2.1. Surfaces

Unless otherwise stated, in this paper, a surface $S=S_{g,n}^k$ is a closed connected oriented differentiable surface of genus $g(S)=g$ from which $n(S)=n$ points and $k(S)=k$ open discs have been removed. A surface $S$ is hyperbolic if its Euler characteristic $\chi (S)$ is negative. For $k=0$ (resp. $n=0$ ), we let $S_{g,n}\;:\!=\;S_{g,n}^0$ (resp. $S_g^k\;:\!=\;S_{g,0}^k$ ). Also, we let $S_g\;:\!=\;S_{g,0}$ . The boundary of $S$ is denoted by $\partial S$ and let ${S}^{^{^{\!\!\!\circ}}}\;:\!=\;S\smallsetminus{\partial S}$ , so that ${S}^{^{^{\!\!\!\circ}}}\cong S_{g,n+k}$ . We let $\overline{S}$ be the closed surface obtained from $S$ filling in the punctures and glueing a closed disc to each boundary component, so that $\overline{S}\cong S_{g}$ .

A marked surface $(S,{\mathcal{P}})$ is a surface $S$ endowed with a set $\mathcal{P}$ of distinct marked points. The marked surface $(S,{\mathcal{P}})$ is hyperbolic if $S\smallsetminus{\mathcal{P}}$ has negative Euler characteristic. We denote by $\vec{\mathcal{P}}$ the set $\mathcal{P}$ with an order fixed on it. We then denote by ${\mathcal{P}}_\circ$ the set of punctures of $S$ and by ${\mathcal{P}}_\partial$ the set of connected components of $\partial S$ .

2.2. Mapping class groups

For a surface $S=S_{g,n}^k$ , we let ${\Gamma }(S)$ and ${\mathrm{P}\Gamma }(S)$ be respectively the mapping class group and the pure mapping class group associated with $S$ . Note that the inclusion ${S}^{^{^{\!\!\!\circ}}}\hookrightarrow S$ induces, by restriction, a natural homomorphism ${\Gamma }(S)\hookrightarrow{\Gamma }({S}^{^{^{\!\!\!\circ}}})$ whose image is the stabilizer of the partition of the punctures of ${S}^{^{^{\!\!\!\circ}}}$ into those which have a boundary in $S$ and those which do not.

The mapping class group ${\Gamma }(S,{\mathcal{P}})$ of the marked surface $(S,{\mathcal{P}})$ is the group of relative, with respect to the subset of points $\mathcal{P}$ , isotopy classes of orientation preserving diffeomorphisms of $S$ . When isotopies are further required to preserve an order on the set $\mathcal{P}$ , we denote the corresponding mapping class group by ${\Gamma }(S,\vec{\mathcal{P}})$ . The pure mapping class group ${\mathrm{P}\Gamma }(S,{\mathcal{P}})$ is the kernel of the natural homomorphism ${\Gamma }(S,{\mathcal{P}})\to \Sigma _{{\mathcal{P}}}\times \Sigma _{{\mathcal{P}}_\circ }\times \Sigma _{{\mathcal{P}}_\partial }$ , where $\Sigma _{\mathcal{P}}$ , $\Sigma _{{\mathcal{P}}_\circ }$ , and $\Sigma _{{\mathcal{P}}_\partial }$ are, respectively, the permutation groups on the sets $\mathcal{P}$ , ${\mathcal{P}}_\circ$ , and ${\mathcal{P}}_\partial$ . There is then a natural isomorphism ${\mathrm{P}\Gamma }(S,{\mathcal{P}})\cong{\mathrm{P}\Gamma }({S}^{^{^{\!\!\!\circ}}}\smallsetminus{\mathcal{P}})$ .

2.3. Mapping class groups of disconnected surfaces

For some applications, it is useful to consider the mapping class group ${\Gamma }(S,{\mathcal{P}})$ of a marked oriented differentiable disconnected surface $(S,{\mathcal{P}})$ . This group can be easily described in terms of the mapping class groups of the connected components of $(S,{\mathcal{P}})$ . Let $S=\coprod _{i=1}^h S^i$ be the decomposition in connected components and let ${\mathcal{P}}^i\;:\!=\;{\mathcal{P}}\cap S^i$ , for $i=1,\ldots,h$ . Let us fix a set of representatives $(S^{i_1},{\mathcal{P}}^{i_1}),\ldots, (S^{i_k},{\mathcal{P}}^{i_k})$ for the orbits of the action of ${\Gamma }(S,{\mathcal{P}})$ on the set of marked connected components of $S$ . Of course, two marked connected components are in the same orbit, if and only if, they are diffeomorphic and support the same number of marked points. Let us denote by $n_i$ the cardinality of the ${\Gamma }(S,{\mathcal{P}})$ -orbit of the component $(S^i,{\mathcal{P}}^i)$ of $(S,{\mathcal{P}})$ , for $i=1,\ldots,h$ . There is then an isomorphism:

(1) \begin{equation} {\Gamma }(S,{\mathcal{P}})\cong \prod _{j=1}^k\left ({\Gamma }(S^{i_j},{\mathcal{P}}^{i_j})\wr \Sigma _{n_{i_j}}\right ), \end{equation}

where ${\Gamma }(S^{i_j},{\mathcal{P}}^{i_j})\wr \Sigma _{n_{i_j}}$ denotes the wreath product of the mapping class group with the symmetric group on $n_{i_j}$ letters.

A similar description holds for the mapping class group ${\Gamma }(S,\vec{\mathcal{P}})$ after replacing the ${\Gamma }(S,{\mathcal{P}})$ -orbits of the marked connected components of $(S,{\mathcal{P}})$ with their ${\Gamma }(S,\vec{\mathcal{P}})$ -orbits. Note that a connected component with a nontrivial marking has a trivial ${\Gamma }(S,\vec{\mathcal{P}})$ -orbit.

By definition, the pure mapping class group of $(S,{\mathcal{P}})$ is instead just the direct product:

(2) \begin{equation} {\mathrm{P}\Gamma }(S,{\mathcal{P}})\cong \prod _{i=1}^h{\mathrm{P}\Gamma }(S^{i},{\mathcal{P}}^{i}). \end{equation}

2.4. Relative mapping class groups

For a marked hyperbolic surface $(S=S_{g,n}^k,{\mathcal{P}})$ , we let ${\Gamma }(S,{\mathcal{P}},{\partial S})$ be the relative mapping class group of the marked surface with boundary $(S,{\mathcal{P}},{\partial S})$ , that is to say the group of relative, with respect to the subspace ${\mathcal{P}}\cup{\partial S}$ , isotopy classes of orientation preserving diffeomorphisms of $S$ . Let $\delta _1,\ldots,\delta _k$ be the connected components of the boundary $\partial S$ . Then, for a hyperbolic marked surface $(S,{\mathcal{P}})$ , the group ${\Gamma }(S,{\mathcal{P}},{\partial S})$ is described by the natural short exact sequence:

\begin{equation*} 1\to \prod _{i=1}^k\tau _{{\delta }_i}^{\mathbb{Z}}\to{\Gamma }(S,{\mathcal{P}},{\partial S})\to{\Gamma }(S,{\mathcal{P}})\to 1. \end{equation*}

The relative pure mapping class group ${\mathrm{P}\Gamma }(S,{\mathcal{P}},{\partial S})$ is the subgroup of ${\Gamma }(S,{\mathcal{P}},{\partial S})$ consisting of those elements which admit representatives fixing pointwise the sets of marked points, punctures and boundary components of $S$ . For a hyperbolic marked surface $(S,{\mathcal{P}})$ , it is described by the short exact sequence:

\begin{equation*} 1\to{\mathrm{P}\Gamma }(S,{\mathcal{P}},{\partial S})\to{\Gamma }(S,{\mathcal{P}},{\partial S})\to \Sigma _{\mathcal{P}}\times \Sigma _n\times \Sigma _k\to 1. \end{equation*}

2.5. Hyperelliptic surfaces

A hyperelliptic surface $(S,{\upsilon })$ is the data consisting of a surface $S$ (cf. Section 2.1) together with a hyperelliptic involution $\upsilon$ on $S$ (i.e. a self-diffeomorphism $\upsilon$ of $S$ such that ${\upsilon }^2=\mathrm{id}_S$ and the quotient surface $S/\langle{\upsilon }\rangle$ has genus $0$ ). A Weierstrass point, resp. puncture, resp. boundary component of $(S,{\upsilon })$ is a point, resp. puncture, resp. boundary component, which is preserved by the hyperelliptic involution $\upsilon$ . We then denote by ${\mathscr{W}}_{\mathcal{P}}(S)$ , resp. ${\mathscr{W}}_\circ (S)$ , resp. ${\mathscr{W}}_\partial (S)$ , the set of Weierstrass points, resp. punctures, resp. boundary components on $(S,{\upsilon })$ . A marked hyperelliptic surface $(S,{\upsilon },{\mathcal{P}})$ is a hyperelliptic surface $(S,{\upsilon })$ endowed with a set $\mathcal{P}$ of distinct marked points.

2.6. Hyperelliptic mapping class groups

For a hyperelliptic surface $(S,{\upsilon })$ , let us also denote by $\upsilon$ the image of the involution $\upsilon$ in the mapping class group ${\Gamma }(S)$ (note that this image is trivial if and only if $g(S)=0$ and $S$ is not hyperbolic). The hyperelliptic mapping class group ${\Upsilon }(S)$ of $(S,{\upsilon })$ is then defined to be the centralizer of $\upsilon$ in ${\Gamma }(S)$ .

Let $S_{/{\upsilon }}\;:\!=\;S/\langle{\upsilon }\rangle$ be the quotient surface, $p_{\upsilon }\colon \thinspace S\to S_{/{\upsilon }}$ be the orbit map and denote by ${\mathcal{B}}_{\upsilon }=p_{\upsilon }({\mathscr{W}}_{\mathcal{P}}(S))$ the branch locus of $p_{\upsilon }$ . Let then ${\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })_{p_{\upsilon }}$ be the subgroup of the mapping class group ${\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ consisting of those elements which preserve the partitions of the sets of punctures and boundary components of $S_{/{\upsilon }}$ into those which are in the image of ${\mathscr{W}}_\circ (S)$ and ${\mathscr{W}}_\partial (S)$ by $p_{\upsilon }$ and those who are not.

For a hyperbolic hyperelliptic surface $(S,{\upsilon })$ , there is then a natural short exact sequence (cf. Corollary 12 in [Reference Maclachlan and Harvey26] and Proposition 3.1 in [Reference Margalit and Winarski27]):

(3) \begin{equation} 1\to \langle{\upsilon }\rangle \to{\Upsilon }(S)\to{\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })_{p_{\upsilon }}\to 1, \end{equation}

and a natural surjective representation:

(4) \begin{equation} \rho _{{\mathscr{W}}}(S)\colon \thinspace{\Upsilon }(S)\to \Sigma _{{\mathscr{W}}_{\mathcal{P}}(S)}\times \Sigma _{{\mathscr{W}}_\circ (S)}\times \Sigma _{{\mathscr{W}}_\partial (S)}, \end{equation}

where $\Sigma _{{\mathscr{W}}_{\mathcal{P}}(S)}$ , $\Sigma _{{\mathscr{W}}_\circ (S)}$ and $\Sigma _{{\mathscr{W}}_\partial (S)}$ are, respectively, the symmetric groups on the sets ${\mathscr{W}}_{\mathcal{P}}(S)$ , ${\mathscr{W}}_\circ (S)$ and ${\mathscr{W}}_\partial (S)$ .

Definition 2.1. The pure hyperelliptic mapping class group of a hyperbolic hyperelliptic surface $(S,{\upsilon })$ is defined to be the intersection inside the mapping class group ${\Gamma }(S)$ :

\begin{equation*}{\mathrm {P}\Upsilon }(S)\;:\!=\;\ker \rho _{{\mathscr{W}}}(S)\cap {\mathrm {P}\Gamma }(S).\end{equation*}

The short exact sequence (3) takes a simpler form for the pure hyperelliptic mapping class group (this in part motivated the definition):

(5) \begin{equation} 1\to \langle{\upsilon }\rangle \cap{\mathrm{P}\Upsilon }(S)\to{\mathrm{P}\Upsilon }(S)\to{\mathrm{P}\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\to 1. \end{equation}

Observe that ${\mathrm{P}\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\cong{\mathrm{P}\Gamma }(S_{/{\upsilon }}\smallsetminus{\mathcal{B}}_{\upsilon })$ . In particular, if $S$ contains at least two symmetric punctures or boundary components, then $\langle{\upsilon }\rangle \cap{\mathrm{P}\Upsilon }(S)=\{1\}$ and there is a series of natural isomorphisms:

\begin{equation*}{\mathrm {P}\Upsilon }(S)\cong {\mathrm {P}\Gamma }(S_{/{\upsilon }},{\mathcal {B}}_{\upsilon })\cong {\mathrm {P}\Gamma }(S_{/{\upsilon }}\smallsetminus {\mathcal {B}}_{\upsilon }).\end{equation*}

2.7. Hyperelliptic mapping class groups of a marked hyperelliptic surface

The hyperelliptic mapping class groups ${\Upsilon }(S,{\mathcal{P}})$ and ${\Upsilon }(S,\vec{{\mathcal{P}}})$ of the marked hyperelliptic surface $(S,{\upsilon },{\mathcal{P}})$ are defined to be, respectively, the inverse images of ${\Upsilon }(S)$ by the natural epimorphisms ${\Gamma }(S,{\mathcal{P}})\to{\Gamma }(S)$ and ${\Gamma }(S,\vec{{\mathcal{P}}})\to{\Gamma }(S)$ .

