Proof. Suppose that $q_{i}$ , $i\in \{1, 2\}$ , is a closed point of $M_{g, n}^{\mathrm {ord}}$ . If $V_{q_{1}} \supseteq _{fe} V_{q_{2}}$ , we see immediately $q_{1} \sim q_{2}$ . Thus, we obtain $U_{X_{q_{1}}} \cong U_{X_{q_{2}}}$ as schemes. This means $q^{\prime }_{1}=q^{\prime }_{2}$ . Conversely, if $V_{q^{\prime }_{1}} \supseteq V_{q^{\prime }_{2}}$ , then we have $q^{\prime }_{1}=q^{\prime }_{2}$ . Thus, we obtain $q_{1}\sim q_{2}$ .

Suppose that $q_{i}$ , $i\in \{1, 2\}$ , is an arbitrary point of $M_{g, n}^{\mathrm {ord}}$ . If $V_{q_{1}} \supseteq _{fe} V_{q_{2}}$ , then the case of closed points implies $V_{q^{\prime }_{1}}^{\mathrm {cl}} \supseteq V_{q^{\prime }_{2}}^{\mathrm { cl}}.$ Since $V_{q^{\prime }_{1}}$ and $V_{q^{\prime }_{2}}$ are irreducible, we obtain $V_{q^{\prime }_{1}} \supseteq V_{q^{\prime }_{2}}.$ Conversely, if $V_{q^{\prime }_{1}} \supseteq V_{q^{\prime }_{2}}$ , we note that $V_{q_{i}}$ is an irreducible component of $(\omega )^{-1}(V_{q^{\prime }_{i}})$ . Then the case of closed points implies $V_{q_{1}} \supseteq _{fe} V_{q_{2}}$ .

Weak Hom-version Conjecture. Let $q_{i} \in M_{g, n}$ (respectively, $q_{i} \in M_{g, n, \mathbb {F}_{p}}$ ), $i\in \{1, 2\}$ , be an arbitrary point. Then we have

$$ \begin{align*}\mathrm{Hom}^{\mathrm{op}}_{\mathrm{pg}}({\pi}_{1}^{\mathrm{t}}(q_{1}), {\pi}_{1}^{\mathrm{t}}(q_{2}))\end{align*} $$

is nonempty if and only if $V_{q_{1}}\supseteq _{fe}V_{q_{2}}$ (respectively, $V_{q_{1}}\supseteq V_{q_{2}}$ ).

Weak Isom-version Conjecture. Let $q_{i} \in M_{g, n}$ (respectively, $q_{i} \in M_{g, n, \mathbb {F}_{p}}$ ), $i\in \{1, 2\}$ , be an arbitrary point. Then we have

$$ \begin{align*}\mathrm{Isom}_{\mathrm{pg}}({\pi}_{1}^{\mathrm{t}}(q_{1}), {\pi}_{1}^{\mathrm{t}}(q_{2}))\end{align*} $$

is nonempty if and only if $V_{q_{1}}=_{fe}V_{q_{2}}$ (respectively, $V_{q_{1}}=V_{q_{2}}$ ).

Proof. Proposition 2.3 was proved by Stevenson when $n=0$ and q is a closed point of $M_{g, 0}$ (cf. [Reference StevensonSte, Proposition 4.2]). Moreover, by similar arguments to the arguments given in the proof of [Reference StevensonSte, Proposition 4.2], Proposition 2.3 also holds for $n\geq 0$ .

Pointed Collection Conjecture. For each $t \in M_{g, n}^{\mathrm {ord, cl}}$ , the set $\mathcal {C}_{t}$ associated to t is a pointed collection. Moreover, let $q\in M_{g, n}^{\mathrm {ord}}$ . Then the natural map

$$ \begin{align*}\mathrm{colle}_{q}: \mathscr{V}_{q}^{\mathrm{ cl}} \rightarrow \mathscr{C}_{q}, \ [t] \mapsto \mathcal{C}_{t},\end{align*} $$

is a bijection, where $[t]$ denotes the image of t in $\mathscr {V}_{q}^{\mathrm {cl}}\stackrel {\mathrm {def}}{=} V_{q}^{\mathrm {cl}}/\sim _{fe}$ .

Proof. Let $\beta , \gamma \in M_{Y}.$ If $\text {Ram}_{g_{\beta }}=\text {Ram}_{g_{\gamma }}$ , then, for each $\lambda , \mu \in \mathbb {F}_{\ell }^{\times }$ for which $\lambda \beta +\mu \gamma \neq 0$ , we have $\text {Ram}_{g_{\lambda \beta +\mu \gamma }}=\text {Ram}_{g_{\beta }}=\text {Ram}_{g_{\gamma }}.$ Thus, we obtain that $\beta \sim \gamma $ . On the other hand, if $\beta \sim \gamma $ , we have $\text {Ram}_{g_{\beta }}=\text {Ram}_{g_{\gamma }}$ . Otherwise, we have $\#\text {Ram}_{g_{\beta +\gamma }}=2d$ . This means that $\beta \sim \gamma \ \text {if and only if} \ \text {Ram}_{g_{\beta }}=\text {Ram}_{g_{\gamma }}.$ Then $\sim $ is an equivalence relation on $M_{Y}\kern-1pt$ .

Let us prove that ${\vartheta }_{X}$ is a bijection. It is easy to see that ${\vartheta }_{X}$ is an injection. On the other hand, for each $e \in D_{X}$ , the structure of the maximal pro- $\ell $ tame fundamental groups implies that we may construct a connected tame Galois covering of $h: (Z, D_{Z}) \rightarrow (Y, D_{Y})$ , such that h is totally tamely ramified over $f^{-1}(e)$ (i.e., the element of $H^{1}_{\text {'et}}(U_Y, \mathbb {F}_{\ell })$ induced by h is contained in $M_{Y}$ ). Then ${\vartheta }_{X}$ is a surjection. This completes the proof of Proposition 3.3.

Proof. Let $e \in D_{X}$ . Write $D_{e} \subseteq D_{Y}$ for the set $f^{-1}(e)$ . Then $M_{Y, e}$ can be naturally regarded as a subset of $H^{1}_{\text {'et}}(Y \setminus D_{e}, \mathbb {F}_{\ell } )$ via the natural open immersion $Y\setminus D_{e} \hookrightarrow Y.$ Write $L_{e}$ for the $\mathbb {F}_{\ell }$ -vector space generated by $M_{Y, e}$ in $H^{1}_{\text {'et}}(Y \setminus D_{e}, \mathbb {F}_{\ell })$ . Then we have $M_{Y, e}=\ L_{e}\setminus H^{1}_{\text {'et}}(Y, \mathbb {F}_{\ell }).$ Write $H_{e}$ for the quotient $L_{e}/H^{1}_{\text {'et}}(Y, \mathbb {F}_{\ell })$ . We have an exact sequence as follows:

$$ \begin{align*}0 \rightarrow H^{1}_{\text{'et}}(Y, \mathbb{F}_{\ell}) \rightarrow L_{e}\rightarrow H_{e}\rightarrow 0.\end{align*} $$

Since the action of $\mu _{d}$ on $f^{-1}(e)$ is transitive, we obtain $\text {dim}_{\mathbb {F}_{\ell }}(H_{e})=1.$ On the other hand, since $\text {dim}_{\mathbb {F}_{\ell }}(H^{1}_{\text {'et}}(Y, \mathbb {F}_{\ell }))=2g_{Y},$ we obtain $\#(M_{Y, e})=\ell ^{2g_{Y}+1}-\ell ^{2g_{Y}}.$ Thus, we have $\#(M_{Y})=n_{X}(\ell ^{2g_{Y}+1}-\ell ^{2g_{Y}}).$ This completes the proof of the lemma.

