = \operatorname{int}(h_i^{-1})\, l_i^{\,pq},\]
LaTeX source
\[
v(l_i) = \operatorname{int}(h_i^{-1})\, l_i^{\,q} \quad \text{donc} \quad
v(l_i)^{p} = \operatorname{int}(h_i^{-1})\, l_i^{\,pq},
\]\[(23)\quad vu(l_i) = \operatorname{int}(v(g_i^{-1})\, h_i^{-1})\, l_i^{\,pq}\]
LaTeX source
\[
(23)\quad vu(l_i) = \operatorname{int}(v(g_i^{-1})\, h_i^{-1})\, l_i^{\,pq}
\]\[(u, (g_i)) \overset{\gamma_i}{\longmapsto} g_i ,\]
LaTeX source
\[
(u, (g_i)) \overset{\gamma_i}{\longmapsto} g_i ,
\]\[(24)\quad \gamma_i(\tilde v\tilde u) =
\struck{\tilde v(\gamma_i(\tilde u))\cdot\gamma_i(\tilde v)}\
\tilde v(\gamma_i(\tilde u))\,\gamma_i(\tilde v)\]
LaTeX source
\[
(24)\quad \gamma_i(\tilde v\tilde u) =
\struck{\tilde v(\gamma_i(\tilde u))\cdot\gamma_i(\tilde v)}\
\tilde v(\gamma_i(\tilde u))\,\gamma_i(\tilde v)
\]\[\Bigl(\hat\pi \times \prod_{i \in \{0,1,\infty\}} L_i\Bigr) \simeq
\hat\pi \times \hat{\mathbb{Z}}^{\{0,1,\infty\}}\]
LaTeX source
\[
\Bigl(\hat\pi \times \prod_{i \in \{0,1,\infty\}} L_i\Bigr) \simeq
\hat\pi \times \hat{\mathbb{Z}}^{\{0,1,\infty\}}
\]\[\struck{\ill{}}\ \struck{\ill{}}\ (g\, l_i^{\alpha_i})\]
LaTeX source
\[
\struck{\ill{}}\ \struck{\ill{}}\ (g\, l_i^{\alpha_i})
\]\[\gamma_i(\rho\tilde u\rho^{-1}) = \gamma_{i+1}(\tilde u)\]
LaTeX source
\[
\gamma_i(\rho\tilde u\rho^{-1}) = \gamma_{i+1}(\tilde u)
\]\[\pi = \pi_{0,3} = \{ l_0, l_1, l_\infty \mid l_\infty l_1 l_0 = 1 \}
= \pi_1(\mathbb{P}^1_{\mathbb{C}} \setminus \{0,1,\infty\}, \ill{}) .\]
LaTeX source
\[
\pi = \pi_{0,3} = \{ l_0, l_1, l_\infty \mid l_\infty l_1 l_0 = 1 \}
= \pi_1(\mathbb{P}^1_{\mathbb{C}} \setminus \{0,1,\infty\}, \ill{}) .