The pure hyperelliptic mapping class group ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})$ is the inverse image of ${\mathrm{P}\Upsilon }(S)$ by the natural epimorphism ${\mathrm{P}\Gamma }(S,{\mathcal{P}})\to{\mathrm{P}\Gamma }(S)$ .

For a hyperbolic marked hyperelliptic surface $(S,{\upsilon },{\mathcal{P}})$ , with these definitions, for a point $Q\in S\smallsetminus{\mathcal{P}}$ , we get, by restriction of the usual Birman short exact sequence for the mapping class group of the marked surface $(S,{\mathcal{P}}\cup \{Q\})$ , the short exact sequences:

(6) \begin{equation} \begin{array}{ll} &1\to \pi _1(S\smallsetminus{\mathcal{P}},Q)\to{\mathrm{P}\Upsilon }(S,{\mathcal{P}}\cup \{Q\})\to{\mathrm{P}\Upsilon }(S,{\mathcal{P}})\to 1,\\ \\ &1\to \pi _1(S\smallsetminus{\mathcal{P}},Q)\to{\Upsilon }(S,\overrightarrow{{\mathcal{P}}\cup \{Q\}})\to{\Upsilon }(S,\vec{\mathcal{P}})\to 1\\ \mbox{and}\;\;\;&\\ &1\to \pi _1(S\smallsetminus{\mathcal{P}},Q)\to{\Upsilon }(S,{\mathcal{P}},Q)\to{\Upsilon }(S,{\mathcal{P}})\to 1, \end{array} \end{equation}

where we let ${\Upsilon }(S,{\mathcal{P}},Q)$ be the stabilizer of the point $Q$ for the action of the hyperelliptic mapping class group ${\Upsilon }(S,{\mathcal{P}}\cup \{Q\})$ on the set of marked points ${\mathcal{P}}\cup \{Q\}$ .

Remark 2.4. Note that, if the set of points $\mathcal{P}$ is preserved by the hyperelliptic involution $\upsilon$ , then $(S\smallsetminus{\mathcal{P}},{\upsilon })$ is a hyperelliptic surface. However, unlike the case of mapping class groups, for ${\mathcal{P}}\neq \emptyset$ and $(S,{\mathcal{P}})$ hyperbolic, we have:

\begin{equation*}{\Upsilon }(S,{\mathcal {P}})\supsetneq {\Upsilon }(S\smallsetminus {\mathcal {P}})\;\;\;\;\;\;\;\;\;\;\;\mbox {and}\;\;\;\;\;\;\;\;\;\;\;{\mathrm {P}\Upsilon }(S,{\mathcal {P}})\supsetneq {\mathrm {P}\Upsilon }(S\smallsetminus {\mathcal {P}}).\end{equation*}

In fact, by definition, ${\Upsilon }(S\smallsetminus{\mathcal{P}})$ and ${\mathrm{P}\Upsilon }(S\smallsetminus{\mathcal{P}})$ are contained in the centralizer of the hyperelliptic involution $\upsilon$ in ${\Gamma }(S\smallsetminus{\mathcal{P}})$ while both ${\Upsilon }(S,{\mathcal{P}})$ and ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})$ contain the kernel of the epimorphism ${\mathrm{P}\Gamma }(S,{\mathcal{P}})\cong{\mathrm{P}\Gamma }(S\smallsetminus{\mathcal{P}})\to{\mathrm{P}\Gamma }(S)$ which does not commute with $\upsilon$ .

For $m\geq 2$ , there is a natural representation $\rho _{[m]}\colon \thinspace{\Upsilon }(S,{\mathcal{P}})\to \operatorname{Sp}(H_1(\overline{S},\mathbb{Z}/m))$ whose kernel we denote by ${\Upsilon }(S,{\mathcal{P}})^{[m]}$ and call the abelian level of order $m$ of ${\Upsilon }(S,{\mathcal{P}})$ . The pure hyperelliptic mapping class group has then the following characterization:

Proposition 2.5. For $S$ a closed surface of genus $\geq 2$ or $S=S_{1,1}$ , we have:

\begin{equation*}{\mathrm {P}\Upsilon }(S,{\mathcal {P}})={\Upsilon }(S,\vec {{\mathcal {P}}})^{[2]}.\end{equation*}

Proof. Proposition 3.3 and Remark 3.4 in [Reference Boggi6] imply that, for an unmarked hyperelliptic surface $(S,{\upsilon })$ as in the hypothesis, since both subgroups ${\mathrm{P}\Upsilon }(S)$ and ${\Upsilon }(S)^{[2]}$ of ${\Upsilon }(S)$ contain the hyperelliptic involution $\upsilon$ , there holds ${\mathrm{P}\Upsilon }(S)={\Upsilon }(S)^{[2]}$ .

For the marked case, let $\pi _{\vec{{\mathcal{P}}}}\colon \thinspace{\Upsilon }(S,\vec{{\mathcal{P}}})\to{\Upsilon }(S)$ be the natural epimorphism. Then, we have that ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})=\pi _{\vec{{\mathcal{P}}}}^{-1}({\mathrm{P}\Upsilon }(S))$ and ${\Upsilon }(S,\vec{{\mathcal{P}}})^{[2]}=\pi _{\vec{{\mathcal{P}}}}^{-1}({\Upsilon }(S)^{[2]})$ . Hence, the conclusion follows also in this case.

2.8. Hyperelliptic mapping class groups of disconnected surfaces

A disconnected hyperelliptic surface $(S,{\upsilon })$ is an oriented differentiable surface $S$ endowed with an involution $\upsilon$ such that all connected components of $S_{/{\upsilon }}\;:\!=\;S/\langle{\upsilon }\rangle$ have genus $0$ . We say that $S$ is hyperbolic if all its connected components are hyperbolic. The sets ${\mathscr{W}}_{\mathcal{P}}(S)$ , ${\mathscr{W}}_\circ (S)$ , and ${\mathscr{W}}_\partial (S)$ of Weierstrass points, punctures, and boundary components of $(S,{\upsilon })$ are defined as in Section 2.6.

The hyperelliptic mapping class group ${\Upsilon }(S)$ of the disconnected surface $(S,{\upsilon })$ is then defined to be the centralizer of $\upsilon$ in the mapping class group ${\Gamma }(S)$ (cf. Section 2.3).

Let us observe that the hyperelliptic involution $\upsilon$ acts on the set of connected components of $S$ and that, if a component is not fixed by this action, it has genus $0$ . As in Section 2.3, let $S=\coprod _{i=1}^h S^i$ be the decomposition in connected components and let $S^{i_1},\ldots, S^{i_k}$ be a set of representatives for their ${\Gamma }(S)$ -orbits. Let us further assume that ${\upsilon }(S^{i_j})=S^{i_j}$ , for $j=1,\ldots,t$ , and ${\upsilon }(S^{i_j})\neq S^{i_j}$ , for $j=t+1,\ldots k$ . From the isomorphism (1), it immediately follows that there is an isomorphism:

(7) \begin{equation} {\Upsilon }(S)\cong \prod _{j=1}^t\left ({\Upsilon }(S^{i_j})\wr \Sigma _{n_{i_j}}\right )\times \prod _{j=t+1}^k\left ({\Gamma }(S^{i_j})\wr \Sigma _{n_{i_j}/2}\right ). \end{equation}

For a hyperbolic disconnected hyperelliptic surface $(S,{\upsilon })$ , the hyperelliptic mapping class group ${\Upsilon }(S)$ then also fits in the natural short exact sequence:

\begin{equation*} 1\to \langle{\upsilon }\rangle \to{\Upsilon }(S)\to{\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })_{p_{\upsilon }}\to 1, \end{equation*}

where $p_{\upsilon }\colon \thinspace S\to S_{{\upsilon }}$ is the natural map, ${\mathcal{B}}_{\upsilon }$ its branch locus and the subgroup ${\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })_{p_{\upsilon }}$ of ${\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ is defined in the same way as in Section 2.6. It follows that there is also a natural surjective representation:

\begin{equation*}\rho _{{\mathscr{W}}}(S)\colon \thinspace {\Upsilon }(S)\to \Sigma _{{\mathscr{W}}_{\mathcal {P}}(S)}\times \Sigma _{{\mathscr{W}}_\circ (S)}\times \Sigma _{{\mathscr{W}}_\partial (S)}.\end{equation*}

The pure hyperelliptic mapping class group of a disconnected hyperbolic hyperelliptic surface $(S,{\upsilon })$ is then defined to be the intersection in the mapping class group ${\Gamma }(S)$ :

\begin{equation*}{\mathrm {P}\Upsilon }(S)\;:\!=\;\ker \rho _{{\mathscr{W}}}(S)\cap {\mathrm {P}\Gamma }(S).\end{equation*}

There is a natural short exact sequence:

\begin{equation*} 1\to \langle{\upsilon }\rangle \cap{\mathrm{P}\Upsilon }(S)\to{\mathrm{P}\Upsilon }(S)\to{\mathrm{P}\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\to 1. \end{equation*}

In order to obtain an explicit description of ${\mathrm{P}\Upsilon }(S)$ , let us put the decomposition of $S$ in connected components in the form:

(8) \begin{equation} S=\left (\coprod _{i=1}^l S^i\right )\coprod \left (\coprod _{i=l+1}^{r} S^i\right )\coprod \left (\coprod _{i=l+1}^r{\upsilon }(S^i)\right ), \end{equation}

where ${\upsilon }(S^i)=S^i$ if and only if $1\leq i\leq l$ and $r\;:\!=\;\frac{h+l}{2}$ . There is then an isomorphism:

(9) \begin{equation} {\mathrm{P}\Upsilon }(S)\cong \prod _{i=1}^l{\mathrm{P}\Upsilon }(S^{i})\times \prod _{i=l+1}^r{\mathrm{P}\Gamma }(S^{i}). \end{equation}

2.9. Hyperelliptic mapping class groups of marked disconnected surfaces

As in Section 2.6, for a marked disconnected hyperelliptic surface $(S,{\upsilon },{\mathcal{P}})$ , the hyperelliptic mapping class groups ${\Upsilon }(S,{\mathcal{P}})$ , ${\Upsilon }(S,\vec{\mathcal{P}})$ and ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})$ are defined to be the inverse images of ${\Upsilon }(S)$ and ${\mathrm{P}\Upsilon }(S)$ under the natural homomorphisms ${\Gamma }(S,{\mathcal{P}})\to{\Gamma }(S)$ , ${\Gamma }(S,\vec{\mathcal{P}})\to{\Gamma }(S)$ and ${\mathrm{P}\Gamma }(S,{\mathcal{P}})\to{\mathrm{P}\Gamma }(S)$ , respectively.

An explicit description of the hyperelliptic mapping class groups associated with a marked disconnected surface is, in general, complicated, but it is simple in the following case:

Proposition 2.6. Let us assume that all connected components of $S$ are fixed by the hyperelliptic involution $\upsilon$ . With the notations of Section 2.3, there is then an isomorphism:

(10) \begin{equation} {\Upsilon }(S,{\mathcal{P}})\cong \prod _{j=1}^k\left ({\Upsilon }(S^{i_j},{\mathcal{P}}^{i_j})\wr \Sigma _{n_{i_j}}\right ), \end{equation}

a similar one for ${\Upsilon }(S,\vec{\mathcal{P}})$ and an isomorphism:

(11) \begin{equation} {\mathrm{P}\Upsilon }(S,{\mathcal{P}})\cong \prod _{i=1}^h{\mathrm{P}\Upsilon }(S^{i},{\mathcal{P}}^i). \end{equation}

Proof. This is an easy consequence of the isomorphisms (7) and (9) and the Birman exact sequences (6).