Proof. (i) Let $\mathfrak {P}$ be the set of prime numbers, and let Q be an arbitrary open subgroup of ${\pi }_{1}^{\mathrm {t}}(U_{X})$ and $r_{Q}$ an integer, such that

$$ \begin{align*}\#\{l \in \mathfrak{P} \ | \ r_{Q}=\text{dim}_{\mathbb{F}_{l}}(Q^{\mathrm{ab}}\otimes\mathbb{F}_{l})\}=\infty.\end{align*} $$

Then we see immediately that the characteristic of k is the unique prime number p, such that there exists an open subgroup $T \subseteq {\pi }_{1}^{\mathrm {t}}(U_{X})$ and $r_{T}\neq \text {dim}_{\mathbb {F}_{p}}(T^{\mathrm {ab}}\otimes \mathbb {F}_{p}).$

(ii) Let H be an arbitrary open normal subgroup of ${\pi }_{1}^{\mathrm {t}}(U_{X})$ . We denote by $(X_{H}, D_{X_{H}})$ the smooth pointed stable curve of type $(g_{X_{H}}, n_{X_{H}})$ over k induced by H, and denote by $f_{H}: (X_{H}, D_{X_{H}}) \rightarrow (X, D_{X})$ the morphism of smooth pointed stable curves over k induced by the natural inclusion $H \hookrightarrow {\pi }_{1}^{\mathrm {t}}(U_{X})$ . We note that $f_{H}$ is étale if and only if $g_{X_{H}}-1=\#({\pi }_{1}^{\mathrm {t}}(U_X)/H)(g_{X}-1).$ We put

$$ \begin{align*}\text{Et}({\pi}_{1}^{\mathrm{t}}(U_X))\stackrel{\mathrm{def}}{=}\{H \subseteq {\pi}_{1}^{\mathrm{t}}(U_X) \ \text{is an open normal subgroup} \ |\ \end{align*} $$

$$ \begin{align*}g_{X_{H}}-1=\#({\pi}_{1}^{\mathrm{t}}(U_X)/H)(g_{X}-1)\}.\end{align*} $$

Moreover, Proposition 3.2 (i) implies that $g_{X_{H}}$ and $g_{X}$ can be mono-anabelian reconstructed from H and ${\pi }_{1}^{\mathrm {t}}(U_X)$ , respectively. Then the set $\text {Et}({\pi }_{1}^{\mathrm {t}}(U_X))$ can be mono-anabelian reconstructed from ${\pi }_{1}^{\mathrm {t}}(U_X)$ . We obtain that

$$ \begin{align*}{\pi}_{1}(X)= {\pi}_{1}^{\mathrm{t}}(U_{X})/\bigcap_{H \in \text{Et}({\pi}_{1}^{\mathrm{t}}(U_{X}))}H .\end{align*} $$

This completes the proof of the lemma.

Proof. We only need to prove the “moreover” part of Proposition 3.6. We maintain the notation introduced in Remark 3.3.2. Note that, for each $e_{X} \in D_{X}$ and each $e_{X_{H}} \in D_{X_{H}}$ , the sets $M_{Y, e_{X}}$ and $M_{Z, e_{X_{H}}}$ can be mono-anabelian reconstructed from ${\pi }_{1}^{\mathrm {t}}(U_{X})$ and H, respectively. Then the “moreover” part follows from Remark 3.3.2.

Proof. Let $C_{X}\stackrel {\mathrm {def}}{=} \{H_{i}\}_{i \in \mathbb {Z}_{> 0}}$ be a set of open normal subgroups of ${\pi }_{1}^{\mathrm {t}}(U_{X})$ , such that $\varprojlim _{i}{\pi }_{1}^{\mathrm {t}}(U_{X})/H_{i} \cong {\pi }_{1}^{\mathrm { t}}(U_{X})$ (i.e., a cofinal system of open normal subgroups).

Let $\widetilde e \in D_{\widetilde X}$ . For each $i \in \mathbb {Z}_{> 0}$ , we write $(X_{H_{i}}, D_{X_{H_{i}}})$ for the smooth pointed stable curve of type $(g_{X_{H_{i}}}, n_{X_{H_{i}}})$ induced by $H_{i}$ and $e_{X_{H_{i}}} \in D_{X_{H_{i}}}$ for the image of $\widetilde e$ . Then we obtain a sequence of marked points

$$ \begin{align*}\mathcal{I}_{\widetilde e}^{C_{X}}: \dots \mapsto e_{X_{H_{2}}}\mapsto e_{X_{H_{1}}}\end{align*} $$

induced by $C_{X}$ . Note that the sequence $\mathcal {I}^{C_{X}}_{\widetilde e}$ admits a natural action of ${\pi }_{1}^{\mathrm {t}}(U_{X})$ . We may identify the inertia subgroup $I_{\widetilde e}$ associated to $\widetilde e$ with the stabilizer of $\mathcal {I}^{C_{X}}_{\widetilde e}$ .

Moreover, since Proposition 3.2 (i) implies that $(g_{X_{H_{i}}}, n_{X_{H_{i}}})$ can be mono-anabelian reconstructed from $H_{i}$ , by choosing a suitable set of open normal subgroups $C_{X}$ , we may assume that $g_{X_{H_{1}}}\geq 2$ . If $n_{X_{H_{1}}}=0$ , Proposition 3.7 is trivial. Then we may assume that $n_{X_{H_{1}}}> 0$ .

On the other hand, Proposition 3.6 implies that, for each $H_{i}$ , $i\in \mathbb {Z}_{> 0}$ , the set $D^{\mathrm {gp}}_{X_{H_{i}}}$ can be mono-anabelian reconstructed from $H_{i}$ . For each $e_{X_{H_{i}}} \in D_{X_{H_{i}}}$ , we denote by

$$ \begin{align*}e^{\mathrm{gp}}_{X_{H_{i}}}\stackrel{\mathrm{def}}{=} {\vartheta}_{X_{H_{i}}}^{-1}(e_{X_{H_{i}}}).\end{align*} $$

Then the sequence of marked points $\mathcal {I}^{C_{X}}_{\widetilde e}$ induces a sequence

$$ \begin{align*}\mathcal{I}^{C_{X}}_{\widetilde e^{\mathrm{gp}}}: \dots \mapsto e^{\mathrm{gp}}_{X_{H_{2}}}\mapsto e^{\mathrm{gp}}_{X_{H_{1}}}.\end{align*} $$

By applying the “moreover” part of Proposition 3.6, we see that $\mathcal {I}^{C_{X}}_{\widetilde e^{\mathrm {gp}}}$ can be mono-anabelian reconstructed from $C_{X}$ . Then Remark 3.6.1 implies that the stabilizer of $\mathcal {I}^{C_{X}}_{\widetilde e^{\mathrm {gp}}}$ is equal to the stabilizer of $\mathcal {I}^{C_{X}}_{\widetilde e}$ . This completes the proof of the proposition.

Proof. (i) is the Tamagawa’s result concerning the limit of p-averages of ${\pi }_{1}^{\mathrm {t}}(U_{X})$ ([Reference TamagawaT4, Theorem 0.5]). Let us prove (ii). The surjection $\phi $ induces a surjection $\phi ^{p'}: \pi ^{\mathrm {t}}_{1}(U_{X_{1}})^{p'} \twoheadrightarrow {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})^{p'},$ where $(-)^{p'}$ denotes the maximal prime-to-p quotient of $(-)$ . Moreover, since ${\pi }_{1}^{\mathrm {t}}(U_{X_{i}})^{p'}$ , $i \in \{1, 2\}$ , is topologically finitely generated, and ${\pi }_{1}^{\mathrm {t}}(U_{X_{1}})^{p'}$ is isomorphic to ${\pi }_{1}^{\mathrm {t}}(U_{X_{2}})^{p'}$ as abstract profinite groups (since the types of $(X_{1}, D_{X_{1}})$ and $(X_{2}, D_{X_{2}})$ are equal to $(g_{X}, n_{X})$ ), we obtain that $\phi ^{p'}$ is an isomorphism (cf. [Reference Fried and JardenFJ, Proposition 16.10.6]).

On the other hand, since $[{\pi }_{1}^{\mathrm {t}}(U_{X_{1}}): H_{1}]=[{\pi }_{1}^{\mathrm {t}}(U_{X_{2}}): H_{2}]$ and $([{\pi }_{1}^{\mathrm {t}}(U_{X_{2}}): H_{2}], p)=1$ , we obtain that the natural homomorphism $\phi _{H}^{p'}: H_{1}^{p'} \twoheadrightarrow H_{2}^{p'}$ induced by $\phi _{H}\stackrel {\mathrm {def}}{=} \phi |_{H_{1}}: H_{1} \twoheadrightarrow H_{2}$ is also an isomorphism. This implies

$$ \begin{align*}\#(H_{1}^{\mathrm{ab}}\otimes \mathbb{Z}/(p^{r}-1)\mathbb{Z})=\#(H_{2}^{\mathrm{ab}}\otimes \mathbb{Z}/(p^{r}-1)\mathbb{Z})\end{align*} $$

for all $r\in \mathbb {N}$ . Let $K_{H_{i}, p^{r-1}}$ , $i\in \{1, 2\}$ , be the kernel of the natural surjection $H_{i} \twoheadrightarrow H_{i}^{\mathrm {ab}}\otimes \mathbb {Z}/(p^{r}-1)\mathbb {Z}.$ Then the surjection $\phi _{H}$ implies

$$ \begin{align*}\text{Avr}_{p}(H_{1}) \stackrel{\mathrm{def}}{=} \lim_{r\rightarrow \infty}\frac{\text{dim}_{\mathbb{F}_{p}}(K^{\mathrm{ab}}_{H_{1}, p^{r}-1}\otimes\mathbb{F}_{p})}{\#(H_{1}^{\mathrm{ab}}\otimes \mathbb{Z}/(p^{r}-1)\mathbb{Z})}\geq \text{Avr}_{p}(H_{2}) \stackrel{\mathrm{def}}{=} \lim_{r\rightarrow \infty}\frac{\text{dim}_{\mathbb{F}_{p}}(K^{\mathrm{ab}}_{H_{2}, p^{r}-1}\otimes\mathbb{F}_{p})}{\#(H_{2}^{\mathrm{ab}}\otimes \mathbb{Z}/(p^{r}-1)\mathbb{Z})}.\end{align*} $$

Thus, (ii) follows from (i).