\]\[\dot\sigma_0 \longmapsto \sigma_0, \qquad \struck{t}\ \tau \longmapsto \tau_0\]
LaTeX source
\[
\dot\sigma_0 \longmapsto \sigma_0, \qquad \struck{t}\ \tau \longmapsto \tau_0
\]\[\struck{F}\ 2 = 0' , \quad -1 = 1' ; \quad \tfrac12 = \infty' ,\]
LaTeX source
\[
\struck{F}\ 2 = 0' , \quad -1 = 1' ; \quad \tfrac12 = \infty' ,
\]\[(\tau\dot\sigma_1)(2) = \tfrac12\]
LaTeX source
\[ (\tau\dot\sigma_1)(2) = \tfrac12 \]
\[U^{\tau\dot\sigma_1} = \{ z \in U \mid z\bar z = 1 \} = \mathbb{U} - \{1\},\]
LaTeX source
\[
U^{\tau\dot\sigma_1} = \{ z \in U \mid z\bar z = 1 \} = \mathbb{U} - \{1\},
\]\[\struck{\ill{}} = \rho\,\struck{\ill{}},\ \tilde{\dot\sigma}_0,\
\tilde{\dot\sigma}_1,\ \tilde{\dot\sigma}_\infty
\qquad (\tilde{\dot\sigma}_0 = \dot\sigma_0\tau,\ \tilde{\dot\sigma}_1 =
\dot\sigma_1\tau,\ \tilde{\dot\sigma}_\infty = \dot\sigma_\infty\tau)\]
LaTeX source
\[
\struck{\ill{}} = \rho\,\struck{\ill{}},\ \tilde{\dot\sigma}_0,\
\tilde{\dot\sigma}_1,\ \tilde{\dot\sigma}_\infty
\qquad (\tilde{\dot\sigma}_0 = \dot\sigma_0\tau,\ \tilde{\dot\sigma}_1 =
\dot\sigma_1\tau,\ \tilde{\dot\sigma}_\infty = \dot\sigma_\infty\tau)
\]\[\left\{
\begin{aligned}
&\tilde{\dot\sigma}_0^{\,2} = \tilde{\dot\sigma}_1^{\,2} =
\tilde{\dot\sigma}_{\uncertain{3}}^{\,2} = 1\\
&\tilde{\dot\sigma}_0\tilde{\dot\sigma}_1 =
\tilde{\dot\sigma}_1\tilde{\dot\sigma}_\infty =
\tilde{\dot\sigma}_\infty\tilde{\dot\sigma}_0 = \rho'^{\,\ill{}}\\
&\rho'^{6} = 1\\
&\rho'^{3} \text{ commute aux } \dot\sigma_i
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
&\tilde{\dot\sigma}_0^{\,2} = \tilde{\dot\sigma}_1^{\,2} =
\tilde{\dot\sigma}_{\uncertain{3}}^{\,2} = 1\\
&\tilde{\dot\sigma}_0\tilde{\dot\sigma}_1 =
\tilde{\dot\sigma}_1\tilde{\dot\sigma}_\infty =
\tilde{\dot\sigma}_\infty\tilde{\dot\sigma}_0 = \rho'^{\,\ill{}}\\
&\rho'^{6} = 1\\
&\rho'^{3} \text{ commute aux } \dot\sigma_i
\end{aligned}
\right.
\]\[\left\{
\begin{aligned}
&\dot\sigma_0^{2} = \dot\sigma_1^{2} = \dot\sigma_\infty^{2} = 1\\
&\dot\sigma_0\dot\sigma_1 = \dot\sigma_1\dot\sigma_\infty =
\dot\sigma_\infty\dot\sigma_0 = \dot\rho\\
&\dot\rho^{3} = 1\\
&\tau^{2} = 1\\
&\tau \text{ commute à } \dot\sigma_0, \dot\sigma_1, \dot\sigma_\infty
\quad (\text{donc aussi à } \dot\rho)
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
&\dot\sigma_0^{2} = \dot\sigma_1^{2} = \dot\sigma_\infty^{2} = 1\\
&\dot\sigma_0\dot\sigma_1 = \dot\sigma_1\dot\sigma_\infty =
\dot\sigma_\infty\dot\sigma_0 = \dot\rho\\
&\dot\rho^{3} = 1\\
&\tau^{2} = 1\\
&\tau \text{ commute à } \dot\sigma_0, \dot\sigma_1, \dot\sigma_\infty
\quad (\text{donc aussi à } \dot\rho)
\end{aligned}
\right.