2.10. Relative hyperelliptic mapping class groups

The relative hyperelliptic mapping class group ${\Upsilon }(S,{\partial S})$ of the hyperelliptic surface with boundary $(S,{\upsilon },{\partial S})$ is the centralizer of the hyperelliptic involution $\upsilon$ in the relative mapping class group ${\Gamma }(S,{\partial S})$ . For $(S,{\upsilon })$ hyperbolic, there is also a natural surjective representation:

\begin{equation*}\rho _{{\mathscr{W}}}(S)\colon \thinspace {\Upsilon }(S)\to \Sigma _{{\mathscr{W}}_{\mathcal {P}}(S)}\times \Sigma _{{\mathscr{W}}_\circ (S)}\times \Sigma _{{\mathscr{W}}_\partial (S)}.\end{equation*}

The relative pure hyperelliptic mapping class group of the hyperbolic hyperelliptic surface $(S,{\upsilon })$ is then the intersection:

\begin{equation*}{\mathrm {P}\Gamma }(S,{\partial S})\;:\!=\;\ker \rho _{{\mathscr{W}}}(S)\cap {\mathrm {P}\Gamma }(S,{\partial S}).\end{equation*}

Let $\delta _1,\ldots,\delta _{h}$ be the connected components of the boundary $\partial S$ which are fixed by $\upsilon$ . The remaining boundary components can then be divided into pairs $\delta _{h+1},{\upsilon }(\delta _{h+1}),\ldots \delta _k,{\upsilon }(\delta _k)$ . The groups ${\Upsilon }(S,{\partial S})$ and ${\mathrm{P}\Upsilon }(S,{\partial S})$ are described by the natural short exact sequences:

\begin{equation*}\begin {array}{ll} &1\to \prod _{i=1}^k\mathbb {Z}\to {\Upsilon }(S,{\partial S})\to {\Upsilon }(S)\to 1\\ \mbox {and}\;\;\;\;\;&\\ &1\to \prod _{i=1}^k\mathbb {Z}\to {\mathrm {P}\Upsilon }(S,{\partial S})\to {\mathrm {P}\Upsilon }(S)\to 1, \end {array}\end{equation*}

where the left hand maps send the $i$ -th standard basis vector of $\prod _{i=1}^k\mathbb{Z}$ to the Dehn twist $\tau _{\delta _i}$ , for $1\leq i\leq h$ , and to the product of Dehn twists $\tau _{\delta _i}\cdot \tau _{{\upsilon }(\delta _i)}$ , for $h+1\leq i\leq k$ .

The relative hyperelliptic mapping class group ${\Upsilon }(S,{\mathcal{P}},{\partial S})$ and relative pure hyperelliptic mapping class group ${\mathrm{P}\Upsilon }(S,{\mathcal{P}},{\partial S})$ of a marked hyperelliptic surface $(S,{\upsilon },{\mathcal{P}})$ are defined to be the inverse images of ${\Upsilon }(S,{\partial S})$ and ${\mathrm{P}\Upsilon }(S,{\partial S})$ , respectively, via the natural epimorphisms ${\Gamma }(S,{\mathcal{P}},{\partial S})\to{\Gamma }(S,{\partial S})$ and ${\mathrm{P}\Gamma }(S,{\mathcal{P}},{\partial S})\to{\mathrm{P}\Gamma }(S,{\partial S})$ .

2.11. The complex of symmetric curves on $(S,{\upsilon })$

For a hyperbolic surface $S$ , the complex of curves $C(S)$ is defined to be the (abstract) simplicial complex whose $k$ -simplices consist of unordered $(k+1)$ -tuples $\{[{\gamma }_0],\ldots,[{\gamma }_k]\}$ of distinct isotopy classes of pairwise disjoint nonperipheral simple closed curves on $S$ . For a hyperelliptic surface $(S,{\upsilon })$ , we then define the complex of symmetric curves $C(S,{\upsilon })$ to be the subcomplex of $C(S)$ consisting of simplices fixed by the hyperelliptic involution $\upsilon$ . More intrinsically and more generally, this can be defined as follows:

There is a natural action of the mapping class group ${\Gamma }(S,{\mathcal{P}})$ of a marked hyperbolic surface $(S,{\mathcal{P}})$ on the curve complex $C(S\smallsetminus{\mathcal{P}})$ . The stabilizer ${\Gamma }(S,{\mathcal{P}})_{\sigma }$ of the simplex $\sigma$ is described by the exact sequence:

\begin{equation*} 1\to \prod _{{\gamma }\in{\sigma }}\tau _{\gamma }^{\mathbb{Z}}\to{\Gamma }(S,{\mathcal{P}})_{\sigma }\to{\Gamma }(S\smallsetminus{\sigma },{\mathcal{P}}), \end{equation*}

where the mapping class group ${\Gamma }(S\smallsetminus{\sigma },{\mathcal{P}})$ of the marked (possibly) disconnected surface $(S\smallsetminus{\sigma },{\mathcal{P}})$ is described by the isomorphism (1) and the image of ${\Gamma }(S,{\mathcal{P}})_{\sigma }$ in this group has finite index. In fact, it consists of those elements of ${\Gamma }(S\smallsetminus{\sigma },{\mathcal{P}})$ which preserve the gluing data on $S\smallsetminus{\sigma }$ induced by its embedding in $S$ as encoded in the dual graph associated with $\sigma$ (the finite connected graph obtained collapsing to a point each connected component of the complement of a tubular neighborhood of $\sigma$ in $S$ and then collapsing to a segment each connected component of the tubular neighborhood).

Let us denote by $\vec{\sigma }$ the simplex $\sigma$ endowed with an order and an orientation on each of its elements. The stabilizer ${\mathrm{P}\Gamma }(S,{\mathcal{P}})_{\sigma }$ of a simplex ${\sigma }\in C(S\smallsetminus{\mathcal{P}})$ for the action of the pure mapping class group ${\mathrm{P}\Gamma }(S,{\mathcal{P}})$ is then described by the short exact sequences:

(12) \begin{equation} \begin{array}{ll} &1\to{\mathrm{P}\Gamma }(S,{\mathcal{P}})_{\vec{\sigma }}\to{\mathrm{P}\Gamma }(S,{\mathcal{P}})_{\sigma }\to \Sigma _{{\sigma }^\pm }\\ \mbox{and}\;\;\;\;\;&\\ &1\to \prod _{{\gamma }\in{\sigma }}\tau _{\gamma }^{\mathbb{Z}}\to{\mathrm{P}\Gamma }(S,{\mathcal{P}})_{\vec{\sigma }}\to{\mathrm{P}\Gamma }(S\smallsetminus{\sigma },{\mathcal{P}})\to 1, \end{array} \end{equation}

where $\Sigma _{{\sigma }^\pm }$ is the symmetric group on the set of oriented simple closed curves ${\sigma }^\pm \;:\!=\;{\sigma }^+\cup{\sigma }^-$ and the group ${\mathrm{P}\Gamma }(S\smallsetminus{\sigma },{\mathcal{P}})$ is described by the isomorphism (2).

From these descriptions, we deduce a similar description for the stabilizers of the action of the hyperelliptic mapping class groups ${\Upsilon }(S,{\mathcal{P}})$ and ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})$ on the complex of symmetric curves $C(S,{\upsilon },{\mathcal{P}})$ of a marked hyperelliptic surface $(S,{\upsilon },{\mathcal{P}})$ .

Let us observe that, for ${\sigma }\in C(S,{\upsilon },{\mathcal{P}})$ , we can choose a set of disjoint representatives on $S\smallsetminus{\mathcal{P}}$ , which we also denote by $\sigma$ , such that there holds ${\upsilon }({\sigma })={\sigma }$ , so that $(S\smallsetminus{\sigma },{\upsilon })$ is a, possibly disconnected, hyperelliptic surface. The stabilizer ${\Upsilon }(S,{\mathcal{P}})_{\sigma }$ of a simplex ${\sigma }\in C(S,{\upsilon },{\mathcal{P}})$ is then described by the exact sequence:

(13) \begin{equation} 1\to \prod _{{\gamma }\in{\sigma }}\tau _{\gamma }^{\mathbb{Z}}\to{\Upsilon }(S,{\mathcal{P}})_{\sigma }\to{\Upsilon }(S\smallsetminus{\sigma },{\mathcal{P}}), \end{equation}

where the hyperelliptic mapping class group ${\Upsilon }(S\smallsetminus{\sigma },{\mathcal{P}})$ of the marked, possibly disconnected, hyperelliptic surface $(S\smallsetminus{\sigma },{\upsilon },{\mathcal{P}})$ is described in Sections 2.8 and 2.9 and the image of ${\Upsilon }(S,{\mathcal{P}})_{\sigma }$ in ${\Upsilon }(S\smallsetminus{\sigma },{\mathcal{P}})$ has finite index.

If ${\sigma }=\{{\gamma }\}$ is a $0$ -simplex, let us denote by $Q_+,Q_-$ the pair of punctures on $S\smallsetminus{\gamma }$ bounded by $\gamma$ and by ${\Upsilon }(S\smallsetminus{\gamma },{\mathcal{P}})_{\{Q_+,Q_-\}}$ the stabilizer of this pair of punctures for the action of the hyperelliptic mapping class group ${\Upsilon }(S\smallsetminus{\gamma },{\mathcal{P}})$ on the set of punctures of $S\smallsetminus{\gamma }$ . Then, there is a short exact sequence:

(14) \begin{equation} 1\to \tau _{\gamma }^{\mathbb{Z}}\to{\Upsilon }(S,{\mathcal{P}})_{\sigma }\to{\Upsilon }(S\smallsetminus{\gamma },{\mathcal{P}})_{\{Q_+,Q_-\}}\to 1. \end{equation}

Let ${\sigma }_{ns}$ and ${\sigma }_{s}$ be the subsets of $\sigma$ consisting, respectively, of nonseparating and of separating curves. Then, the stabilizer ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\sigma }$ is described by the exact sequences:

(15) \begin{equation} \begin{array}{ll} &1\to{\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\vec{\sigma }}\to{\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\sigma }\to \Sigma _{{\sigma }^\pm }\\ \mbox{and}\;\;\;\;\;&\\ &1\to \prod _{{\gamma }\in{\sigma }_{ns}}\tau _{\gamma }^{2\mathbb{Z}}\times \prod _{{\gamma }\in{\sigma }_{s}}\tau _{\gamma }^{\mathbb{Z}}\to{\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\vec{\sigma }}\to{\mathrm{P}\Upsilon }(S\smallsetminus{\sigma },{\mathcal{P}})\to 1, \end{array} \end{equation}

where the pure hyperelliptic mapping class group ${\mathrm{P}\Upsilon }(S\smallsetminus{\sigma },{\mathcal{P}})$ of the marked, possibly disconnected, hyperelliptic surface $(S\smallsetminus{\sigma },{\upsilon },{\mathcal{P}})$ above is described in Sections 2.8 and 2.9.

3.1. Moduli stacks of complex curves of a fixed topological type

Let $S=S_{g,n}$ be a hyperbolic surface without boundary. We can then associate with $S$ the moduli stack ${\mathcal{M}}(S)$ of $n$ -punctured, genus $g$ smooth curves. The stack ${\mathcal{M}}(S)$ represents the functor of smooth curves ${\mathscr{C}}\to X$ such that each geometric fiber of the family has a complex model homeomorphic to the surface $S$ . This is a smooth, irreducible, Deligne-Mumford (briefly DM) stack over $\operatorname{Spec}\mathbb{Z}$ . Fixing an order on the set of punctures of the geometric fibers of the curves ${\mathscr{C}}\to X$ defines an étale Galois covering of DM stacks $\mathrm{P}{\mathcal{M}}(S)\to{\mathcal{M}}(S)$ .

The moduli stack ${\mathcal{M}}(S)$ admits a natural compactification ${\overline{\mathcal{M}}}(S)$ , called the DM compactification. This is the stack which represents $n$ -punctured, genus $g$ stable curves, that is to say, flat nodal curves ${\mathscr{C}}\to X$ such that each geometric fiber of the family has a complex model homeomorphic to the, possibly singular, topological surface obtained from $S$ collapsing to points the connected components of a, possibly empty, essential multicurve.

The stack ${\overline{\mathcal{M}}}(S)$ is a smooth, proper DM stack also defined over $\operatorname{Spec}\mathbb{Z}$ and, fixing an order on the set of punctures of the fibers of the relative curves ${\mathscr{C}}\to X$ , defines an étale Galois covering of DM stacks $\mathrm{P}{\overline{\mathcal{M}}}(S)\to{\overline{\mathcal{M}}}(S)$ .

In this paper, we are essentially interested in the complex theory of these objects. Therefore, without further specification, from now on, we will denote by ${\mathcal{M}}(S)$ , $\mathrm{P}{\mathcal{M}}(S)$ , ${\overline{\mathcal{M}}}(S)$ and $\mathrm{P}{\overline{\mathcal{M}}}(S)$ the corresponding complex DM stacks.

The mapping class groups of the surface $S$ which we introduced in Section 2.2 have a natural interpretation as the topological fundamental groups of the above complex DM stacks. Indeed, a marking $S\to C$ of an $n$ -punctured, genus $g$ complex smooth curve determines isomorphisms ${\Gamma }(S)\cong \pi _1({\mathcal{M}}(S),[C])$ and ${\mathrm{P}\Gamma }(S)\cong \pi _1(\mathrm{P}{\mathcal{M}}(S),[C])$ .