Proof. Let $f_{1}: (Y_{1}, D_{Y_{1}}) \rightarrow (X_{1}, D_{X_{1}})$ be an étale covering of degree $\ell $ over $k_{1}$ . Write $f_{2}: (Y_{2}, D_{Y_{2}}) \rightarrow (X_{2}, D_{X_{2}})$ for the connected Galois tame covering of degree $\ell $ over $k_{2}$ induced by $\phi ^{p'}$ . Then we will prove that $f_{2}$ is also an étale covering over $k_{2}$ .

Write $g_{Y_{1}}$ and $g_{Y_{2}}$ for the genus of $Y_{1}$ and $Y_{2}$ , respectively. Since $f_{1}$ is an étale covering of degree $\ell $ , the Riemann-Hurwitz formula implies $g_{Y_{1}}=\ell (g_{X_{1}}-1)+1.$ On the other hand, the Riemann-Hurwitz formula implies $g_{Y_{2}}=\ell (g_{X_{2}}-1)+1+\frac {1}{2}(\ell -1)\#(\text {Ram}_{f_{2}}).$ By applying Proposition 3.8 (ii), the surjection $\phi $ implies $g_{Y_{1}} \geq g_{Y_{2}}.$ This means $\#(\text {Ram}_{f_{2}})=0.$ So $f_{2}$ is an étale covering over $k_{2}$ . Then the morphism $(\phi ^{p'})^{-1}$ induces an injection

$$ \begin{align*}\psi_{X}^{\ell}: \mathrm{Hom}({\pi}_{1}(X_{1}), \mathbb{F}_{\ell}) \hookrightarrow \mathrm{Hom}({\pi}_{1}(X_{2}), \mathbb{F}_{\ell}).\end{align*} $$

Furthermore, since $\text {dim}_{\mathbb {F}_{\ell }}(\mathrm {Hom}({\pi }_{1}(X_{1}), \mathbb {F}_{\ell })) =\text {dim}_{\mathbb {F}_{\ell }}( \mathrm {Hom}({\pi }_{1}(X_{2}), \mathbb {F}_{\ell }))=2g_{X}$ , we obtain that $\psi _{X}^{\ell }$ is a bijection. This completes the proof of the lemma.

Proof. Let $(\ell , d, y_{2})$ be an mp-triple associated to ${\pi }_{1}^{\mathrm {t}}(U_{X_{2}})$ (see Section 3.3.4). Then Lemma 3.9 implies that $\phi $ induces an mp-triple $(\ell , d, y_{1})$ associated to ${\pi }_{1}^{\mathrm {t}}(U_{X_{1}})$ , where $y_{1}\stackrel {\mathrm {def}}{=}(\psi ^{d}_{X})^{-1}(y_{2}) \in \mathrm {Hom}({\pi }_{1}(X_{1}), \mu _{d})$ .

Let $f_{i}: (Y_{i}, D_{Y_i}) \rightarrow (X_{i}, D_{X_{i}})$ , $i \in \{1, 2\}$ , be the étale covering of degree d over $k_{i}$ induced by $y_{i}$ . Then the mp-triple $(\ell , d, y_{i})$ associated to ${\pi }_{1}^{\mathrm {t}}(U_{X_{i}})$ determine an mp-triple

$$ \begin{align*}(\ell, d, f_{i}: (Y_{i}, D_{Y_{i}}) \rightarrow (X_{i}, D_{X_{i}}))\end{align*} $$

associated to $(X_{i}, D_{X_{i}})$ over $k_{i}$ . Note that the types of $(Y_{1}, D_{Y_{1}})$ and $(Y_{2}, D_{Y_{2}})$ are equal.

Write ${\pi }_{1}^{\mathrm {t}}(U_{Y_{i}})$ , $i \in \{1, 2\}$ , for the kernel of ${\pi }_{1}^{\mathrm {t}}(U_{X_{i}}) \twoheadrightarrow {\pi }_{1}(X_{i}) \overset {y_{i}}\twoheadrightarrow \mu _{d}$ . By replacing $(X_{i}, D_{X_{i}})$ by $(Y_{i}, D_{Y_{i}})$ , Lemma 3.9 implies that $(\phi |_{{\pi }_{1}^{\mathrm {t}}(U_{Y_{1}})}^{p'})^{-1}$ induces a commutative diagram as follows:

where all the vertical arrows are isomorphisms. We note that $H^{1}_{\text {'et}}(Y_{i}, \mathbb {F}_{\ell })$ , $H^{1}_{\text {'et}}(U_{Y_{i}}, \mathbb {F}_{\ell })$ , and $\text {Div}^{0}_{D_{Y_{i}}}(Y_{i})\otimes \mathbb {F}_{\ell }$ , $i, \in \{1, 2\}$ , are naturally isomorphic to $\text {Hom}({\pi }_{1}(Y_{i}), \mathbb {F}_{\ell })$ , $\text {Hom}({\pi }_{1}^{\mathrm {t}}(U_{Y_{i}}), \mathbb {F}_{\ell })$ , and $\text {Hom}({\pi }_{1}^{\mathrm {t}}(U_{Y_{i}}), \mathbb {F}_{\ell })/\text {Hom}({\pi }_{1}(Y_{i}), \mathbb {F}_{\ell }),$ respectively. Then Lemma 3.5 implies that the commutative diagram above can be mono-anabelian reconstructed from $\phi |_{{\pi }_{1}^{\mathrm {t}}(U_{Y_{1}})}: {\pi }_{1}^{\mathrm {t}}(U_{Y_{1}}) \twoheadrightarrow {\pi }_{1}^{\mathrm {t}}(U_{Y_{2}})$ .

Write $M_{Y_{i}}\subseteq M_{Y_{i}}^{*}$ for the subsets of $H^{1}_{\text {'et}}(U_{Y_{i}}, \mathbb {F}_{\ell })$ defined in Section 3.2.4. Since the actions of $\mu _{d}$ on the exact sequences are compatible with the isomorphisms appearing in the commutative diagram above, we have $\psi _{Y}^{\mathrm {t, \ell }}(M_{Y_{1}}^{*})=M_{Y_{2}}^{*}.$ Next, we prove $\psi _{Y}^{\mathrm {t, \ell }}(M_{Y_{1}})=M_{Y_{2}}.$

Let $\alpha _{1} \in M_{Y_{1}}$ and $g_{\alpha _{1}}: (Y_{\alpha _{1}}, D_{Y_{\alpha _{1}}}) \rightarrow (Y_{1}, D_{Y_{1}})$ the Galois tame covering of degree $\ell $ over $k_{1}$ induced by $\alpha _{1}$ . Write $g_{\alpha _{2}}: (Y_{\alpha _{2}}, D_{Y_{\alpha _{2}}}) \rightarrow (Y_{2}, D_{Y_{2}})$ for the Galois tame covering of degree $\ell $ over $k_{2}$ induced by $\alpha _{2}\stackrel {\mathrm {def}}{=}\psi ^{\mathrm {t, \ell }}_{Y}(\alpha _{1})$ . Write $g_{Y_{\alpha _{1}}}$ and $g_{Y_{\alpha _{2}}}$ for the genus of $Y_{\alpha _{1}}$ and $Y_{\alpha _{2}}$ , respectively. Then Proposition 3.8 (ii) and the Riemann-Hurwitz formula imply that $g_{Y_{\alpha _{1}}}-g_{Y_{\alpha _{2}}}=\frac {1}{2}(d-\#(\text {Ram}_{g_{\alpha _{2}}}))(\ell -1)\geq 0.$ This means $d-\#(\text {Ram}_{g_{\alpha _{2}}}) \geq 0.$ Since $\alpha _{2} \in M_{Y_{2}}^{*}$ , we have $d \ | \ \#(\text {Ram}_{g_{\alpha _{2}}})$ . Thus, either $\#(\text {Ram}_{g_{\alpha _{2}}})=0$ or $\#(\text {Ram}_{g_{\alpha _{2}}})=d$ holds.