\]\[\left\{
\begin{aligned}
&\sigma_0^{2} = \sigma_1^{2} = \sigma_\infty^{2} = \tau_0^{2}
= [\tau_1^{2} = \tau_\infty^{2}] = 1\\
&\rho^{3} = 1\\
&\rho\tau_0\rho^{-1} = \tau_1, \quad \rho\tau_1\rho^{-1} = \tau_\infty,
\quad \rho\tau_\infty\rho^{-1} = \tau_0\\
&\tau_0\sigma_0 = \sigma_0\tau_0, \quad \tau_1\sigma_1 = \sigma_1\tau_1,
\quad \tau_\infty\sigma_\infty = \sigma_\infty\tau_\infty\\
&\sigma_0\sigma_1 = \tau_\infty\tau_1\rho, \quad \sigma_1\sigma_\infty =
\tau_0\tau_\infty\rho, \quad \sigma_\infty\sigma_0 = \tau_1\tau_0\rho
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
&\sigma_0^{2} = \sigma_1^{2} = \sigma_\infty^{2} = \tau_0^{2}
= [\tau_1^{2} = \tau_\infty^{2}] = 1\\
&\rho^{3} = 1\\
&\rho\tau_0\rho^{-1} = \tau_1, \quad \rho\tau_1\rho^{-1} = \tau_\infty,
\quad \rho\tau_\infty\rho^{-1} = \tau_0\\
&\tau_0\sigma_0 = \sigma_0\tau_0, \quad \tau_1\sigma_1 = \sigma_1\tau_1,
\quad \tau_\infty\sigma_\infty = \sigma_\infty\tau_\infty\\
&\sigma_0\sigma_1 = \tau_\infty\tau_1\rho, \quad \sigma_1\sigma_\infty =
\tau_0\tau_\infty\rho, \quad \sigma_\infty\sigma_0 = \tau_1\tau_0\rho
\end{aligned}
\right.
\]\[\left\{
\begin{aligned}
&\text{a)}\ \sigma_0^{2} = \tau_0^{2} = 1, \quad \tau_0\sigma_0 =
\sigma_0\tau_0 \quad (\text{i.e.\ } (\underbrace{\sigma_0\tau_0}_{\tilde\sigma_0})^{2} = 1)\\
&\text{b)}\ \rho^{3} = 1\\
&\underbrace{\sigma_0\rho\sigma_0^{-1}\rho^{-1}}_{\uncertain{\sigma_1}}
= \underbrace{\rho^{2}\tau_0\rho^{-2}}_{\tau_\infty}\,
\underbrace{\rho\tau_0\rho^{-1}}_{\tau_1}\,\rho
\quad \text{i.e.} \quad \sigma_0\rho\sigma_0^{-1}\rho^{-1} =
\rho^{-1}\tau_0\rho\tau_0\\
&\rho\sigma_0\rho^{-1}\sigma_0 = \tau_0\rho^{-1}\tau_0\rho
\quad \text{\uncertain{soit}} \quad \struck{\operatorname{int}(\rho)\sigma_0}\
[\rho, \sigma_0] = [\tau_0, \rho^{-1}]\\
&\text{\uncertain{soit}} \quad [\rho, \sigma_0][\rho^{-1}, \tau_0] = 1
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
&\text{a)}\ \sigma_0^{2} = \tau_0^{2} = 1, \quad \tau_0\sigma_0 =
\sigma_0\tau_0 \quad (\text{i.e.\ } (\underbrace{\sigma_0\tau_0}_{\tilde\sigma_0})^{2} = 1)\\
&\text{b)}\ \rho^{3} = 1\\
&\underbrace{\sigma_0\rho\sigma_0^{-1}\rho^{-1}}_{\uncertain{\sigma_1}}
= \underbrace{\rho^{2}\tau_0\rho^{-2}}_{\tau_\infty}\,
\underbrace{\rho\tau_0\rho^{-1}}_{\tau_1}\,\rho
\quad \text{i.e.} \quad \sigma_0\rho\sigma_0^{-1}\rho^{-1} =
\rho^{-1}\tau_0\rho\tau_0\\
&\rho\sigma_0\rho^{-1}\sigma_0 = \tau_0\rho^{-1}\tau_0\rho
\quad \text{\uncertain{soit}} \quad \struck{\operatorname{int}(\rho)\sigma_0}\
[\rho, \sigma_0] = [\tau_0, \rho^{-1}]\\
&\text{\uncertain{soit}} \quad [\rho, \sigma_0][\rho^{-1}, \tau_0] = 1
\end{aligned}
\right.