3.2. Moduli stacks of marked complex curves

For a marked hyperbolic surface $(S,{\mathcal{P}})$ , let ${\mathcal{M}}(S,{\mathcal{P}})$ be the moduli stack which represents the functor of smooth curves ${\mathscr{C}}\to X$ , endowed with sections $s_Q\colon \thinspace X\to{\mathscr{C}}$ , labeled by the points $Q\in{\mathcal{P}}$ , such that, for every geometric point $x\in X$ , the fiber $({\mathscr{C}}_x,\{s_Q(x)\}_{Q\in{\mathcal{P}}})$ has a complex model homeomorphic to the marked surface $(S,{\mathcal{P}})$ . The moduli stack ${\mathcal{M}}(S,\vec{\mathcal{P}})$ is defined by fixing an order on $\mathcal{P}$ (and then on the set of sections). These are smooth, irreducible, DM stacks over $\operatorname{Spec}\mathbb{Z}$ . There is a natural smooth morphism ${\mathcal{M}}(S,{\mathcal{P}})\to{\mathcal{M}}(S)$ (resp. ${\mathcal{M}}(S,\vec{\mathcal{P}})\to{\mathcal{M}}(S)$ ) whose fiber ${\mathcal{M}}(S,{\mathcal{P}})_{[C]}$ (resp. ${\mathcal{M}}(S,\vec{\mathcal{P}})_{[C]}$ ), for a geometric point $[C]\in{\mathcal{M}}(S)$ , identifies with the configuration spaces of $n$ distinct (resp. distinct and ordered) points on $C$ .

Let then $\mathrm{P}{\mathcal{M}}(S,{\mathcal{P}})$ be the fibered product of $\mathrm{P}{\mathcal{M}}(S)$ and ${\mathcal{M}}(S,\vec{\mathcal{P}})$ over ${\mathcal{M}}(S)$ . There is a natural isomorphism of DM stacks $\mathrm{P}{\mathcal{M}}(S,{\mathcal{P}})\cong \mathrm{P}{\mathcal{M}}(S\smallsetminus{\mathcal{P}})$ .

The DM compactification ${\overline{\mathcal{M}}}(S,{\mathcal{P}})$ is the moduli stack of flat nodal curves ${\mathscr{C}}\to X$ , endowed with disjoint sections $s_Q\colon \thinspace X\to{\mathscr{C}}$ , for $Q\in{\mathcal{P}}$ , such that, for every geometric point $x\in X$ , the fiber ${\mathscr{C}}_x\smallsetminus \{s_Q(x)\}_{Q\in{\mathcal{P}}}$ has a complex model homeomorphic to the surface obtained from $S\smallsetminus{\mathcal{P}}$ collapsing to points the connected components of a, possibly empty, essential multicurve. The DM stacks ${\overline{\mathcal{M}}}(S,\vec{\mathcal{P}})$ and $\mathrm{P}{\overline{\mathcal{M}}}(S,{\mathcal{P}})$ are then defined similarly.

As above, we are interested in the complex theory of these objects and we will maintain the same notation for the corresponding complex DM stacks. A marking $\phi \colon \thinspace S\to C$ then determines the isomorphisms:

\begin{equation*}\begin {array}{ll} &{\Gamma }(S,{\mathcal {P}})\cong \pi _1({\mathcal {M}}(S,{\mathcal {P}}),[C,\phi ({\mathcal {P}})]),\;\;\;\;\;\;\;\;\;\;\;{\Gamma }(S,\vec {\mathcal {P}})\cong \pi _1({\mathcal {M}}(S,\vec {\mathcal {P}}),[C,\phi ({\mathcal {P}})])\\ \mbox {and}\;\;\;\;\;&\\ &{\mathrm {P}\Gamma }(S,{\mathcal {P}})\cong \pi _1(\mathrm {P}{\mathcal {M}}(S,{\mathcal {P}}),[C,\phi ({\mathcal {P}})]). \end {array}\end{equation*}

3.3. Moduli stacks of hyperelliptic curves of a fixed topological type

Let $(S,{\upsilon })$ be a hyperbolic hyperelliptic surface without boundary. Then, a hyperelliptic complex curve $(C,\iota )$ of topological type $(S,{\upsilon })$ is a smooth complex curve $C$ together with a hyperelliptic involution $\iota \in \operatorname{Aut}(C)$ such that there exists an orientation preserving homeomorphism $f\colon \thinspace C\to S$ with the property that $f\circ \iota \circ f^{-1}={\upsilon }$ .

Hyperelliptic complex curves of topological type $(S,{\upsilon })$ are parameterized by a closed smooth irreducible substack ${\mathcal{M}}(S,{\upsilon })$ of the DM stack ${\mathcal{M}}(S)$ and an orientation preserving homeomorphism $f\colon \thinspace C\to S$ determines an isomorphism (cf. Proposition 1 in [Reference González-Díez, Harvey, Harvey and Maclachlan19]):

\begin{equation*}{\Upsilon }(S)\cong \pi _1({\mathcal {M}}(S,{\upsilon }),[C,\iota ]).\end{equation*}

We let $\mathrm{P}{\mathcal{M}}(S,{\upsilon })\to{\mathcal{M}}(S,{\upsilon })$ be the étale Galois covering associated with the normal finite index subgroup ${\mathrm{P}\Upsilon }(S)$ of ${\Upsilon }(S)$ . In particular, by definition, there is an isomorphism:

\begin{equation*}{\mathrm {P}\Upsilon }(S)\cong \pi _1(\mathrm {P}{\mathcal {M}}(S,{\upsilon }),[C,\iota ]).\end{equation*}

The assignment $(C,\iota )\mapsto (C/\iota,{\mathcal{B}}_\iota )$ , where ${\mathcal{B}}_\iota$ is the branch locus of the natural morphism $C\to C/\iota$ , defines natural morphisms of DM stacks:

(16) \begin{equation} {\mathcal{M}}(S,{\upsilon })\to{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\;\;\;\;\;\;\;\;\;\;\;\mbox{ and }\;\;\;\;\;\;\;\;\;\;\;\mathrm{P}{\mathcal{M}}(S,{\upsilon })\to \mathrm{P}{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon }), \end{equation}

where, as we observed above, there holds $\mathrm{P}{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\cong \mathrm{P}{\mathcal{M}}(S_{/{\upsilon }}\smallsetminus{\mathcal{B}}_{\upsilon })$ . These morphisms are, respectively, a $\langle{\upsilon }\rangle$ -gerbe and a $(\langle{\upsilon }\rangle \cap{\mathrm{P}\Upsilon }(S))$ -gerbe over their base and induce on fundamental groups the maps ${\Upsilon }(S)\to{\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ and ${\mathrm{P}\Upsilon }(S)\to{\mathrm{P}\Gamma }(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ which appear in the short exact sequences (3) and (5), respectively. In particular, if $S$ has at least two symmetric punctures, there are natural isomorphisms:

\begin{equation*}\mathrm {P}{\mathcal {M}}(S,{\upsilon })\cong \mathrm {P}{\mathcal {M}}(S_{/{\upsilon }},{\mathcal {B}}_{\upsilon })\cong \mathrm {P}{\mathcal {M}}(S_{/{\upsilon }}\smallsetminus {\mathcal {B}}_{\upsilon }).\end{equation*}

The DM compactification ${\overline{\mathcal{M}}}(S,{\upsilon })$ is the closure of ${\mathcal{M}}(S,{\upsilon })$ in the proper DM stack ${\overline{\mathcal{M}}}(S)$ . The DM compactification $\mathrm{P}{\overline{\mathcal{M}}}(S,{\upsilon })$ is then the normalization of the DM stack ${\overline{\mathcal{M}}}(S,{\upsilon })$ in $\mathrm{P}{\mathcal{M}}(S,{\upsilon })$ .

3.4. Moduli stacks of marked hyperelliptic curves

To a marked hyperelliptic surface $(S,{\upsilon },{\mathcal{P}})$ , we can associate the DM stacks ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ , ${\mathcal{M}}(S,{\upsilon },\vec{\mathcal{P}})$ and $\mathrm{P}{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ . These are just the inverse images of ${\mathcal{M}}(S,{\upsilon })$ and $\mathrm{P}{\mathcal{M}}(S,{\upsilon })$ via the natural morphisms ${\mathcal{M}}(S,{\mathcal{P}})\to{\mathcal{M}}(S)$ , ${\mathcal{M}}(S,\vec{\mathcal{P}})\to{\mathcal{M}}(S)$ and $\mathrm{P}{\mathcal{M}}(S,{\mathcal{P}})\to \mathrm{P}{\mathcal{M}}(S)$ , respectively.

As above, a marking $\phi \colon \thinspace (S,{\upsilon })\to (C,\iota )$ determines isomorphisms:

\begin{equation*}\begin {array}{ll} &{\Upsilon }(S,{\mathcal {P}})\cong \pi _1({\mathcal {M}}(S,{\upsilon },{\mathcal {P}}),[C,\iota,\phi ({\mathcal {P}})]),\;\;\;\;\;\;{\Upsilon }(S,\vec {\mathcal {P}})\cong \pi _1({\mathcal {M}}(S,{\upsilon },\vec {\mathcal {P}}),[C,\iota,\phi ({\mathcal {P}})])\\ \mbox {and}\;\;\;\;\;&\\ &{\mathrm {P}\Upsilon }(S,{\mathcal {P}})\cong \pi _1(\mathrm {P}{\mathcal {M}}(S,{\upsilon },{\mathcal {P}}),[C,\iota,\phi ({\mathcal {P}})]). \end {array}\end{equation*}

Let $Q\in S\smallsetminus{\mathcal{P}}$ . The Birman short exact sequences (6) are then induced by the following smooth families of curves (with fibers homeomorphic to $S\smallsetminus{\mathcal{P}}$ ):

\begin{equation*}\begin {array}{ll} &\mathrm {P}{\mathcal {M}}(S,{\upsilon },{\mathcal {P}}\cup \{Q\})\to \mathrm {P}{\mathcal {M}}(S,{\upsilon },{\mathcal {P}}),\;\;\;\;\;\;{\mathcal {M}}(S,{\upsilon },\overrightarrow {{\mathcal {P}}\cup \{Q\}})\to {\mathcal {M}}(S,{\upsilon },\vec {\mathcal {P}})\\ \mbox {and}\;\;\;\;\;&\\ &{\mathcal {M}}(S,{\upsilon },{\mathcal {P}},Q)\to {\mathcal {M}}(S,{\upsilon },{\mathcal {P}}). \end {array}\end{equation*}

where ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)\to{\mathcal{M}}(S,{\upsilon },{\mathcal{P}}\cup \{Q\})$ is the étale covering associated with the subgroup ${\Upsilon }(S,{\mathcal{P}},Q)$ of ${\Upsilon }(S,{\mathcal{P}}\cup \{Q\})$ or, equivalently, ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)$ parameterizes curves marked by the set $\mathcal{P}$ plus a distinguished marked point $Q$ .

3.5. The DM boundary of moduli stacks of stable curves

A classical result by Knudsen (cf. [Reference Knudsen25]) describes the DM boundary of the moduli stack $\mathrm{P}{\overline{\mathcal{M}}}(S,{\mathcal{P}})$ associated with a hyperbolic marked surface $(S,{\mathcal{P}})$ . From our perspective and with our notation, we can formulate this result as follows.

Let us denote by $S/{\sigma }$ the singular topological surface obtained collapsing on $S$ a multicurve, supported on $S\smallsetminus{\mathcal{P}}$ , in the isotopy class of $\sigma$ . We associate with a simplex ${\sigma }\in C(S\smallsetminus{\mathcal{P}})$ the closed boundary stratum ${\delta }_{\sigma }$ of $\mathrm{P}{\overline{\mathcal{M}}}(S,{\mathcal{P}})$ whose generic point parameterizes marked stable complex curves of topological type $(S/{\sigma },{\mathcal{P}})$ . We then let $\dot{{\delta }}_{\sigma }$ be the open substack of ${\delta }_{\sigma }$ whose points parameterize stable complex curves of topological type $(S/{\sigma },{\mathcal{P}})$ .

Let $S\smallsetminus{\sigma }=\coprod _{i=1}^h S^i$ be the decomposition in connected components and let ${\mathcal{P}}^i\;:\!=\;{\mathcal{P}}\cap S^i$ , for $i=1,\ldots,h$ . There is then a natural finite unramified morphism of DM stacks:

\begin{equation*}b_{\sigma }\colon \thinspace \prod _{i=1}^h\mathrm {P}{\overline {\mathcal {M}}}(S^i,{\mathcal {P}}^i)\to \mathrm {P}{\overline {\mathcal {M}}}(S,{\mathcal {P}}),\end{equation*}

with image the closed boundary stratum ${\delta }_{\sigma }$ and whose restriction to $\prod _{i=1}^h\mathrm{P}{\mathcal{M}}(S^i,{\mathcal{P}}^i)$ is étale onto $\dot{{\delta }}_{\sigma }$ . Moreover, if we denote by $\tilde{{\delta }}_{\sigma }$ the normalization of ${\delta }_{\sigma }$ , the morphism $b_s$ factors through a Galois finite étale covering:

\begin{equation*}\tilde {b}_{\sigma }\colon \thinspace \prod _{i=1}^h\mathrm {P}{\overline {\mathcal {M}}}(S^i,{\mathcal {P}}^i)\to \tilde {{\delta }}_{\sigma },\end{equation*}

whose Galois group identifies with the image of the stabilizer ${\mathrm{P}\Gamma }(S,{\mathcal{P}})_{\sigma }$ in the symmetric group $\Sigma _{{\sigma }^\pm }$ (cf. the upper exact sequence (12)).