If $\#(\text {Ram}_{g_{\alpha _{2}}})=0$ , then $g_{\alpha _{2}}$ is an étale covering over $k_{2}$ . Then Lemma 3.9 implies that $g_{\alpha _{1}}$ is an étale covering over $k_{1}$ . This provides a contradiction to the fact that $\alpha _{1} \in M_{Y_{1}}$ . Then we have $\#(\text {Ram}_{g_{\alpha _{2}}})=d$ . This means $\alpha _{2} \in M_{Y_{2}}$ . Thus, we obtain $\psi _{Y}^{\mathrm {t, \ell }}(M_{Y_{1}})\subseteq M_{Y_{2}}.$ On the other hand, Lemma 3.4 implies $\#(M_{Y_{1}})=\#(M_{Y_{2}}).$ We have $\psi _{Y}^{\mathrm {t, \ell }}: M_{Y_{1}} \stackrel {\sim }{\rightarrow } M_{Y_{2}}.$ Then Proposition 3.3 implies that $\psi ^{\mathrm {t, \ell }}_{Y}$ induces a bijection

$$ \begin{align*}\rho_{\phi}: D^{\mathrm{gp}}_{X_{1}} \stackrel{\sim}{\rightarrow} D^{\mathrm{gp}}_{X_{2}}.\end{align*} $$

Moreover, since $M_{Y_{i}}$ and $M_{Y_{i}}^{*}$ can be mono-anabelian reconstructed from ${\pi }_{1}^{\mathrm {t}}(U_{Y_{i}})$ , the bijection $\rho _{\phi }$ can be mono-anabelian reconstructed from $\phi $ . This completes the proof of the lemma.

Proof. Proposition 3.6 and Lemma 3.10 imply the diagram

can be mono-anabelian reconstructed from the commutative diagram of profinite groups

To verify Lemma 3.11, it is sufficient to check that the diagram is commutative.

Let $e^{\mathrm {gp}}_{X_{H_{1}}} \in D^{\mathrm {gp}}_{X_{H_{1}}}$ , $e^{\mathrm {gp}}_{X_{H_{2}}}\stackrel {\mathrm {def}}{=} \rho _{\phi |_{H_{1}}}(e^{\mathrm {gp}}_{X_{H_{1}}}) \in D^{\mathrm {gp}}_{X_{H_{2}}}$ , $e^{\mathrm { gp}}_{X_1}\stackrel {\mathrm {def}}{=} \gamma _{H_{1}, {\pi }_{1}^{\mathrm {t}}(U_{X_{1}})}(e^{\mathrm {gp}}_{X_{H_{1}}}) \in D^{\mathrm {gp}}_{X_{1}}$ , $e^{\mathrm {gp}}_{X_2}\stackrel {\mathrm {def}}{=} (\gamma _{H_{2}, {\pi }_{1}^{\mathrm { t}}(U_{X_{2}})}\circ \rho _{\phi |_{H_{1}}})(e^{\mathrm {gp}}_{X_{H_{1}}}) \in D^{\mathrm {gp}}_{X_{2}}$ , and $e^{\mathrm {gp, *}}_{X_1}\stackrel {\mathrm {def}}{=} \rho ^{-1}_{\phi }(e^{\mathrm {gp}}_{X_2}) \in D^{\mathrm {gp}}_{X_{1}}$ . Let us prove

$$ \begin{align*}e^{\mathrm{gp}}_{X_1}=e^{\mathrm{gp, *}}_{X_1}.\end{align*} $$

We put $S^{\mathrm {gp}}_{X_{H_{1}}} \stackrel {\mathrm {def}}{=} \gamma _{H_{1}, {\pi }_{1}^{\mathrm {t}}(U_{X_{1}})}^{-1}(e_{X_{1}}^{\mathrm {gp, *}})$ and $S^{\mathrm { gp}}_{X_{H_{2}}}\stackrel {\mathrm {def}}{=} \gamma _{H_{2}, {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})}^{-1}(e^{\mathrm {gp}}_{X_2})$ , respectively. Note that $e^{\mathrm {gp}}_{X_{H_{2}}} \in S^{\mathrm {gp}}_{X_{H_{2}}}$ . To verify $e^{\mathrm { gp}}_{X_1}=e^{\mathrm {gp, *}}_{X_1}$ , it is sufficient to prove that $e^{\mathrm {gp}}_{X_{H_{1}}} \in S^{\mathrm {gp}}_{X_{H_{1}}}$ . Moreover, for each $i\in \{1, 2\}$ , we put

$$ \begin{align*}e_{X_{i}} \stackrel{\mathrm{def}}{=} {\vartheta}_{X_{i}}(e^{\mathrm{gp}}_{X_{i}}), \ e_{X_{H_i}} \stackrel{\mathrm{def}}{=} {\vartheta}_{X_{H_i}}(e^{\mathrm{gp}}_{X_{i}}), \ e^{*}_{X_{1}} \stackrel{\mathrm{def}}{=} {\vartheta}_{X_{1}}(e_{X_{1}}^{\mathrm{gp, *}}), \ S_{X_{i}} \stackrel{\mathrm{def}}{=} S^{\mathrm{gp}}_{X_{i}}, \ S_{X_{H_{i}}} \stackrel{\mathrm{def}}{=} S_{X_{H_i}}^{\mathrm{gp}}.\end{align*} $$

Then to verify the lemma, we only need to prove that $e_{X_{H_{1}}} \in {\vartheta }_{X_{H_1}}(S_{X_{H_{1}}})$ .

Let $(\ell , d, y_{2})$ be an mp-triple associated to ${\pi }_{1}^{\mathrm {t}}(U_{X_{2}})$ . Then Lemma 3.9 implies that $\phi $ induces an mp-triple $(\ell , d, y_{1})$ associated to ${\pi }_{1}^{\mathrm {t}}(U_{X_{1}})$ , where $y_{1}\stackrel {\mathrm {def}}{=} (\psi ^{d}_{X})^{-1}(y_{2}) \in \mathrm {Hom}({\pi }_{1}(X_{1}), \mu _{d})$ . Let $f_{i}: (Y_{i}, D_{Y_i}) \rightarrow (X_{i}, D_{X_{i}})$ , $i \in \{1, 2\}$ , be the tame covering of degree d over $k_{i}$ induced by $y_{i}$ . Then the mp-triple $(\ell , d, y_{i})$ associated to ${\pi }_{1}^{\mathrm {t}}(U_{X_{i}})$ induces an mp-triple

$$ \begin{align*}(\ell, d, f_{i}: (Y_{i}, D_{Y_{i}}) \rightarrow (X_{i}, D_{X_{i}}))\end{align*} $$

associated to $(X_{i}, D_{X_{i}})$ over $k_{i}$ . Note that since $f_{1}$ and $f_{2}$ are étale, the types of $(Y_{1}, D_{Y_{1}})$ and $(Y_{2}, D_{Y_{2}})$ are equal. On the other hand, we have an mp-triple

$$ \begin{align*}(\ell, d, g_{2}: (Z_{2}, D_{Z_{2}})\stackrel{\mathrm{def}}{=} (Y_{2}, D_{Y_{2}}) \times_{(X_{2}, D_{X_{2}})} (X_{H_2}, D_{X_{H_2}}) \rightarrow (X_{H_2}, D_{X_{H_2}}))\end{align*} $$

associated to $(X_{H_2}, D_{X_{H_2}})$ induced by the natural inclusion $H_{2} \hookrightarrow {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})$ and the mp-triple $(\ell , d, f_{2}: (Y_{2}, D_{Y_{2}}) \rightarrow (X_{2}, D_{X_{2}}))$ . By Lemma 3.9 again, we obtain an mp-triple

$$ \begin{align*}(\ell, d, g_{1}: (Z_{1}, D_{Z_{1}})\stackrel{\mathrm{def}}{=} (Y_{1}, D_{Y_{1}}) \times_{(X_{1}, D_{X_{1}})} (X_{H_1}, D_{X_{H_1}}) \rightarrow (X_{H_1}, D_{X_{H_1}}))\end{align*} $$

associated to $(X_{H_1}, D_{X_{H_1}})$ induced by $\phi |_{H_{1}}$ and the triple $(\ell , d, g_{2}: (Z_{2}, D_{Z_{2}})\rightarrow (X_{H_2}, D_{X_{H_2}}))$ .