\]\[(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}) * \mathbb{Z}/3\mathbb{Z},\]
LaTeX source
\[
(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}) * \mathbb{Z}/3\mathbb{Z},
\]\[S\mathcal{E} = S\Gamma \times_{\Gamma} \mathcal{E} .\]
LaTeX source
\[
S\mathcal{E} = S\Gamma \times_{\Gamma} \mathcal{E} .
\]\[(1)\quad (p, \sigma, g_0, g_1, g_\infty) = \tilde u, \qquad
p \in \hat{\mathbb{Z}}^{*},\ \sigma \in \mathfrak{S}_3,\ g_0, g_1, g_\infty
\in \hat\pi,\]
LaTeX source
\[
(1)\quad (p, \sigma, g_0, g_1, g_\infty) = \tilde u, \qquad
p \in \hat{\mathbb{Z}}^{*},\ \sigma \in \mathfrak{S}_3,\ g_0, g_1, g_\infty
\in \hat\pi,
\]\[(2)\quad \operatorname{int}(g_\infty)(l_{\sigma(\infty)}^{\,p}) \cdot
\operatorname{int}(g_1)(l_{\sigma(1)}^{\,p}) \cdot
\operatorname{int}(g_0)(l_{\sigma(0)}^{\,p}) = 1\]
LaTeX source
\[
(2)\quad \operatorname{int}(g_\infty)(l_{\sigma(\infty)}^{\,p}) \cdot
\operatorname{int}(g_1)(l_{\sigma(1)}^{\,p}) \cdot
\operatorname{int}(g_0)(l_{\sigma(0)}^{\,p}) = 1
\]\[(3)\quad
\left\{
\begin{aligned}
u(l_0) &= \operatorname{int}(g_0)\, l_{\sigma(0)}^{\,p}\\
u(l_1) &= \operatorname{int}(g_1)\, l_{\sigma(1)}^{\,p}\\
u(l_\infty) &= \operatorname{int}(g_\infty)\, l_{\sigma(\infty)}^{\,p}
\end{aligned}
\right.
\qquad \text{ce qui explicite l'homomorphisme}\]
LaTeX source
\[
(3)\quad
\left\{
\begin{aligned}
u(l_0) &= \operatorname{int}(g_0)\, l_{\sigma(0)}^{\,p}\\
u(l_1) &= \operatorname{int}(g_1)\, l_{\sigma(1)}^{\,p}\\
u(l_\infty) &= \operatorname{int}(g_\infty)\, l_{\sigma(\infty)}^{\,p}
\end{aligned}
\right.
\qquad \text{ce qui explicite l'homomorphisme}
\]\[(4)\quad S\hat{\hat{\mathcal{E}}} \xrightarrow{\ p\ } \hat{\hat{\mathcal{E}}} =
\operatorname{Aut}_{\mathrm{lac}}(\hat\pi) .\]
LaTeX source
\[
(4)\quad S\hat{\hat{\mathcal{E}}} \xrightarrow{\ p\ } \hat{\hat{\mathcal{E}}} =
\operatorname{Aut}_{\mathrm{lac}}(\hat\pi) .
\]\[(5)\quad i : \hat\pi \times \hat{\mathbb{Z}}^{\{0,1,\infty\}} \longrightarrow
S\hat{\hat{\mathcal{E}}}\]
LaTeX source
\[
(5)\quad i : \hat\pi \times \hat{\mathbb{Z}}^{\{0,1,\infty\}} \longrightarrow
S\hat{\hat{\mathcal{E}}}
\]\[(6)\quad i(g; \alpha_0, \alpha_1, \alpha_\infty) = (1, 1, g\,l_0^{\alpha_0},
g\,l_1^{\alpha_1}, g\,l_\infty^{\alpha_\infty})\]
LaTeX source
\[
(6)\quad i(g; \alpha_0, \alpha_1, \alpha_\infty) = (1, 1, g\,l_0^{\alpha_0},
g\,l_1^{\alpha_1}, g\,l_\infty^{\alpha_\infty})
\]\[(7)\quad p\,i(g; \alpha_0, \alpha_1, \alpha_\infty) = \operatorname{int} g .\]
LaTeX source
\[
(7)\quad p\,i(g; \alpha_0, \alpha_1, \alpha_\infty) = \operatorname{int} g .