3.6. The DM boundary of moduli stacks of hyperelliptic stable curves

A similar result holds for moduli stacks of hyperelliptic curves. As we observed in Section 2.11, for ${\sigma }\in C(S,{\upsilon },{\mathcal{P}})$ , there is a set of disjoint representatives on $S\smallsetminus{\mathcal{P}}$ , which we also denote by $\sigma$ , such that there holds ${\upsilon }({\sigma })={\sigma }$ and $(S\smallsetminus{\sigma },u)$ is a, possibly disconnected, hyperelliptic topological surface. We then associate with $\sigma$ the closed boundary stratum ${\delta }_{\sigma }^{\upsilon }$ (resp. the open stratum $\dot{{\delta }}_{\sigma }^{\upsilon }$ ) of $\mathrm{P}{\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}})$ whose generic point (resp. points) parameterizes marked stable complex hyperelliptic curves of topological type $(S/{\sigma },{\upsilon },{\mathcal{P}})$ .

For ${\mathcal{P}}=\emptyset$ , we can describe explicitly the strata ${\delta }_{\sigma }^{\upsilon }$ and $\dot{{\delta }}_{\sigma }^{\upsilon }$ in the following way. As in Section 2.8, let us put the decomposition of $S$ in connected components in the form (8) and let us denote by ${\upsilon }_i$ the restriction of the hyperelliptic involution $\upsilon$ to $S^i$ for $i=1,\ldots,l$ . There is then a natural finite unramified morphism of DM stacks:

(17) \begin{equation} b_{\sigma }^{\upsilon }\colon \thinspace \prod _{i=1}^l\mathrm{P}{\overline{\mathcal{M}}}(S^i,{\upsilon }_i)\times \prod _{i=l+1}^r\mathrm{P}{\overline{\mathcal{M}}}(S^i)\to \mathrm{P}{\overline{\mathcal{M}}}(S,{\upsilon }), \end{equation}

with image ${\delta }_{\sigma }^{\upsilon }$ and whose restriction to $\prod _{i=1}^l\mathrm{P}{\mathcal{M}}(S^i,{\upsilon }_i)\times \prod _{i=l+1}^r\mathrm{P}{\mathcal{M}}(S^i)$ is étale onto $\dot{{\delta }}_{\sigma }^{\upsilon }$ . Moreover, if we let $\tilde{{\delta }}_{\sigma }^{\upsilon }$ be the normalization of ${\delta }_{\sigma }^{\upsilon }$ , the morphism $b_s^{\upsilon }$ factors through a finite étale Galois covering:

\begin{equation*}\tilde {b}_{\sigma }^{\upsilon }\colon \thinspace \prod _{i=1}^l\mathrm {P}{\overline {\mathcal {M}}}(S^i,{\upsilon }_i)\times \prod _{i=l+1}^r\mathrm {P}{\overline {\mathcal {M}}}(S^i)\to \tilde {{\delta }}_{\sigma }^{\upsilon },\end{equation*}

whose Galois group identifies with the image of the stabilizer ${\mathrm{P}\Upsilon }(S)_{\sigma }$ in the symmetric group $\Sigma _{{\sigma }^\pm }$ (cf. the upper exact sequence (15)).

For ${\mathcal{P}}\neq \emptyset$ , we can describe explicitly these strata only under the further assumption that $\upsilon$ preserves all connected components of $S\smallsetminus{\sigma }$ . With the notations of Section 3.5, let us also denote by ${\upsilon }_i$ the restriction of the hyperelliptic involution $\upsilon$ to $S^i$ for $i=1,\ldots,h$ . There is then a natural finite unramified morphism of DM stacks:

\begin{equation*}b_{\sigma }^{\upsilon }\colon \thinspace \prod _{i=1}^h\mathrm {P}{\overline {\mathcal {M}}}(S^i,{\upsilon }_i,{\mathcal {P}}^i)\to \mathrm {P}{\overline {\mathcal {M}}}(S,{\upsilon },{\mathcal {P}}),\end{equation*}

with image the closed boundary stratum ${\delta }_{\sigma }^{\upsilon }$ and whose restriction to $\prod _{i=1}^h\mathrm{P}{\mathcal{M}}(S^i,{\upsilon }_i,{\mathcal{P}}^i)$ is étale onto $\dot{{\delta }}_{\sigma }^{\upsilon }$ . As above, if we denote by $\tilde{{\delta }}_{\sigma }^{\upsilon }$ the normalization of ${\delta }_{\sigma }^{\upsilon }$ , the morphism $b_s^{\upsilon }$ factors through a Galois finite étale covering:

\begin{equation*}\tilde {b}_{\sigma }^{\upsilon }\colon \thinspace \prod _{i=1}^h\mathrm {P}{\overline {\mathcal {M}}}(S^i,{\upsilon }_i,{\mathcal {P}}^i)\to \tilde {{\delta }}_{\sigma },\end{equation*}

whose Galois group identifies with the image of the stabilizer ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\sigma }$ in the symmetric group $\Sigma _{{\sigma }^\pm }$ (cf. the exact sequence (15)).

3.7. Levels and level structures

A level ${\Gamma }^{\lambda }$ is a normal finite index subgroup of the mapping class group ${\Gamma }(S,{\mathcal{P}})$ . The level structure ${\mathcal{M}}(S,{\mathcal{P}})^{\lambda }$ is the finite étale Galois covering of the moduli stack ${\mathcal{M}}(S,{\mathcal{P}})$ associated with this subgroup.

The abelian level ${\Gamma }(S,{\mathcal{P}})^{[m]}$ (or simply ${\Gamma }^{[m]}$ ) of order $m$ , for an integer $m\geq 2$ , is the kernel of the natural representation $\rho ^{[m]}\colon \thinspace{\Gamma }(S,{\mathcal{P}})\to \operatorname{Sp}(H_1(\overline{S},\mathbb{Z}/m)$ . We denote by ${\mathcal{M}}(S,{\mathcal{P}})^{[m]}$ the associated abelian level structure.

For ${\Gamma }^{\lambda }$ a level of ${\Gamma }(S,{\mathcal{P}})$ , the DM compactification ${\overline{\mathcal{M}}}(S,{\mathcal{P}})^{\lambda }$ of the level structure ${\mathcal{M}}(S,{\mathcal{P}})^{\lambda }$ is defined to be the normalization of the moduli stack ${\overline{\mathcal{M}}}(S,{\mathcal{P}})$ in ${\mathcal{M}}(S,{\mathcal{P}})^{\lambda }$ .

Similarly, a level ${\Upsilon }^{\lambda }$ of the hyperelliptic mapping class group is a normal finite index subgroup of ${\Upsilon }(S,{\mathcal{P}})$ and the associated finite étale Galois covering of the moduli stack ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ is denoted by ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{\lambda }$ and also called a level structure.

As above, to an integer $m\geq 2$ , we associate the abelian level of order $m$ defined by ${\Upsilon }(S,{\mathcal{P}})^{[m]}\;:\!=\;{\Upsilon }(S,{\mathcal{P}})\cap{\Gamma }(S,{\mathcal{P}})^{[m]}$ and the abelian level structure ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{[m]}$ . We also let ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})^{[m]}\;:\!=\;{\mathrm{P}\Upsilon }(S,{\mathcal{P}})\cap{\Gamma }(S,{\mathcal{P}})^{[m]}$ and ${\mathrm{P}\Upsilon }(S,\vec{{\mathcal{P}}})^{[m]}\;:\!=\;{\mathrm{P}\Upsilon }(S,\vec{{\mathcal{P}}})\cap{\Gamma }(S,{\mathcal{P}})^{[m]}$ .

For ${\Upsilon }^{\lambda }$ a level of ${\Upsilon }(S,{\mathcal{P}})$ , the DM compactification ${\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}})^{\lambda }$ of the level structure ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{\lambda }$ is defined to be the normalization of the moduli stack ${\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}})$ in ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{\lambda }$ .

3.8. The hyperelliptic Birman exact sequence

Besides the Birman short exact sequences (6), which are a consequence of the standard one for the mapping class group, there are short exact sequences which are peculiar to the hyperelliptic mapping class group.

For a hyperbolic hyperelliptic surface without boundary $(S,{\upsilon })$ , let $Q\in S\smallsetminus{\mathscr{W}}_{\mathcal{P}}(S)$ and $R\;:\!=\;{\upsilon }(Q)$ . Let then $S^\circ \;:\!=\;S\smallsetminus \{Q,R\}$ and label by $Q$ (resp. $R$ ) the puncture obtained by removing $Q$ (resp. $R$ ) from $S$ . For a marking $(S^\circ,{\mathcal{P}})$ , let ${\Upsilon }(S^\circ,{\mathcal{P}})_\circ$ and ${\Upsilon }(S^\circ,\vec{\mathcal{P}})_\circ$ be the index $2$ subgroups of ${\Upsilon }(S^\circ,{\mathcal{P}})$ and ${\Upsilon }(S^\circ,\vec{\mathcal{P}})$ , respectively, consisting of those elements which do not swap the two punctures labeled by $Q$ and $R$ . We then have:

Theorem 3.1. With the above notations, there are natural short exact sequences:

(18) \begin{equation} \begin{array}{ll} &1\to N_\circ \to{\Upsilon }(S^\circ,{\mathcal{P}})_\circ \to{\Upsilon }(S,{\mathcal{P}},Q)\to 1,\\ \\ &1\to N_\circ \to{\Upsilon }(S^\circ,\vec{\mathcal{P}})_\circ \to{\Upsilon }(S,\overrightarrow{{\mathcal{P}}\cup \{Q\}})\to 1\\ \mbox{and}\;\;\;\;\;&\\ &1\to N_\circ \to{\mathrm{P}\Upsilon }(S^\circ,{\mathcal{P}})\to{\mathrm{P}\Upsilon }(S,{\mathcal{P}}\cup \{Q\})\to 1, \end{array} \end{equation}

where $N_\circ$ is the normal subgroup generated by the Dehn twists about symmetric separating simple closed curves on $S^\circ$ bounding a disc which contains no marked points and only the two punctures labeled by $Q$ and $R$ .

Proof. Let ${\mathcal{M}}(S^\circ,{\upsilon },{\mathcal{P}})_\circ \to{\mathcal{M}}(S^\circ,{\upsilon },{\mathcal{P}})$ be the étale covering associated with the subgroup ${\Upsilon }(S^\circ,{\mathcal{P}})_\circ$ of ${\Upsilon }(S^\circ,{\mathcal{P}})$ , so that ${\mathcal{M}}(S^\circ,{\upsilon },{\mathcal{P}})_\circ$ parameterizes marked complex hyperelliptic curves with an order on the set of punctures labeled by $Q$ and $R$ .

Let ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\mathscr{W}}$ be the open substack of ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)$ which parameterizes marked complex hyperelliptic curves $[C,\iota,{\mathcal{P}},Q]$ such that the distinguished point labeled by $Q$ is not a Weierstrass point of the curve $C$ .

Note that the complement $\partial _{\mathscr{W}}\;:\!=\;{\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)\smallsetminus{\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\mathscr{W}}$ is a smooth irreducible divisor of ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)$ and that the assignment $[C,\iota,{\mathcal{P}}, Q]\mapsto [C\smallsetminus \{Q,\iota (Q)\},\iota,{\mathcal{P}}]$ defines a natural isomorphism of moduli stacks:

\begin{equation*}{\mathcal {M}}(S,{\upsilon },{\mathcal {P}},Q)^{\mathscr{W}}\stackrel {\sim }{\to }{\mathcal {M}}(S^\circ,{\upsilon },{\mathcal {P}})_\circ .\end{equation*}

Let ${\mathcal{M}}(S,{\mathcal{P}},Q,R)$ be the étale covering of ${\mathcal{M}}(S,{\mathcal{P}}\cup \{Q,R\})$ defined by taking $Q$ and $R$ as distinguished and ordered marked points. The assignment $[C,\iota,{\mathcal{P}}, Q]\mapsto [C,{\mathcal{P}},Q,\iota (Q)]$ then defines a natural embedding ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\mathscr{W}}\hookrightarrow{\mathcal{M}}(S,{\mathcal{P}},Q,R)$ . This extends to an embedding ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)\hookrightarrow{\overline{\mathcal{M}}}(S,{\mathcal{P}},Q,R)$ which maps the divisor $\partial _{\mathscr{W}}$ to the irreducible component of the DM boundary of ${\overline{\mathcal{M}}}(S,{\mathcal{P}},Q,R)$ parameterizing stable curves with a rational tail marked by $Q$ and $R$ .

Therefore, an element of the fundamental group $\pi _1({\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\mathscr{W}})$ , which can be represented by a small loop around the divisor $\partial _{\mathscr{W}}$ , corresponds in the hyperelliptic mapping class group ${\Upsilon }(S^\circ,{\upsilon },{\mathcal{P}})_\circ \lt{\Gamma }(S^\circ,{\mathcal{P}})_\circ ={\Gamma }(S,{\mathcal{P}},Q,R)$ to a Dehn twist about a symmetric simple closed curve on $S^\circ$ bounding a disc which contains no marked points and only the punctures labeled by $Q$ and $R$ .

The embedding ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\mathscr{W}}\subset{\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)$ induces an epimorphism on fundamental groups with kernel generated by small loops around the divisor $\partial _{\mathscr{W}}$ . This clearly implies all the claims of the theorem.