Let $\alpha _{2} \in M_{Y_{2}, e_{X_2}}$ . The final paragraph of the proof of Lemma 3.10 implies that we have a bijection $M_{Y_{1}}=\bigsqcup _{e \in D_{X_{1}}} M_{Y_{1}, e} \stackrel {\sim }{\rightarrow } M_{Y_{2}}=\bigsqcup _{e \in D_{X_{2}}} M_{Y_{2}, e}$ induced by $\phi $ . Then $\alpha _{2}$ induces an element $\alpha _{1} \in M_{Y_{1}, e^{*}_{X_1}}.$ Write $(Y_{\alpha _{1}}, D_{Y_{\alpha _{1}}})$ and $(Y_{\alpha _{2}}, D_{Y_{\alpha _{2}}})$ for the smooth pointed stable curves over $k_{1}$ and $k_{2}$ induced by $\alpha _{1}$ and $\alpha _{2}$ , respectively. Consider the connected Galois tame covering

$$ \begin{align*}(Y_{\alpha_{2}}, D_{Y_{\alpha_{2}}}) \times_{(X_{2}, D_{X_{2}})} (X_{H_2}, D_{X_{H_2}}) \rightarrow (Z_{2}, D_{Z_{2}})\end{align*} $$

of degree $\ell $ over $k_{2}$ , and write $\beta _{2}$ for an element of $M_{Z_{2}}^{*}$ corresponding to this connected Galois tame covering. Then we have

$$ \begin{align*}\beta_{2}=\sum_{c_{2}\in S_{X_{H_{2}}}} t_{c_{2}}\beta_{c_{2}},\end{align*} $$

where $t_{c_{2}} \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ and $\beta _{c_{2}} \in M_{Z_{2}, c_{2}}$ . On the other hand, by the proof concerning $\psi _{Y}^{\mathrm {t, \ell }}(M_{Y_{1}}^{*})=M_{Y_{2}}^{*}$ in the fourth paragraph of the proof of Lemma 3.10, $\beta _{2}$ induces an element

$$ \begin{align*}\beta_{1}\stackrel{\mathrm{def}}{=} \sum_{c_{2}\in S_{X_{H_{2}}}\setminus \{e_{X_{H_{2}}}\}} t_{c_{2}}\beta_{\rho^{-1}_{\phi|_{H_{1}}}(c_{2})} +t_{e_{X_{H_{2}}}}\beta_{\rho^{-1}_{\phi|_{H_{1}}}(e_{X_{H_2}})} \\ =\sum_{c_{2}\in S_{X_{H_{2}}}\setminus \{e_{X_{H_{2}}}\}} t_{c_{2}}\beta_{\rho^{-1}_{\phi|_{H_{1}}}(c_{2})} +t_{e_{X_{H_{2}}}}\beta_{e_{X_{H_1}}}\in M_{Z_{1}}^{*}.\end{align*} $$

Then we have that the coefficient $t_{e_{X_{H_{2}}}}$ of $\beta _{e_{X_{H_1}}}$ is not equal to $0$ . Thus, the composition

$$ \begin{align*}(Y_{\alpha_{1}}, D_{Y_{\alpha_{1}}}) \times_{(X_{1}, D_{X_{1}})} (X_{H_1}, D_{X_{H_1}}) \rightarrow (Z_{1}, D_{Z_{1}}) \overset{g_{1}}\rightarrow (X_{H_1}, D_{X_{H_{1}}})\end{align*} $$

is tamely ramified over $e_{X_{H_1}}$ . This means that $e_{X_{H_{1}}}$ is contained in $S_{X_{H_1}}$ . We complete the proof of the lemma.

Proof. We write m for $\#G$ and put $K_{2}\stackrel {\mathrm {def}}{=} \text {ker}({\pi }_{1}^{\mathrm {t}}(U_{X_{2}}) \twoheadrightarrow {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})^{\mathrm {ab}}\otimes \mathbb {Z}/m\mathbb {Z}).$ Then we see immediately that $K_{2}$ is contained in $H_{2}$ . Let $K_{1}\stackrel {\mathrm {def}}{=} \phi ^{-1}(K_{2}) \subseteq H_{1}$ . Write $(X_{K_{i}}, D_{X_{K_{i}}})$ for the smooth pointed stable curves of type $(g_{X_{K_{i}}}, n_{X_{K_{i}}})$ over $k_{i}$ induced by $K_{i}$ and $f_{K_{i}}: (X_{K_{i}}, D_{X_{K_{i}}}) \rightarrow (X_{i}, D_{X_{i}})$ for the tame covering over $k_{i}$ induced by the inclusion $K_{i}\hookrightarrow {\pi }_{1}^{\mathrm { t}}(U_{X_{i}})$ . We identify ${\pi }_{1}^{\mathrm {t}}(U_{X_{1}})/K_{1}$ with ${\pi }_{1}^{\mathrm {t}}(U_{X_{2}})/K_{2}$ via the isomorphism induced by $\phi $ , and denote by $A\stackrel {\mathrm {def}}{=} {\pi }_{1}^{\mathrm { t}}(U_{X_{1}})/K_{1} \cong {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})/K_{2}$ .

Since each p-Galois tame covering is étale (i.e., Galois tame coverings whose Galois group is a p-group), we have that $g_{X_{K_{1}}}=g_{X_{K_{2}}}$ follows from the Riemann-Hurwitz formula, and that $n_{X_{K_{1}}}=\#(A)n_{X}=n_{X_{K_{2}}}$ . Then we obtain $(g_{X_{K_{1}}}, n_{X_{K_{1}}})=(g_{X_{K_{2}}}, n_{X_{K_{2}}}).$ Thus, Lemma 3.11 implies that the commutative diagram

of profinite groups induces a commutative diagram

Moreover, Remark 3.11.1 implies that the commutative diagram above admits a natural action of A. Then, for each $e^{\mathrm {gp}}_{X_{K_{1}}} \in D^{\mathrm {gp}}_{X_{K_{1}}}$ , the inertia subgroup $I_{e^{\mathrm { gp}}_{X_{K_{1}}}}$ in A associated to $e^{\mathrm {gp}}_{X_{K_{1}}}$ (i.e., the stabilizer of $e^{\mathrm {gp}}_{X_{K_{1}}}$ under the action of A) is equal to the inertia subgroup $I_{e^{\mathrm {gp}}_{X_{K_{2}}}}$ in A associated to $e^{\mathrm {gp}}_{X_{K_{2}}}\stackrel {\mathrm {def}}{=} \rho _{\phi |_{K_{1}}}(e^{\mathrm {gp}}_{X_{K_{1}}}) \in D_{X_{K_{2}}}^{\mathrm {gp}}$ . On the other hand, write F for the kernel of the natural morphism $A \twoheadrightarrow G$ induced by the inclusion $K_{i} \hookrightarrow H_{i}$ , $i \in \{1, 2\}.$ Since $(X_{H_{i}}, D_{X_{H_{i}}}) \cong (X_{K_{i}}, D_{X_{K_{i}}})/F$ , the set of ramification indices of the Galois tame covering $(X_{K_{i}}, D_{X_{K_{i}}}) \rightarrow (X_{H_{i}}, D_{X_{H_{i}}})$ with Galois group F are equal to $\{\#(F \cap I_{e^{\mathrm {gp}}_{X_{K_{i}}}})\}_{e^{\mathrm {gp}}_{X_{K_{i}}} \in D^{\mathrm {gp}}_{X_{K_{i}}}}$ . Then, by the Riemann-Hurwitz formula, we have $(g_{X_{H_{1}}}, n_{X_{H_{1}}})=(g_{X_{H_{2}}}, n_{X_{H_{2}}}).$ This completes the proof of the lemma.

Proof. First, suppose that G is a simple finite group. By applying Lemma 3.12, we may assume that G is nonabelian. Moreover, we claim that we may assume that $n_{X}$ is a positive even number. Let us prove this claim. Suppose $p\neq 2$ . Let $R_{2} \subseteq {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})$ be an open subgroup, such that $\#({\pi }_{1}^{\mathrm {t}}(U_{X_{2}})/R_{2})=2$ , and that $R_{2} \supseteq \text {ker}({\pi }_{1}^{\mathrm { t}}(U_{X_{2}}) \twoheadrightarrow {\pi }_{1}(X_{2}))$ (i.e., the cyclic Galois tame covering corresponding to $R_{2}$ is étale). Let $R_{1} \stackrel {\mathrm {def}}{=} \phi ^{-1}(R_{2}) \subseteq {\pi }_{1}^{\mathrm {t}}(U_{X_{1}})$ . Then we have that $\#({\pi }_{1}^{\mathrm {t}}(U_{X_{1}})/R_{1})=2$ , and that Lemma 3.9 implies $R_{1} \supseteq \text {ker}({\pi }_{1}^{\mathrm {t}}(U_{X_{1}}) \twoheadrightarrow {\pi }_{1}(X_{1}))$ . By replacing $H_{i}$ and ${\pi }_{1}^{\mathrm {t}}(U_{X_{i}})$ , $i\in \{1, 2\}$ , by $H_{i} \cap R_{i}$ and $R_{i}$ , respectively, we may assume that $n_{X}$ is a positive even number. Suppose that $p=2$ . Let $\ell $ be a prime number, such that $(\ell , 2)=(\ell , \#G)=1$ . By [Reference RaynaudR1, Théorème 4.3.1], there exists an open subgroup $R^{*}_{2} \subseteq {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})$ , such that $\#({\pi }_{1}^{\mathrm {t}}(U_{X_{2}})/R^{*}_{2})=\ell $ , that $R^{*}_{2} \supseteq \text {ker}({\pi }_{1}^{\mathrm {t}}(U_{X_{2}}) \twoheadrightarrow {\pi }_{1}(X_{2}))$ , and that

$$ \begin{align*}\text{dim}_{\mathbb{F}_{p}}(R_{2}^{*, \mathrm{ab}} \otimes \mathbb{F}_{p})>0.\end{align*} $$