\]\[(8)\quad \underbrace{(p, \sigma, g_0, g_1, g_\infty)}_{\tilde u}
\underbrace{(p', \sigma', g'_0, g'_1, g'_\infty)}_{\tilde u'} =
(pp', \sigma\sigma', u(g'_0)\,g_{\sigma'(0)}, u(g'_1)\,g_{\sigma'(1)},
u(g'_\infty)\,g_{\sigma'(\infty)})\]
LaTeX source
\[
(8)\quad \underbrace{(p, \sigma, g_0, g_1, g_\infty)}_{\tilde u}
\underbrace{(p', \sigma', g'_0, g'_1, g'_\infty)}_{\tilde u'} =
(pp', \sigma\sigma', u(g'_0)\,g_{\sigma'(0)}, u(g'_1)\,g_{\sigma'(1)},
u(g'_\infty)\,g_{\sigma'(\infty)})
\]\[(9)\quad i(g, \alpha_0, \alpha_1, \alpha_\infty)\,(p, \sigma, g_0, g_1,
g_\infty) = (p, \sigma, g\,g_0\,l_{\sigma(0)}^{\alpha_{\sigma(0)}},
g\,g_1\,l_{\sigma(1)}^{\alpha_{\sigma(1)}},
g\,g_\infty\,l_{\sigma(\infty)}^{\alpha_{\sigma(\infty)}})\]
LaTeX source
\[
(9)\quad i(g, \alpha_0, \alpha_1, \alpha_\infty)\,(p, \sigma, g_0, g_1,
g_\infty) = (p, \sigma, g\,g_0\,l_{\sigma(0)}^{\alpha_{\sigma(0)}},
g\,g_1\,l_{\sigma(1)}^{\alpha_{\sigma(1)}},
g\,g_\infty\,l_{\sigma(\infty)}^{\alpha_{\sigma(\infty)}})
\]\[(9')\quad i(g, \alpha_0, \alpha_1, \alpha_\infty)\,(p, 1, g_0, g_1,
g_\infty) = (p, 1, g\,g_0\,l_0^{\alpha_0}, g\,g_1\,l_1^{\alpha_1},
g\,g_\infty\,l_\infty^{\alpha_\infty}) .\]
LaTeX source
\[
(9')\quad i(g, \alpha_0, \alpha_1, \alpha_\infty)\,(p, 1, g_0, g_1,
g_\infty) = (p, 1, g\,g_0\,l_0^{\alpha_0}, g\,g_1\,l_1^{\alpha_1},
g\,g_\infty\,l_\infty^{\alpha_\infty}) .
\]\[(10)\quad \tilde u\, i(g; \alpha_0, \alpha_1, \alpha_\infty)\,\tilde u^{-1}
= i(u(g); p\,\alpha_{\sigma^{-1}(0)}, p\,\alpha_{\sigma^{-1}(1)},
p\,\alpha_{\sigma^{-1}(\infty)})\]
LaTeX source
\[
(10)\quad \tilde u\, i(g; \alpha_0, \alpha_1, \alpha_\infty)\,\tilde u^{-1}
= i(u(g); p\,\alpha_{\sigma^{-1}(0)}, p\,\alpha_{\sigma^{-1}(1)},
p\,\alpha_{\sigma^{-1}(\infty)})
\]\[(11)\quad \varphi : \Gamma^{*}_{P} \longrightarrow S\mathcal{E}\]
LaTeX source
\[
(11)\quad \varphi : \Gamma^{*}_{P} \longrightarrow S\mathcal{E}
\]\[(12)\quad
\begin{aligned}
\dot\rho = (\dot\rho, 1) &\longmapsto (1, \rho, 1, 1, 1)
\overset{\text{déf}}{=} \tilde\rho\\
\dot\sigma'_i = (\dot\sigma_i, -1) &\longmapsto (-1, \dot\sigma_i, 1, 1, 1)
\overset{\text{déf}}{=} \tilde\sigma'_i \qquad \bigl(i \in \{0, 1,
\infty\}\bigr)
\end{aligned}\]
LaTeX source
\[
(12)\quad
\begin{aligned}
\dot\rho = (\dot\rho, 1) &\longmapsto (1, \rho, 1, 1, 1)
\overset{\text{déf}}{=} \tilde\rho\\
\dot\sigma'_i = (\dot\sigma_i, -1) &\longmapsto (-1, \dot\sigma_i, 1, 1, 1)
\overset{\text{déf}}{=} \tilde\sigma'_i \qquad \bigl(i \in \{0, 1,
\infty\}\bigr)
\end{aligned}
\]\[(13)\quad l_{\sigma(0)}^{\,p}\, l_{\sigma(1)}^{\,p}\, l_{\sigma(\infty)}^{\,p} = 1 .\]
LaTeX source
\[
(13)\quad l_{\sigma(0)}^{\,p}\, l_{\sigma(1)}^{\,p}\, l_{\sigma(\infty)}^{\,p} = 1 .