There is a natural epimorphism $p_\circ \colon \thinspace{\Upsilon }(S^\circ,{\mathcal{P}})\to{\Upsilon }(S,{\mathcal{P}})$ induced by filling back with $Q$ and $R$ the two symmetric punctures of $S^\circ$ . For ${\mathcal{P}}=\emptyset$ , from the short exact sequence (3) and the standard genus $0$ Birman short exact sequence (6), it follows that the kernel of this epimorphism is naturally isomorphic to the fundamental group $\pi _1(S_{/{\upsilon }}\smallsetminus{\mathcal{B}}_{\upsilon },\,p_{\upsilon }(Q))$ , where a simple loop around a puncture of $S_{/{\upsilon }}\smallsetminus{\mathcal{B}}_{\upsilon }$ is sent to a half twist about a symmetric simple closed curve of $S^\circ$ bounding a disc which only contains the punctures labeled by $Q$ and $R$ . Therefore, we get the hyperelliptic Birman short exact sequence (cf. Theorem 3.2 in [Reference Brendle and Margalit14] for the case in which $S$ is a closed surface):

(19) \begin{equation} 1\to \pi _1(S_{/{\upsilon }}\smallsetminus{\mathcal{B}}_{\upsilon },\,p_{\upsilon }(Q))\to{\Upsilon }(S^\circ )\stackrel{p_\circ }{\to }{\Upsilon }(S)\to 1. \end{equation}

Let us then consider the epimorphism ${\Upsilon }(S^\circ,{\mathcal{P}})_\circ \to{\Upsilon }(S,{\mathcal{P}})$ obtained by restriction of $p_\circ$ to the index $2$ subgroup ${\Upsilon }(S^\circ,{\mathcal{P}})_\circ$ of ${\Upsilon }(S^\circ,{\mathcal{P}})$ . This epimorphism factors through the epimorphism ${\Upsilon }(S^\circ,{\mathcal{P}})_\circ \to{\Upsilon }(S,{\mathcal{P}},Q)$ of Theorem 3.1 and the natural epimorphism ${\Upsilon }(S,{\mathcal{P}},Q)\to{\Upsilon }(S,{\mathcal{P}})$ of the Birman short exact sequence (6).

The epimorphism ${\Upsilon }(S^\circ,{\mathcal{P}})_\circ \to{\Upsilon }(S,{\mathcal{P}})$ is induced by the natural morphism of moduli stacks ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\mathscr{W}}\to{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ (defined by forgetting the marked point $Q$ ).

If ${\mathcal{P}}=\emptyset$ , this morphism is a smooth curve over ${\mathcal{M}}(S,{\upsilon })$ with fibers homeomorphic to the surface $S\smallsetminus{\mathscr{W}}_{\mathcal{P}}(S)$ . Since, as a topological stack, ${\mathcal{M}}(S,{\upsilon })$ is a $K(\pi,1)$ , from the associated long exact sequence of homotopy groups, it then follows:

Theorem 3.2 (The hyperelliptic Birman short exact sequence). There are natural short exact sequences:

(20) \begin{equation} \begin{array}{ll} &1\to \pi _1(S\smallsetminus{\mathscr{W}}_{\mathcal{P}}(S),Q)\to{\Upsilon }(S^\circ )_\circ \to{\Upsilon }(S)\to 1\\ \mbox{\textit{and}}\;\;\;\;\;&\\ &1\to \pi _1(S\smallsetminus{\mathscr{W}}_{\mathcal{P}}(S),Q)\to{\mathrm{P}\Upsilon }(S^\circ )\to{\mathrm{P}\Upsilon }(S)\to 1. \end{array} \end{equation}

Moreover, the subgroup $N_\circ$ , defined in Theorem 3.1, identifies with the kernel of the natural epimorphism of fundamental groups $\pi _1(S\smallsetminus{\mathscr{W}}_{\mathcal{P}}(S),Q)\to \pi _1(S,Q)$ . In particular, $N_\circ$ is a free group of infinite rank.

1. 1. Note that the fundamental group $\pi _1(S\smallsetminus{\mathscr{W}}_{\mathcal{P}}(S),Q)$ identifies with the index $2$ subgroup of $\pi _1(S_{/{\upsilon }}\smallsetminus{\mathcal{B}}_{\upsilon },\,p_{\upsilon }(Q))$ associated with the unramified double covering $p_{\upsilon }\colon \thinspace S\smallsetminus{\mathscr{W}}_{\mathcal{P}}(S)\to S_{/{\upsilon }}\smallsetminus{\mathcal{B}}_{\upsilon }$ .

2. 2. As we already observed, there is a natural isomorphism ${\mathrm{P}\Upsilon }(S^\circ )\cong{\mathrm{P}\Gamma }(S^\circ _{/{\upsilon }}\smallsetminus{\mathcal{B}}_{\upsilon })$ . Thus, for $S=S_g$ , we get the isomorphism ${\mathrm{P}\Upsilon }(S^\circ )\cong{\mathrm{P}\Gamma }(S_{0,2g+3})$ . The natural epimorphism ${\mathrm{P}\Gamma }(S_{0,2g+3})\to{\mathrm{P}\Gamma }(S_{0,2g+2})$ , induced by filling in a puncture of $S_{0,2g+3}$ , then factors through the epimorphism ${\mathrm{P}\Gamma }(S_{0,2g+3})\to{\mathrm{P}\Upsilon }(S_g)$ of Theorem 3.2 and the epimorphism ${\mathrm{P}\Upsilon }(S_g)\to{\mathrm{P}\Gamma }(S_{0,2g+2})$ provided by Birman-Hilden theory.

4.1. Generators for the pure hyperelliptic mapping class group

For genus $\geq 2$ , there is a simple generating set for the pure hyperelliptic mapping class group.

Let us consider first the case ${\mathcal{P}}=\emptyset$ and $S$ either a closed surface of genus $\geq 2$ or $S=S_{1,1}$ . The following analogue for hyperelliptic mapping class groups of a result by Humphries for mapping class groups (cf. Proposition 2.1 in [Reference Humphries24]) then holds:

Proof. Let $(\hat{S},{\upsilon })$ be a hyperelliptic surface of genus $g=g(S)$ with a single boundary component $\gamma$ , fixed by $\upsilon$ , and no punctures. The relative hyperelliptic mapping class group ${\Upsilon }(\hat{S},\partial \hat{S})$ is described by the Birman-Hilden isomorphism (cf. Theorem 9.2 in [Reference Farb and Margalit16]):

\begin{equation*}{\Upsilon }(\hat {S},\partial \hat {S})\cong {\Gamma }(\hat {S}_{/{\upsilon }},{\mathcal {B}}_{\upsilon },\partial \hat {S}_{/{\upsilon }})=\mathrm {B}_{2g+1},\end{equation*}

where $\mathrm{B}_{2g+1}$ is the braid group in $2g+1$ strands and the isomorphism sends the Dehn twist $\tau _{\gamma }\in{\Upsilon }(\hat{S},\partial \hat{S})$ to the square of a generator of the center of $\mathrm{B}_{2g+1}$ .

(i): By Proposition 2.5, we have ${\mathrm{P}\Upsilon }(S)={\Upsilon }(S)^{[2]}$ . Hence, the abelian level ${\Upsilon }(S)^{[2]}$ and ${\mathcal{Q}}(S,{\upsilon })$ both have for image under the natural epimorphism ${\Upsilon }(S)\to{\Gamma }(S_{/{\upsilon }},{\mathcal{B}})$ the pure mapping class group ${\mathrm{P}\Gamma }(S_{/{\upsilon }},{\mathcal{B}})\cong{\mathrm{P}\Gamma }(S_{0,2g+2})$ . Therefore, ${\Upsilon }(S)^{[2]}=\langle{\upsilon }\rangle \cdot{\mathcal{Q}}(S,{\upsilon })$ . Hence, in order to prove the first item of the proposition, it is enough to show that ${\upsilon }\in{\mathcal{Q}}(S,{\upsilon })$ . We borrow here an argument from the proof of Proposition 2.1 in [Reference Humphries24].

An identity, due to Birman and Hilden (cf. Section 4.4 in [Reference Birman2]), implies that ${\upsilon }\in{\mathcal{Q}}(S,{\upsilon })$ . More precisely, let ${\gamma }_1,\ldots,{\gamma }_{2g+1}$ be a maximal chain of symmetric nonseparating simple closed curves on $S$ . Then, we have:

\begin{equation*}{\upsilon }=\tau _{{\gamma }_1}\tau _{{\gamma }_2}\cdot \ldots \cdot \tau _{{\gamma }_{2g}}\tau _{{\gamma }_{2g+1}}^2\tau _{{\gamma }_{2g}}\cdot \ldots \cdot \tau _{{\gamma }_2}\tau _{{\gamma }_1}.\end{equation*}

The above identity easily implies that $\upsilon$ is the identity element modulo the normal subgroup ${\mathcal{Q}}(S,{\upsilon })$ and so that ${\upsilon }\in{\mathcal{Q}}(S,{\upsilon })$ .

(ii): For $S=S_{1,1}$ , the relative hyperelliptic mapping class group ${\Upsilon }(\hat{S},\partial \hat{S})$ is the universal central extension of the mapping class groups ${\Gamma }(S)$ :

\begin{equation*} 1\to \tau _{\gamma }^{\mathbb{Z}}\to{\Upsilon }(\hat{S},\partial \hat{S})\to{\Gamma }(S)\to 1. \end{equation*}

The Birman-Hilden isomorphism ${\Upsilon }(\hat{S},\partial \hat{S})\cong B_3$ then identifies ${\Gamma }(S)$ with the quotient of the braid group $B_3$ by the square of its center, so that a generator of the center of $B_3$ descends to the hyperelliptic involution $\upsilon$ of ${\Gamma }(S)$ .

The pure braid group $\mathrm{PB}_n$ contains the center of $\mathrm{B}_n$ , splits over it, and there is a natural isomorphism $\mathrm{PB}_n\cong \mathbb{Z}\times{\mathrm{P}\Gamma }(S_{0,n+1})$ (cf. Section 9.3 in [Reference Farb and Margalit16]). By means of the Birman-Hilden isomorphism above, the pure hyperelliptic mapping class group ${\mathrm{P}\Upsilon }(S)$ is then identified with the quotient of $\mathrm{PB}_3$ by the square of its center and so there is a natural isomorphism ${\mathrm{P}\Upsilon }(S)\cong \mathbb{Z}/2\times{\mathrm{P}\Gamma }(S_{0,4})$ which sends the hyperelliptic involution to the factor $\mathbb{Z}/2$ .

The Dehn twists about essential simple closed curves in $S_{0,4}$ naturally lift to squares of nonseparating Dehn twists in ${\mathrm{P}\Upsilon }(S)$ and these generate a subgroup ${\mathcal{Q}}(S,{\upsilon })$ isomorphic to ${\mathrm{P}\Gamma }(S_{0,4})$ . This completes the proof of the second item of the proposition.

For the general case, we need to introduce some notation. We say that a Dehn twist about a symmetric, resp. nonseparating, resp. separating simple closed curve on $S$ is symmetric, resp. nonseparating, resp. separating. A symmetric bitwist is the product of two Dehn twists about a symmetric pair of simple closed curves. A symmetric bounding pair map is a bounding pair map about a symmetric pair of simple closed curves. We then have:

Proof. Since there is a natural isomorphism ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})\cong{\mathrm{P}\Upsilon }({S}^{^{^{\!\!\!\circ}}},{\mathcal{P}})$ , it is not restrictive to assume that $\partial S=\emptyset$ . For a similar reason, we may also assume that, for $g(S)\geq 2$ , we have ${\mathscr{W}}_\circ (S)=\emptyset$ and, for $g(S)=1$ , we have $\sharp{\mathscr{W}}_\circ (S)=1$ .

If $S$ has $n$ pairs of symmetric punctures, by Theorem 3.1 and a simple induction, there is a natural epimorphism ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})\to{\mathrm{P}\Upsilon }(\overline{S},{\mathcal{P}}\cup \{Q_1,\ldots,Q_n\})$ whose kernel is generated by Dehn twists about symmetric separating simple closed curves on $S$ bounding discs which contain no marked points and at least a pair of symmetric punctures. Thus, we are reduced to prove Theorem 4.2 for $S$ a closed surface of genus $\geq 2$ and for $S=S_{1,1}$ .

A bounding pair map $\tau _{{\gamma }_0}\tau _{{\gamma }_1}^{-1}$ about a symmetric pair ${\gamma }_0,{\gamma }_1$ of simple closed curves which bound a marked but unpunctured annulus is the product of the symmetric bitwist $\tau _{{\gamma }_0}\tau _{{\gamma }_1}$ with the inverse of $\tau _{{\gamma }_1}^{2}$ and these are both part of our generating set. Since the kernel of the natural epimorphism ${\mathrm{P}\Gamma }(S,{\mathcal{P}})\to{\mathrm{P}\Gamma }(S)$ is generated by bounding pair maps about symmetric pairs of simple closed curves which bound a marked unpunctured annulus, the theorem is reduced to the case ${\mathcal{P}}=\emptyset$ . The conclusion then follows from Proposition 4.1.