Let $R^{*}_{1} \stackrel {\mathrm {def}}{=} \phi ^{-1}(R^{*}_{2}) \subseteq {\pi }_{1}^{\mathrm {t}}(U_{X_{1}})$ . Then we have that $\#({\pi }_{1}^{\mathrm {t}}(U_{X_{1}})/R^{*}_{1})=\ell $ , that $\text {dim}_{\mathbb {F}_{p}}(R_{1}^{*, \mathrm {ab}} \otimes \mathbb {F}_{p})>0$ , and that Lemma 3.9 implies $R^{*}_{1} \supseteq \text {ker}({\pi }_{1}^{\mathrm {t}}(U_{X_{1}}) \twoheadrightarrow {\pi }_{1}(X_{1}))$ . Thus, we may take an open subgroup $R^{\prime }_{2} \subseteq R^{*}_{2}$ , such that

$$ \begin{align*}{\pi}_{1}^{\mathrm{t}}(U_{X_{2}})/R^{\prime}_{2} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/\ell\mathbb{Z},\end{align*} $$

and that $R^{\prime }_{2} \supseteq \text {ker}({\pi }_{1}^{\mathrm {t}}(U_{X_{2}}) \twoheadrightarrow {\pi }_{1}(X_{2}))$ . We put $R_{1}' \stackrel {\mathrm {def}}{=} \phi ^{-1}(R_{2}')$ . Then the construction of $R_{1}'$ implies ${\pi }_{1}^{\mathrm {t}}(U_{X_{1}})/R^{\prime }_{1} \cong \mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/\ell \mathbb {Z}$ and $R^{\prime }_{1} \supseteq \text {ker}({\pi }_{1}^{\mathrm {t}}(U_{X_{1}}) \twoheadrightarrow {\pi }_{1}(X_{1}))$ . By replacing $H_{i}$ and ${\pi }_{1}^{\mathrm {t}}(U_{X_{i}})$ , $i\in \{1, 2\}$ , by $H_{i} \cap R^{\prime }_{i}$ and $R^{\prime }_{i}$ , respectively, we may assume that $n_{X}$ is a positive even number. This completes the proof of the claim.

Let $\#G\stackrel {\mathrm {def}}{=} p^{t}m'$ , such that $(m', p)=1$ . Since $n_{X}$ is a positive even number, we may choose a Galois tame covering

$$ \begin{align*}f_{2}: (Y_{2}, D_{Y_{2}}) \rightarrow (X_{2}, D_{X_{2}})\end{align*} $$

over $k_{2}$ with Galois group $\mathbb {Z}/m'\mathbb {Z}$ , such that $f_{2}$ is totally ramified over every marked point of $D_{X_{2}}$ . Write $(g_{Y_{2}}, n_{Y_{2}})$ for the type of $(Y_{2}, D_{Y_{2}})$ , $Q_{2} \subseteq {\pi }_{1}^{\mathrm { t}}(U_{X_{2}})$ for the open normal subgroup induced by $f_{2}$ , $Q_{1}\stackrel {\mathrm {def}}{=} \phi ^{-1}(Q_{2}) \subseteq {\pi }_{1}^{\mathrm {t}}(U_{X_{1}})$ ,

$$ \begin{align*}f_{1}: (Y_{1}, D_{Y_{1}}) \rightarrow (X_{1}, D_{X_{1}})\end{align*} $$

for the Galois tame covering over $k_{1}$ with Galois group $\mathbb {Z}/m'\mathbb {Z}$ induced by the natural inclusion $Q_{1} \hookrightarrow {\pi }_{1}^{\mathrm {t}}(U_{X_{1}})$ , and $(g_{Y_{1}}, n_{Y_{1}})$ for the type of $(Y_{1}, D_{Y_{1}})$ . Then Lemma 3.12 implies that $(g_{Y_{1}}, n_{Y_{1}})=(g_{Y_{2}}, n_{Y_{2}})$ and $f_{1}$ is also totally ramified over every marked point of $D_{X_{1}}$ .

We consider the Galois tame covering

$$ \begin{align*}(Z_{i}, D_{Z_{i}}) \stackrel{\mathrm{def}}{=} (X_{H_i}, D_{X_{H_{i}}}) \times_{(X_{i}, D_{X_{i}})} (Y_{i}, D_{Y_{i}}) \rightarrow (X_{i}, D_{X_{i}}), \ i \in \{1, 2\},\end{align*} $$

over $k_{i}$ with Galois group $G \times \mathbb {Z}/m'\mathbb {Z}$ which is the composition of $(Z_{i}, D_{Z_{i}}) \rightarrow (Y_{i}, D_{Y_{i}})$ and $(Y_{i}, D_{Y_{i}}) \rightarrow (X_{i}, D_{X_{i}})$ . Note that since G is a nonabelian simple finite group, $(Z_{i}, D_{Z_{i}})$ is connected. Moreover, by Abhyankar’s lemma, we obtain that $(Z_{i}, D_{Z_{i}}) \rightarrow (Y_{i}, D_{Y_{i}})$ is an étale covering over $k_{i}$ . Since $(g_{Y_{1}}, n_{Y_{1}})=(g_{Y_{2}}, n_{Y_{2}})$ and $(Z_{i}, D_{Z_{i}}) \rightarrow (Y_{i}, D_{Y_{i}})$ is unramified, the Riemann-Hurwitz formula implies $(g_{Z_{1}}, n_{Z_{1}})=(g_{Z_{2}}, n_{Z_{2}}).$

Next, let us prove the lemma in the case where G is an arbitrary finite group. Let $G_{1} \subseteq G_{2} \subseteq \dots \subseteq G_{n}\stackrel {\mathrm {def}}{=} G$ be a sequence of subgroups of G, such that $G_{i}/G_{i-1}$ is a simple group for all $i\in \{2, \dots n\}$ . In order to verify the lemma, we see that it is sufficient to prove the lemma when $n=2$ . Let $N_{2}$ be the kernel of the natural homomorphism ${\pi }_{1}^{\mathrm { t}}(U_{X_{2}}) \twoheadrightarrow G \twoheadrightarrow G_{1}$ and $N_{1} \stackrel {\mathrm {def}}{=} \phi ^{-1}(N_{2})$ . Then by replacing G by $G_{1}$ and by applying the lemma for the simple group $G_{1}$ , we obtain an open normal subgroup $M_{2} \subseteq {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})$ which is contained in $N_{2}$ , such that $(g_{X_{M_{1}}}, n_{X_{M_{1}}})=(g_{X_{M_{2}}}, n_{X_{M_{2}}}),$ where $M_{1} \stackrel {\mathrm {def}}{=} \phi ^{-1}(M_{2})$ , and $(g_{X_{M_{i}}}, n_{X_{M_{i}}})$ , $i\in \{1, 2\}$ , denotes the type of the smooth pointed stable curve corresponding to $M_{i}$ .

If $M_{i} \subseteq H_{i}$ , $i\in \{1, 2\}$ , then we may put $P_{i} \stackrel {\mathrm {def}}{=} M_{i}.$ If $H_{i}$ , $i\in \{1, 2\}$ , does not contain $M_{i}$ , we put $O_{i} \stackrel {\mathrm {def}}{=} M_{i} \cap H_{i}$ . Then we have $M_{i}/O_{i}\cong G/G_{1}.$ Note that $G/G_{1}$ is a simple group. Then the lemma follows from the lemma when we replace $(X_{i}, D_{X_{i}})$ and G by $(X_{M_{i}}, D_{X_{M_{i}}})$ and the simple group $G/G_{1}$ , respectively. This completes the proof of the lemma.