\]\[(14)\quad l_\infty^{\,p}\, l_1^{\,p}\, l_0^{\,p} = 1\]
LaTeX source
\[
(14)\quad l_\infty^{\,p}\, l_1^{\,p}\, l_0^{\,p} = 1
\]\[\begin{aligned}
(15)&\quad (\operatorname{int}\tilde\rho)(p, \sigma, g_0, g_1, g_\infty) =
(p, \dot\rho\sigma\dot\rho^{-1}, \rho(g_\infty), \rho(g_0), \rho(g_1))\\
(16)&\quad \operatorname{int}(\tilde\sigma_0)(p, \sigma, g_0, g_1, g_\infty) =
(p, \dot\sigma_0\sigma\dot\sigma_0^{-1}, \sigma_0(g_0), \sigma_0(g_\infty),
\sigma_0(g_1))
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
(15)&\quad (\operatorname{int}\tilde\rho)(p, \sigma, g_0, g_1, g_\infty) =
(p, \dot\rho\sigma\dot\rho^{-1}, \rho(g_\infty), \rho(g_0), \rho(g_1))\\
(16)&\quad \operatorname{int}(\tilde\sigma_0)(p, \sigma, g_0, g_1, g_\infty) =
(p, \dot\sigma_0\sigma\dot\sigma_0^{-1}, \sigma_0(g_0), \sigma_0(g_\infty),
\sigma_0(g_1))
\end{aligned}
\]\[(17)\quad g_1 = \rho(g_0), \quad g_\infty = \rho(g_1), \quad g_0 =
\rho(g_\infty) .\]
LaTeX source
\[ (17)\quad g_1 = \rho(g_0), \quad g_\infty = \rho(g_1), \quad g_0 = \rho(g_\infty) . \]
\[g_1 = \rho(g_0), \quad g_\infty = \rho(g_1) = \rho^{2}(g_0) =
\rho^{-1}(g_0)\]
LaTeX source
\[
g_1 = \rho(g_0), \quad g_\infty = \rho(g_1) = \rho^{2}(g_0) =
\rho^{-1}(g_0)
\]\[(18)\quad \operatorname{int}(\rho^{2}(g_0))(l_{\sigma(\infty)}^{\,p}) \cdot
\operatorname{int}(\rho(g_0))(l_{\sigma(1)}^{\,p}) \cdot
\operatorname{int}(g_0)(l_{\sigma(0)}^{\,p}) = 1\]
LaTeX source
\[
(18)\quad \operatorname{int}(\rho^{2}(g_0))(l_{\sigma(\infty)}^{\,p}) \cdot
\operatorname{int}(\rho(g_0))(l_{\sigma(1)}^{\,p}) \cdot
\operatorname{int}(g_0)(l_{\sigma(0)}^{\,p}) = 1
\]\[g_0 = \sigma_0(g_0), \quad g_\infty = \sigma_0(g_1) \quad (\text{donc } g_1 =
\sigma_0(g_\infty))\]
LaTeX source
\[
g_0 = \sigma_0(g_0), \quad g_\infty = \sigma_0(g_1) \quad (\text{donc } g_1 =
\sigma_0(g_\infty))
\]