Proof. This follows from the geometric interpretation of the generators of the fundamental group of $\mathrm{P}{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ as loops about the irreducible components of its DM boundary and Theorem 4.2.

We can determine the gerbe structure of the moduli stacks considered above:

Proof. (i): This is well known (see, for instance, item (h) of Theorem 1.4 in [Reference Ghigi and Tamborini18]) but also an almost immediate consequence of item (i) of Proposition 4.1.

(ii): Assigning to a smooth projective genus $0$ curve $C$ endowed with an ordered set of marked points ${\mathcal{B}}_{\upsilon }$ , the hyperelliptic curve $C'$ such that $C'/\langle{\upsilon }\rangle \cong C$ with branch locus ${\mathcal{B}}_{\upsilon }$ , defines a morphism $\bar s\colon \thinspace \mathrm{P}{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\to{\mathcal{M}}(S,{\upsilon })$ (cf. item (h) of Theorem 1.4 in [Reference Ghigi and Tamborini18]), which, composed with the natural morphism ${\mathcal{M}}(S,{\upsilon })\to{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ , is just the natural morphism $\mathrm{P}{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\to{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ .

Since ${\mathcal{M}}(S)^{[2]}\cong{\mathcal{M}}(S,{\upsilon })\times _{{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })}\mathrm{P}{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ , the morphism $\bar s$ lifts to an étale morphism $s\colon \thinspace \mathrm{P}{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\to{\mathcal{M}}(S)^{[2]}$ which is a section of ${\mathcal{M}}(S,{\upsilon })^{[2]}\to \mathrm{P}{\mathcal{M}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ and so trivializes this $\mathbb{Z}/2$ -gerbe.

(iii): Let us assume, by contradiction, that the gerbe ${\overline{\mathcal{M}}}(S,{\upsilon })^{[2]}\to \mathrm{P}{\overline{\mathcal{M}}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })$ is trivial. A splitting of this gerbe is a (representable) morphism $\iota \colon \thinspace \mathrm{P}{\overline{\mathcal{M}}}(S_{/{\upsilon }},{\mathcal{B}}_{\upsilon })\to{\overline{\mathcal{M}}}(S,{\upsilon })^{[2]}$ which composed with the latter is the identity and so is étale. By (i) of Corollary 4.3, the stack ${\overline{\mathcal{M}}}(S,{\upsilon })^{[2]}$ is simply connected which implies that $\bar \iota$ and then $\iota$ is an isomorphism which gives a contradiction.

4.2. Generators for the full hyperelliptic mapping class group

We have the following generalizations of a well-known result by Birman and Hilden (cf. [Reference Birman and Hilden3]):

Proof. The kernel of the natural epimorphism ${\Upsilon }(S,{\mathcal{P}})\to{\Upsilon }(S)$ identifies with the fundamental group of the configuration space of $\sharp{\mathcal{P}}$ points on the surface $S$ , and it is well known that this group is generated by bounding pair maps and half twists about simple closed curves which bound a disc containing a pair of marked points. We are then reduced to prove the proposition for the hyperelliptic mapping class group ${\Upsilon }(S)$ .

It is easy to reduce to the case when $\partial S=\emptyset$ . By the hyperelliptic Birman short exact sequence (19), we can then reduce further to the case when $S$ is a closed surface, that is to say to the case treated by Birman and Hilden.

Proof. This is essentially the same proof as that of Proposition 4.5. We only need to observe that the kernel of the natural epimorphism ${\Upsilon }(S,\vec{\mathcal{P}})\to{\Upsilon }(S)$ identifies with the fundamental group of the configuration space of $\sharp{\mathcal{P}}$ ordered points on the surface $S$ . The latter, for $g(S)\geq 1$ , is generated by bounding pair maps which are products of powers of nonseparating symmetric Dehn twists.

6.1. The Putman-Wieland conjecture

In [Reference Putman and Wieland30], Putman and Wieland made a conjecture about virtual linear representations of mapping class groups which we present here in a slightly different but equivalent formulation. Let $(S,{\mathcal{P}})$ be a hyperbolic marked closed surface and let $Q\in S\smallsetminus{\mathcal{P}}$ . We then have the Birman exact sequence:

\begin{equation*} 1\to \pi _1(S\smallsetminus{\mathcal{P}},Q)\to{\Gamma }(S,{\mathcal{P}},Q)\stackrel{p_Q}{\to }{\Gamma }(S,{\mathcal{P}})\to 1. \end{equation*}

For a finite index subgroup ${\Gamma }^{\lambda }$ of ${\Gamma }(S,{\mathcal{P}},Q)$ , let $\Pi ^{\lambda }\;:\!=\;\pi _1(S\smallsetminus{\mathcal{P}},Q)\cap{\Upsilon }$ . By restriction, we then get a short exact sequence:

\begin{equation*} 1\to \Pi ^{\lambda }\to{\Gamma }^{\lambda }\to p_Q({\Gamma }^{{\lambda }})\to 1. \end{equation*}

Let $(S\smallsetminus{\mathcal{P}})^{\lambda }\to S\smallsetminus{\mathcal{P}}$ be the unramified covering associated with the subgroup $\Pi ^{\lambda }$ of $\pi _1(S\smallsetminus{\mathcal{P}},Q)$ and let $S^{\lambda }\to S$ be the (possibly ramified) covering obtained from this filling in all punctures. The outer representation associated with the above short exact sequence induces a linear representation: $p_Q({\Gamma }^{{\lambda }})\to \operatorname{Sp}(H^1(S^{\lambda },\mathbb{Q}))$ . The Putman-Wieland conjecture states that, for $g(S)\geq 2$ , all nontrivial orbits of this representation are infinite.

6.2. The hyperelliptic Putman-Wieland problem

From now to the end of Section 6, we assume that $(S,{\upsilon },{\mathcal{P}})$ is a hyperbolic marked hyperelliptic closed surface. Let us consider the Birman exact sequence (6):

\begin{equation*} 1\to \pi _1(S\smallsetminus{\mathcal{P}},Q)\to{\Upsilon }(S,{\mathcal{P}},Q)\stackrel{f_Q}{\to }{\Upsilon }(S,{\mathcal{P}})\to 1. \end{equation*}

For a finite index subgroup ${\Upsilon }^{\lambda }$ of ${\Upsilon }(S,{\mathcal{P}},Q)$ , let $\Pi ^{\lambda }\;:\!=\;\pi _1(S\smallsetminus{\mathcal{P}},Q)\cap{\Upsilon }^{\lambda }$ . There is then a short exact sequence:

(21) \begin{equation} 1\to \Pi ^{\lambda }\to{\Upsilon }^{\lambda }\to f_Q({\Upsilon }^{\lambda })\to 1 \end{equation}

and an associated representation $\rho ^{\lambda }\colon \thinspace f_Q({\Upsilon }^{\lambda })\to \operatorname{Out}(\Pi ^{\lambda })$ .

Let $(S\smallsetminus{\mathcal{P}})^{\lambda }\to S\smallsetminus{\mathcal{P}}$ be the unramified covering associated with the subgroup $\Pi ^{\lambda }$ of $\pi _1(S\smallsetminus{\mathcal{P}},Q)$ and let $S^{\lambda }\to S$ be the (possibly ramified) covering obtained from this filling in all punctures. The representation $\rho ^{\lambda }$ then induces a linear representation:

(22) \begin{equation} L\rho ^{\lambda }\colon \thinspace f_Q({\Upsilon }^{\lambda })\to \operatorname{Sp}(H^1(S^{\lambda },\mathbb{Q})). \end{equation}

The hyperelliptic Putman-Wieland problem asks whether, for every finite index subgroup ${\Upsilon }^{\lambda }$ of ${\Upsilon }(S,{\mathcal{P}},Q)$ , all nontrivial orbits of $L\rho ^{\lambda }$ are infinite.

In Corollary 6.9, we will show that the answer is no for $g(S)\geq 2$ and $\sharp{\mathcal{P}}\geq 4$ . In particular, since for $g(S)=2$ , there holds ${\Upsilon }(S,{\mathcal{P}},Q)={\Gamma }(S,{\mathcal{P}},Q)$ , this will also provide a counterexample to Putman-Wieland conjecture in genus $2$ .

6.3. Geometric interpretation of the hyperelliptic Putman-Wieland problem

The hyperelliptic Putman-Wieland problem can be reformulated as follows.

Let ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\lambda }\to{\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)$ be the finite étale covering associated with the subgroup ${\Upsilon }^{\lambda }$ of ${\Upsilon }(S,{\upsilon },{\mathcal{P}},Q)$ . For simplicity, let us denote by ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })}\to{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ the finite étale covering associated with the subgroup $f_Q({\Upsilon }^{\lambda })$ of ${\Upsilon }(S,{\upsilon },{\mathcal{P}})$ . This coincides with the normalization of ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ in ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\lambda }$ . The natural morphism $\phi _Q\colon \thinspace{\mathcal{M}}(S,{\upsilon },{\mathcal{P}},Q)^{\lambda }\to{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })}$ is then a connected smooth curve with fibers homeomorphic to the surface $(S\smallsetminus{\mathcal{P}})^{\lambda }$ .

Extending the morphism $\phi _Q$ to DM compactifications, we obtain a projective flat morphism $\bar \phi _Q\colon \thinspace{\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}},Q)^{\lambda }\to{\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })}$ , with fibers semistable curves, such that its restriction over ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })}$ :

(23) \begin{equation} \tilde{\phi }_Q\colon \thinspace{\mathscr{C}}^{\lambda }\to{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })} \end{equation}

is a smooth proper curve with fibers homeomorphic to the surface $S^{\lambda }$ defined above. The representation (22) $L\rho ^{\lambda }\colon \thinspace f_Q({\Upsilon }^{\lambda })\to \operatorname{Sp}(H^1(S^{\lambda },\mathbb{Q}))$ is then the monodromy representation associated with the proper smooth curve (23).

This interpretation is important because it allows to implement Hodge theoretic techniques. Indeed, since moduli stacks of (hyperelliptic) curves are $K(\pi,1)$ -spaces, the homology and cohomology of (hyperelliptic) mapping class groups with rational coefficients identify with the singular homology and cohomology of smooth DM stacks, as such they are endowed with a natural mixed Hodge structure. The same holds for finite index subgroups of (hyperelliptic) mapping class groups. A first interesting consequence is the following lemma:

Lemma 6.1. Let $(S,{\upsilon },{\mathcal{P}})$ be a hyperbolic marked hyperelliptic closed surface and ${\Upsilon }^{\lambda }$ a finite index subgroup of ${\mathrm{P}\Upsilon }(S,{\mathcal{P}}\cup \{Q\})$ . Associated with the short exact sequence ( 21 ), there is a short exact sequence of polarized Hodge structures of weight one:

(24) \begin{equation} 0\to W_1H^1(f_Q({\Upsilon }^{\lambda }),\mathbb{Q})\to W_1H^1({\Upsilon }^{\lambda },\mathbb{Q})\to H^1(S^{\lambda },\mathbb{Q}))^{f_Q({\Upsilon }^{\lambda })}\to 0. \end{equation}

Proof. By a theorem of Deligne (cf. [Reference Deligne15]), the associated Leray spectral sequence associated with the proper curve $\tilde{\phi }_Q\colon \thinspace{\mathscr{C}}^{\lambda }\to{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })}$ degenerates at the $E_2$ level and there is a short exact sequence:

\begin{equation*} 0\to H^1({\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })},\mathbb{Q})\stackrel{f_\ast }{\to }H^1({\mathscr{C}}^{\lambda },\mathbb{Q})\to H^0({\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })},R^1\tilde{\phi }_{Q\ast }\mathbb{Q})\to 0, \end{equation*}

where the space $H^0({\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })},R^1\tilde{\phi }_{Q\ast }\mathbb{Q})$ identifies with the space of monodromy invariants $H^1(\tilde{\phi }_Q^{-1}(z),\mathbb{Q})^{f_Q({\Upsilon }^{\lambda })}$ , for all $z\in{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })}$ .

It follows that, after identifying $H^1(\tilde{\phi }_Q^{-1}(z),\mathbb{Q})^{f_Q({\Upsilon }^{\lambda })}$ with $H^1(S^{\lambda },\mathbb{Q}))^{f_Q({\Upsilon }^{\lambda })}$ , there is a short exact sequence:

\begin{equation*} 0\to H^1({\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })},\mathbb{Q})\stackrel{\tilde{\phi }_Q^\ast }{\to }H^1({\mathscr{C}}^{\lambda },\mathbb{Q})\to H^1(S^{\lambda },\mathbb{Q}))^{f_Q({\Upsilon }^{\lambda })}\to 0. \end{equation*}

This sequence remains exact passing to its weight $1$ part:

\begin{equation*} 0\to W_1 H^1({\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{f_Q({\lambda })},\mathbb{Q})\stackrel{\tilde{\phi }_Q^\ast }{\to }W_1 H^1({\mathscr{C}}^{\lambda },\mathbb{Q})\to H^1(S^{\lambda },\mathbb{Q}))^{f_Q({\Upsilon }^{\lambda })}\to 0. \end{equation*}

The conclusion of the lemma then follows observing that we have:

\begin{equation*}\begin {array}{ll} &H^1({\mathcal {M}}(S,{\upsilon },{\mathcal {P}})^{f_Q({\lambda })},\mathbb {Q})=H^1(f_Q({\Upsilon }^{\lambda }),\mathbb {Q})\\ \mbox {and}\;\;\;\;\;&\\ &W_1 H^1({\mathscr{C}}^{\lambda },\mathbb {Q})=W_1 H^1({\mathcal {M}}(S,{\upsilon },{\mathcal {P}},Q)^{\lambda },\mathbb {Q})=W_1 H^1({\Upsilon }^{\lambda },\mathbb {Q}). \end {array}\end{equation*}

6.4. The pure cohomology of hyperelliptic mapping class groups

In this subsection, we collect more Hodge theoretic results. Some are of independent interest and some we will need later.

Proof. By the first item of Corollary 4.3, we have that $H^1(\mathrm{P}{\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}}),\mathbb{Q})=\{0\}$ and, since $\mathrm{P}{\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}})$ is a smooth compactification of $\mathrm{P}{\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ , there is an isomorphism:

\begin{equation*}H^1(\mathrm {P}{\overline {\mathcal {M}}}(S,{\upsilon },{\mathcal {P}}),\mathbb {Q})\cong W_1 H^1(\mathrm {P}{\mathcal {M}}(S,{\upsilon },{\mathcal {P}}),\mathbb {Q}).\end{equation*}

Proof. It is enough to prove the proposition for $g(S)=1$ , $n(S)=1$ , and $g(S)\geq 2$ and $n(S)=0$ , since, if $p\colon \thinspace{\Upsilon }(S,{\mathcal{P}})\to{\Upsilon }(S)$ is the natural epimorphism, then, for a finite index subgroup ${\Upsilon }^{\lambda }$ of ${\Upsilon }(S,{\mathcal{P}})$ , the natural homomorphism $p^\ast \colon \thinspace W_1H^1(p({\Upsilon }^{\lambda }))\to W_1H^1({\Upsilon }^{\lambda })$ is injective.

For ${\Gamma }(S_{1,1})=\operatorname{SL}_2(\mathbb{Z})$ , the claim is well-known since $W_1H^1({\Upsilon }^{\lambda })$ is just the first cohomology group of the projective modular curve associated with ${\Upsilon }^{\lambda }$ .

Let us assume then $n(S)=0$ and $g(S)\geq 2$ . Let ${\mathcal{T}}(S,{\upsilon })$ be the universal cover of the moduli stack of hyperelliptic curves ${\mathcal{M}}(S,{\upsilon })$ . For $g(S)=2$ , this is just the ordinary genus $2$ Teichmüller space ${\mathcal{T}}(S)$ associated with $S$ , while, for $g(S)\geq 3$ , it can be identified with a closed contractible submanifold of the Teichmüller space ${\mathcal{T}}(S)$ . Since there is a natural étale covering ${\mathcal{M}}(S,{\upsilon })\to{\mathcal{M}}(S_{0,2g+2})$ , we have a natural isomorphism ${\mathcal{T}}(S,{\upsilon })\cong{\mathcal{T}}(S_{0,2g+2})$ . There is also a natural epimorphism $p\colon \thinspace{\mathrm{P}\Gamma }(S_{0,2g+2})\to{\mathrm{P}\Gamma }(S_{0,4})$ and a corresponding $p$ -equivariant surjective map $\phi \colon \thinspace{\mathcal{T}}(S_{0,2g+2})\to{\mathcal{T}}(S_{0,4})$ .

Let us assume that ${\Upsilon }^{\lambda }$ is contained in the abelian level ${\Upsilon }(S)^{[2]}$ . By Proposition 3.3 in [Reference Boggi6], there is a natural epimorphism $r\colon \thinspace{\Upsilon }(S)^{[2]}\to{\mathrm{P}\Gamma }(S_{0,2g+2})$ . It follows that the map $\phi$ induces a surjective map ${\mathcal{T}}(S,{\upsilon })/{\Upsilon }^{\lambda }\to{\mathcal{T}}(S_{0,4})/p(r({\Upsilon }^{\lambda }))$ , which induces an epimorphism on fundamental groups. Therefore, we have a natural injective map:

\begin{equation*}W_1H^1({\mathcal {T}}(S_{0,4})/p(r({\Upsilon }^{\lambda })))\hookrightarrow W_1H^1({\mathcal {T}}(S,{\upsilon })/{\Upsilon }^{\lambda })=W_1H^1({\Upsilon }^{\lambda }).\end{equation*}

Since $\mathrm{P}{\mathcal{M}}(S_{0,4})\cong{\mathbb P}^1\smallsetminus \{0,1,\infty \}$ , there is a normal finite index subgroup $N$ of ${\mathrm{P}\Gamma }(S_{0,4})$ such that the associated covering of $\mathrm{P}{\mathcal{M}}(S_{0,4})$ has genus $\geq 1$ .

For ${\Upsilon }^{\lambda }$ any normal finite index subgroup of ${\Upsilon }(S,{\mathcal{P}})$ which is contained in $r^{-1}(p^{-1}(N))$ , we then have $W_1H^1({\mathcal{T}}(S_{0,4})/p(r({\Upsilon }^{\lambda })))\neq \{0\}$ and the claim of the proposition follows.

A consequence of independent interest of Proposition 6.3 is a slight improvement of Corollary 1.1 in [Reference Serre32] (cf. [Reference Funar17] and [Reference Masbaum29] for the mapping class group version of this result):

Proof. There is a normal subgroup $N$ of ${\mathrm{P}\Gamma }(S_{0,4})$ of index a power $(m/2)^k$ such that the associated covering of $\mathrm{P}{\mathcal{M}}(S_{0,4})$ has genus $\geq 1$ . Then, ${\Upsilon }^{\lambda }\;:\!=\;r^{-1}(p^{-1}(N))$ is a normal finite index subgroup of ${\Upsilon }(S)^{[2]}$ of index $(m/2)^k$ . Since ${\Upsilon }(S)^{[2]}$ contains all squares of symmetric multitwists, it follows that all $m^k$ -powers of symmetric multitwists belong to ${\Upsilon }^{\lambda }$ . Let us denote by $K$ the normal subgroup of ${\Upsilon }^{\lambda }$ generated by these elements and let ${\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}})^{\lambda }$ be the DM compactification of the level structure ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})^{\lambda }$ over ${\mathcal{M}}(S,{\upsilon },{\mathcal{P}})$ associated with ${\Upsilon }^{\lambda }$ . There is an epimorphism ${\Upsilon }^{\lambda }/K\twoheadrightarrow H_1({\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}})^{\lambda },\mathbb{Z})$ , and, as we showed in the proof of Proposition 6.3, there holds $H^1({\overline{\mathcal{M}}}(S,{\upsilon },{\mathcal{P}})^{\lambda },\mathbb{Z})\otimes \mathbb{Q}\cong W_1H^1({\Upsilon }^{\lambda })\neq \{0\}$ .

For ${\sigma }\in C(S,{\upsilon },{\mathcal{P}})$ , where $S$ is a closed surface, the stabilizer ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\vec{\sigma }}$ of the oriented simplex $\vec{\sigma }$ for the action of ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})$ is described by the short exact sequence (15):

\begin{equation*} 1\to \prod _{{\gamma }\in{\sigma }_{ns}}\tau _{\gamma }^{2\mathbb{Z}}\times \prod _{{\gamma }\in{\sigma }_{s}}\tau _{\gamma }^{\mathbb{Z}}\to{\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\vec{\sigma }}\stackrel{q_{\sigma }}{\to }{\mathrm{P}\Upsilon }(S\smallsetminus{\sigma },{\mathcal{P}})\to 1. \end{equation*}

Since powers of Dehn twists have weight $-2$ in homology, we then have:

Lemma 6.5. For a finite index subgroup ${\Upsilon }^{\lambda }_{\vec{\sigma }}$ of ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\vec{\sigma }}$ , the restriction map:

(25) \begin{equation} W_1 H^1(q_{\sigma }({\Upsilon }^{\lambda }_{\vec{\sigma }}),\mathbb{Q})\stackrel{q_{\sigma }^\ast }{\to } W_1 H^1({\Upsilon }^{\lambda }_{\vec{\sigma }},\mathbb{Q}) \end{equation}

is an isomorphism.

6.5. A counterexample to the hyperelliptic Putman-Wieland problem

Let us fix some notation. For $\gamma$ a nonseparating simple closed curve on $S$ , let $S_{\gamma }\;:\!=\;S\smallsetminus{\gamma }$ , let $\overline{S}_{\gamma }$ be the closed surface obtained from $S_{\gamma }$ filling in the punctures and put $\{Q,R\}\;:\!=\;\overline{S}_{\gamma }\smallsetminus S_{\gamma }$ .

We then denote by $\overline{{\Upsilon }}_{\vec{\gamma }}^{\lambda }$ the image of ${\Upsilon }^{\lambda }\cap{\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\vec{\gamma }}$ in ${\mathrm{P}\Upsilon }(\overline{S}_{\gamma },{\mathcal{P}}\cup \{Q\})$ by the composition of the natural epimorphism $q_{\gamma }\colon \thinspace{\mathrm{P}\Upsilon }(S,{\mathcal{P}})_{\vec{\gamma }}\to{\mathrm{P}\Upsilon }(S_{\gamma },{\mathcal{P}})$ (cf. the short exact sequence (15)) with the natural epimorphism ${\mathrm{P}\Upsilon }(S_{\gamma },{\mathcal{P}})\to{\mathrm{P}\Upsilon }(\overline{S}_{\gamma },{\mathcal{P}}\cup \{Q\})$ (cf. the last one of the short exact sequences (18)). There is a Birman short exact sequence:

\begin{equation*} 1\to \pi _1(\overline{S}_{\gamma }\smallsetminus{\mathcal{P}},Q)\to{\mathrm{P}\Upsilon }(\overline{S}_{\gamma },{\mathcal{P}}\cup \{Q\})\stackrel{f_Q}{\to }{\mathrm{P}\Upsilon }(\overline{S}_{\gamma },{\mathcal{P}})\to 1. \end{equation*}

Let $\overline{S}_{\gamma }^{\lambda }\to \overline{S}_{\gamma }$ be the covering of closed surfaces obtained filling in the punctures of the unramified covering $(\overline{S}_{\gamma }\smallsetminus{\mathcal{P}})^{\lambda }\to \overline{S}_{\gamma }\smallsetminus{\mathcal{P}}$ associated with the subgroup $\pi _1(\overline{S}_{\gamma }\smallsetminus{\mathcal{P}},Q)\cap \overline{{\Upsilon }}_{\vec{\gamma }}^{\lambda }$ of $\pi _1(\overline{S}_{\gamma }\smallsetminus{\mathcal{P}},Q)$ . Let then:

\begin{equation*}L\overline {\rho }_{\gamma }^{\lambda }\colon \thinspace f_Q(\overline {{\Upsilon }}^{\lambda }_{\vec {\gamma }})\to \operatorname {Sp}(H^1(\overline {S}_{\gamma }^{\lambda },\mathbb {Q}))\end{equation*}

be the corresponding linear representation.

We then have:

Proof. The proof is by contraposition. So let us assume that, for a given finite index subgroup ${\Upsilon }^{\lambda }$ of ${\mathrm{P}\Upsilon }(S,{\mathcal{P}})$ , which satisfies the hypotheses of the theorem, and some nonseparating simple closed curve $\gamma$ on $S$ , the linear representation $L\overline{\rho }_{\gamma }^{\lambda }\colon \thinspace f_Q(\overline{{\Upsilon }}^{\lambda }_{\vec{\gamma }})\to \operatorname{Sp}(H^1(\overline{S}_{\gamma }^{\lambda },\mathbb{Q}))$ has only infinite nontrivial orbits. Note that, since ${\Upsilon }^{\lambda }$ is normal in ${\Upsilon }(S,\vec{\mathcal{P}})$ and nonseparating simple closed curve on $S$ form a single orbit under the latter group, this property then holds for all of them. We will show that this implies that $W_1 H^1({\Upsilon }^{\lambda })= 0$ .

By Proposition 6.2, we have that $W_1H^1({\Upsilon }^{\lambda },\mathbb{Q})^{{\Upsilon }(S,\vec{\mathcal{P}})}=W_1H^1({\Upsilon }(S,\vec{\mathcal{P}}),\mathbb{Q})=0$ . So it is enough to prove that our hypothesis implies that ${\Upsilon }(S,\vec{\mathcal{P}})$ acts trivially on $W_1 H^1({\Upsilon }^{\lambda })$ . Since, by Proposition 4.6, the hyperelliptic mapping class group ${\Upsilon }(S,\vec{\mathcal{P}})$ is generated by symmetric nonseparating Dehn twists, we just need to show that, for any symmetric nonseparating simple closed curve $\gamma$ on $S$ , the Dehn twist $\tau _{\gamma }$ acts trivially on $W_1 H^1({\Upsilon }^{\lambda })$ .