Proof. If $n_{X}=0$ , then the theorem is trivial. We suppose $n_{X}>0$ . Let $m>>0$ be an integer number, such that $(m, p)=1$ . We put $K_{i}\stackrel {\mathrm {def}}{=} \text {ker}({\pi }_{1}^{\mathrm {t}}(U_{X_{i}}) \twoheadrightarrow {\pi }_{1}^{\mathrm {t}}(U_{X_{i}})^{\mathrm {ab}}\otimes \mathbb {Z}/m\mathbb {Z})$ , $i \in \{1, 2\}$ . Write $(X_{K_{i}}, D_{K_{i}})$ for the smooth pointed stable curve of type $(g_{X_{K_{i}}}, n_{X_{K_{i}}})$ over $k_{i}$ induced by $K_{i}$ . Moreover, the condition $m>>0$ implies $g_{X_{K_{1}}}=g_{X_{K_{2}}} \geq 2, \ n_{X_{K_{1}}}=n_{X_{K_{2}}}\geq 2.$

By applying Lemma 3.13, we may choose a set of open subgroups $C_{X_{2}}\stackrel {\mathrm {def}}{=} \{H_{2, j}\}_{j \in \mathbb {Z}_{> 0}}$ of ${\pi }_{1}^{\mathrm {t}}(U_{X_{2}})$ , such that the following conditions hold: (a) $H_{2, 1}=K_{2}$ ; (b) $\varprojlim _{j}{\pi }_{1}^{\mathrm {t}}(U_{X_{2}})/H_{2, j} \cong {\pi }_{1}^{\mathrm {t}}(U_{X_{2}})$ (i.e., $C_{X_{2}}$ is a cofinal system); (c) write $\{H_{1, j}\stackrel {\mathrm {def}}{=} \phi ^{-1}(H_{2, j})\}_{j \in \mathbb {Z}_{> 0}}$ for the set of open subgroups of ${\pi }_{1}^{\mathrm {t}}(U_{X_{1}})$ induced by $\phi $ , and, for each $j \in \mathbb {Z}_{> 0}$ , write $(X_{H_{i, j}}, D_{X_{H_{i, j}}})$ , $i\in \{1, 2\}$ , for the smooth pointed stable curve of type $(g_{X_{H_{i,j}}}, n_{X_{H_{i,j}}})$ over $k_{i}$ induced by $H_{i, j}$ ; then we have $(g_{X_{H_{1, j}}}, n_{X_{H_{1, j}}})=(g_{X_{H_{2, j}}}, n_{X_{H_{2, j}}})$ .

For each $j \in \mathbb {Z}_{> 0}$ , we write $e_{X_{H_{2, j}}} \in D_{X_{H_{2, j}}}$ for the image of $\widetilde e_{2}$ . Then we obtain a sequence of marked points

$$ \begin{align*}\mathcal{I}_{\widetilde e_{2}}^{C_{X_{2}}}: \dots \mapsto e_{H_{2, 2}}\mapsto e_{H_{2, 1}}.\end{align*} $$

Proposition 3.6 implies that, for each $H_{2, j}$ , $j\in \mathbb {Z}_{> 0}$ , the set $D^{\mathrm {gp}}_{X_{H_{2, j}}}$ can be mono-anabelian reconstructed from $H_{2, j}$ . For each $e_{X_{H_{2, j}}} \in D_{X_{H_{2, j}}}$ , we denote by

$$ \begin{align*}e^{\mathrm{gp}}_{X_{H_{2, j}}}\stackrel{\mathrm{def}}{=} {\vartheta}_{X_{H_{2, j}}}^{-1}(e_{X_{H_{2, j}}}).\end{align*} $$

Then the sequence of marked points $\mathcal {I}^{C_{X}}_{\widetilde e_{2}}$ induces a sequence

$$ \begin{align*}\mathcal{I}^{C_{X}}_{\widetilde e_{2}^{\mathrm{gp}}}: \dots \mapsto e^{\mathrm{gp}}_{X_{H_{2, 2}}}\mapsto e^{\mathrm{ gp}}_{X_{H_{2,1}}}.\end{align*} $$

Then Remark 3.6.1 implies that the inertia subgroup associated to $\widetilde e_{2}$ is equal to the stabilizer of $\mathcal {I}^{C_{X}}_{\widetilde e_{2}^{\mathrm {gp}}}$ .

By Lemma 3.11 and Lemma 3.13, $\mathcal {I}_{\widetilde e^{\mathrm {gp}}_{2}}^{C_{X_{2}}}$ induces a sequence as follows:

$$ \begin{align*}\dots \mapsto e^{\mathrm{gp}}_{X_{H_{1, 2}}}\stackrel{\mathrm{def}}{=} \rho^{-1}_{\phi|_{H_{1, 2}}}(e^{\mathrm{gp}}_{X_{H_{2, 2}}}) \in D^{\mathrm{gp}}_{X_{H_{1,2}}}\mapsto e^{\mathrm{gp}}_{X_{H_{1, 1}}}\stackrel{\mathrm{def}}{=} \rho^{-1}_{\phi|_{H_{1, 1}}}(e^{\mathrm{gp}}_{X_{H_{2, 1}}})\in D^{\mathrm{ gp}}_{X_{H_{1, 1}}}\end{align*} $$

with an action of $I_{\widetilde e_{2}}$ . Then Proposition 3.7 implies that we have a sequence

$$ \begin{align*}\dots \mapsto e_{X_{H_{1, 2}}}\stackrel{\mathrm{def}}{=} {\vartheta}_{X_{H_{1, 2}}}(e^{\mathrm{gp}}_{X_{H_{1, 2}}})\in D_{X_{H_{1,2}}} \mapsto e_{X_{H_{1, 1}}}\stackrel{\mathrm{def}}{=} {\vartheta}_{X_{H_{1, 1}}}(e^{\mathrm{gp}}_{X_{H_{1, 1}}})\in D_{X_{H_{1,1}}}\end{align*} $$

with an action of $I_{\widetilde e_{2}}.$

Let $K_{\text {ker}(\phi )}$ be the subfield of $\widetilde K$ induced by the closed subgroup $\text {ker}(\phi )$ of ${\pi }_{1}^{\mathrm {t}}(U_{X_{1}})$ , $\widetilde X_{1, \text {ker}(\phi )}$ the normalization of $X_{1}$ in $K_{\text {ker}(\phi )}$ , and $D_{\widetilde X_{1, \text {ker}(\phi )}}$ the inverse image of $D_{X_{1}}$ in $\widetilde X_{1, \text {ker}(\phi )}$ . Then the sequence

$$ \begin{align*}\dots \mapsto e_{X_{H_{1, 2}}} \mapsto e_{X_{H_{1, 1}}}\end{align*} $$

determines a point $\widetilde e_{1, \text {ker}(\phi )} \in D_{\widetilde X_{1, \text {ker}(\phi )}}.$ We choose a point of $\widetilde e_{1} \in D_{\widetilde X_{1}}$ , such that the image of $\widetilde e_{1}$ in $D_{\widetilde X_{1, \text {ker}(\phi )}}$ is $\widetilde e_{1, \text {ker}(\phi )}$ . Then we have $\phi (I_{\widetilde e_{1}})=I_{\widetilde e_{2}}$ . Moreover, since $I_{\widetilde e_{1}}$ and $I_{\widetilde e_{2}}$ are isomorphic to $\widehat {\mathbb {Z}}(1)^{p'}$ , the restriction homomorphism $\phi |_{I_{\widetilde e_{1}}}$ is an isomorphism. This completes the proof of the theorem.

Proof. First, we claim that we may assume $n_{X}=3.$ If $g_{X}=0$ , then $n_{X}\geq 3$ . Suppose that $g_{X}\geq 1$ . Theorem 3.14 implies that $\phi : \pi _{1}^{\mathrm {t}}(U_{X_{1}}) \twoheadrightarrow \pi _{1}^{\mathrm { t}}(U_{X_{2}})$ induces an open continuous surjection $\phi ^{\text {'et}}: \pi _{1}(X_{1}) \twoheadrightarrow \pi _{1}(X_{2}).$ Let $H^{\prime }_{2} \subseteq \pi _{1}(X_{2})$ be an open normal subgroup, such that $\#(\pi _{1}(X_{2})/H^{\prime }_{2})\geq 3$ and $H^{\prime }_{1} \stackrel {\mathrm {def}}{=} (\phi ^{\text {'et}})^{-1}(H_{2}')$ . Write $H_{i}\subseteq \pi _{1}^{\mathrm {t}}(U_{X_{i}})$ , $i \in \{1, 2\}$ , for the inverse image of $H_{i}'$ of the natural surjection $\pi _{1}^{\mathrm {t}}(U_{X_{i}})\twoheadrightarrow \pi _{1}(X_{i})$ , and $(X_{H_{i}}, D_{X_{H_i}})$ for the smooth pointed stable curve of type $(g_{X_{H_i}}, n_{X_{H_{i}}})$ over $k_{i}$ induced by $H_{i}$ . Note that $g_{X_{H_1}}=g_{X_{H_2}}\geq 2$ and $n_{X_{H_{1}}}=n_{X_{H_{2}}}\geq 3$ . By replacing $(X_{i}, D_{X_{i}})$ by $(X_{H_{i}}, D_{X_{H_i}})$ , we may assume $g_{X}\geq 2$ and $n_{X} \geq 3$ . The surjection $\phi $ induces a bijection

$$ \begin{align*}D_{X_{1}} \overset{\vartheta_{X_{1}}^{-1}}{\stackrel{\sim}{\rightarrow}} D_{X_{1}}^{\mathrm{gp}} \overset{\rho_{\phi}}{\stackrel{\sim}{\rightarrow}} D_{X_{2}}^{\mathrm{gp}} \overset{\vartheta_{X_{2}}}{\stackrel{\sim}{\rightarrow}} D_{X_{2}}.\end{align*} $$

Let $D^{\prime }_{X_1}\stackrel {\mathrm {def}}{=} \{e_{1, 1}, e_{1, 2}, e_{1, 3}\} \subseteq D_{X_{1}}$ and $D^{\prime }_{X_2} \stackrel {\mathrm {def}}{=} \{e_{2, 1}\stackrel {\mathrm {def}}{=} \vartheta _{X_{2}}\circ \rho _{\phi }\circ \vartheta _{X_{1}}^{-1}(e_{1, 1}), e_{2, 2}\stackrel {\mathrm {def}}{=} \vartheta _{X_{2}}\circ \rho _{\phi }\circ \vartheta _{X_{1}}^{-1}(e_{1, 2}), e_{2, 3}\stackrel {\mathrm {def}}{=} \vartheta _{X_{2}}\circ \rho _{\phi }\circ \vartheta _{X_{1}}^{-1}(e_{1, 3})\} \subseteq D_{X_{2}}$ . Then $(X_{i}, D^{\prime }_{X_i})$ , $i\in \{1, 2\}$ , is a smooth pointed stable curve of type $(g_{X}, 3)$ over $k_{i}$ . Write $I_{i}$ , $i \in \{1, 2\}$ , for the closed subgroup of $\pi _{1}^{\mathrm {t}}(U_{X_i})$ generated by the inertia subgroups associated to the elements of $D_{\widetilde X_{i}}$ whose images in $D_{X_{i}}$ are contained in $D_{X_{i}} \setminus D^{\prime }_{X_i}$ . Then we have an isomorphism

$$ \begin{align*}\pi_{1}^{\mathrm{t}}(X_{i} \setminus D^{\prime}_{X_i}) \cong \pi_{1}^{\mathrm{t}}(U_{X_i})/I_{i}, \ i \in \{1, 2\}.\end{align*} $$

Moreover, Theorem 3.14 implies that $\phi $ induces an open continuous surjective homomorphism

$$ \begin{align*}\phi': \pi_{1}^{\mathrm{t}}(X_{1} \setminus D^{\prime}_{X_1}) \twoheadrightarrow \pi_{1}^{\mathrm{t}}(X_{2} \setminus D^{\prime}_{X_2}).\end{align*} $$

Thus, by replacing $(X_{i}, D_{X_{i}})$ , $\pi _{1}^{\mathrm {t}}(U_{X_{i}})$ , and $\phi $ by $(X_{i}, D^{\prime }_{X_i})$ , $\pi _{1}^{\mathrm {t}}(X_{i} \setminus D^{\prime }_{X_i})$ , and $\phi '$ , respectively, we may assume $n_{X}=3.$

Let $r \in \mathbb {N}$ . We denote by $\mathbb {F}_{p^{r}, \widetilde e_{i}}, \ i \in \{1, 2\},$ the unique subfield of $\mathbb {F}_{\widetilde e_{i}}$ whose cardinality is equal to $p^{r}$ . On the other hand, we fix any finite field $\mathbb {F}_{p^{r}}$ of cardinality $p^{r}$ and an algebraic closure $\overline {\mathbb {F}}_{p}$ of $\mathbb {F}_{p}$ . By Proposition 3.7, we have $\mathbb {F}_{p^{r}, \widetilde e_{i}}^{\times }=I_{\widetilde e_{i}}/(p^{r}-1)$ can be mono-anabelian reconstructed from ${\pi }_{1}^{\mathrm {t}}(U_{X_{i}})$ . Then reconstructing the field structure of $\mathbb {F}_{p^{r}, \widetilde e_{i}}$ is equivalent to reconstructing $\text {Hom}_{\mathrm {fields}}(\mathbb {F}_{p^{r}, \widetilde e_{i}}, \mathbb {F}_{p^{r}})$ as a subset of $\text {Hom}_{\mathrm {group}}(\mathbb {F}_{p^{r}, \widetilde e_{i}}^{\times }, \mathbb {F}_{p^{r}}^{\times })$ . Note that, in order to reconstruct the field structure of $\mathbb {F}_{\widetilde e_{i}}$ , it is sufficient to reconstruct the subset $\text {Hom}_{\mathrm {fields}}(\mathbb {F}_{p^{r}, \widetilde e_{i}}, \mathbb {F}_{p^{r}})$ for r in a cofinal subset of $\mathbb {N}$ with respect to division.

Let $\chi _{i} \in \text {Hom}_{\mathrm {groups}}({\pi }_{1}^{\mathrm {t}}(U_{X_{i}})^{\mathrm {ab}}\otimes \mathbb {Z}/(p^{r}-1)\mathbb {Z}, \mathbb {F}^{\times }_{p^{r}}).$ Write $H_{\chi _{i}}$ for the kernel of ${\pi }_{1}^{\mathrm { t}}(U_{X_{i}}) \twoheadrightarrow {\pi }_{1}^{\mathrm {t}}(U_{X_{i}})^{\mathrm {ab}}\otimes \mathbb {Z}/(p^{r}-1)\mathbb {Z} \overset {\chi _{i}}\rightarrow \mathbb {F}_{p^{r}}^{\times }$ , $M_{\chi _{i}}$ for $H^{\mathrm {ab}}_{\chi _{i}} \otimes \mathbb {F}_{p}$ , and $(X_{H_{\chi _{i}}}, D_{X_{H_{\chi _{i}}}})$ for the smooth pointed stable curve over $k_{i}$ induced by $H_{\chi _{i}}$ . We define

$$ \begin{align*}M_{\chi_{i}}[\chi_{i}]\stackrel{\mathrm{def}}{=} \{a \in M_{\chi_{i}} \otimes_{\mathbb{F}_{p}} \overline{\mathbb{F}}_{p} \ | \ \sigma(a)=\chi_{i}(\sigma)a \ \text{for all} \ \sigma \in {\pi}_{1}^{\mathrm{t}}(U_{X_{i}})^{\mathrm{ab}}\otimes \mathbb{Z}/(p^{r}-1)\mathbb{Z} \}\end{align*} $$

and $\gamma _{\chi _{i}}(M_{\chi _{i}})\stackrel {\mathrm {def}}{=} \text {dim}_{\overline {\mathbb {F}}_{p}}(M_{\chi _{i}}[\chi _{i}])$ (i.e., a generalized Hasse-Witt invariant (see [Reference YangY5, Section 2.2])). Then [Reference TamagawaT4, Remark 3.7] implies $\gamma _{\chi _{i}}(M_{\chi _{i}}) \leq g_{X}+1$ . Moreover, we define two maps

$$ \begin{align*}\text{Res}_{i, r}: \text{Hom}_{\mathrm{groups}}({\pi}_{1}^{\mathrm{t}}(U_{X_{i}})^{\mathrm{ ab}}\otimes \mathbb{Z}/(p^{r}-1)\mathbb{Z}, \mathbb{F}^{\times}_{p^{r}}) \rightarrow \text{Hom}_{\mathrm{groups}}(\mathbb{F}^{\times}_{p^{r}, \widetilde e_{i}}, \mathbb{F}_{p^{r}}^{\times}),\end{align*} $$

$$ \begin{align*}\Gamma_{i, r}: \text{Hom}_{\mathrm{groups}}({\pi}_{1}^{\mathrm{t}}(U_{X_{i}})^{\mathrm{ab}}\otimes \mathbb{Z}/(p^{r}-1)\mathbb{Z}, \mathbb{F}^{\times}_{p^{r}}) \rightarrow \mathbb{Z}_{\geq 0}, \ \chi_{i}\mapsto\gamma_{\chi_{i}}(M_{\chi_{i}}),\end{align*} $$

where the map $\text {Res}_{i, r}$ is the restriction with respect to the natural inclusion $\mathbb {F}^{\times }_{p^{r}, \widetilde e_{i}} \hookrightarrow {\pi }_{1}^{\mathrm {t}}(U_{X_{i}})^{\mathrm {ab}}\otimes \mathbb {Z}/(p^{r}-1)\mathbb {Z}.$

Let $m_{0}$ be the product of all prime numbers $\leq p-2$ if $p\neq 2, 3$ and $m_{0}=1$ if $p=2, 3$ . Let $r_{0}$ be the order of p in the multiplicative group $(\mathbb {Z}/m_{0}\mathbb {Z})^{\times }$ . Then [Reference TamagawaT4, Claim 5.4] implies the following result holds: