Linear representations of hyperelliptic mapping class groups

Let $p:S\to S_g$ be a finite $G$-covering of a closed surface of genus $g\geq 1$ and let $B$ its branch locus. To this data, it is associated a representation of a finite index subgroup of the mapping class group $\operatorname{Mod}(S_g\smallsetminus B)$ in the centralizer of the group $G$ in the symplectic group $\operatorname{Sp}(H_1(S,{\mathbb Q}))$. They are called \emph{virtual linear representations} of the mapping class group and are related, via a conjecture of Putman and Wieland, to a question of Kirby and Ivanov on the abelianization of finite index subgroup of the mapping class group. The purpose of this paper is to study the restriction of such representations to the hyperelliptic mapping class group $\operatorname{Mod}(S_g,B)^\iota$, which is a subgroup of $\operatorname{Mod}(S_g\smallsetminus B)$ associated to a given hyperelliptic involution $\iota$ on $S_g$. We extend to hyperelliptic mapping class groups some previous results on virtual linear representations of the mapping class group. We then show that, for all $g\geq 1$, there are virtual linear representations of hyperelliptic mapping class groups with nontrivial finite orbits. In particular, we show that there is such a representation associated to an unramified $G$-covering $S\to S_2$, thus providing a counterexample to the genus $2$ case of the Putman-Wieland conjecture.Comment: 22 page

Curves with prescribed symmetry and associated representations of mapping class groups

Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map from the group algebra QG to the algebra of Q-endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation acts Q-irreducibly in a G-isogeny space of H^1(C; Q)and with image often a Q-almost simple group

Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves

Let ${\cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected \'etale cover {\cal M}^\l of ${\cal M}_{g,[n]}$, defined over a sub-$p$-adic field $k$, is "almost" anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let \pi_1({\cal M}^\l_{\ol{k}}) be the geometric algebraic fundamental group of {\cal M}^\l and let {Out}^*(\pi_1({\cal M}^\l_{\ol{k}})) be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of {\cal M}^\l (this is the "$\ast$-condition" motivating the "almost" above). Let us denote by {Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})) the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack {\cal M}^\l has a trivial automorphisms group. Then, there is a natural isomorphism: {Aut}_k({\cal M}^\l)\cong{Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})). This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.Comment: This paper has been withdrawn because of a flaw in the paper "Profinite Teichm\"uller theory" of the first author, on which this paper built o

Fundamental groups of moduli stacks of stable curves of compact type

Let $\widetilde{\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, stable complex curves of compact type. Various characterizations and properties are obtained of both the algebraic and topological fundamental groups of the stack $\widetilde{\cal M}_{g,n}$. Let $\Gamma_{g,n}$, for $2g-2+n>0$, be the Teichm\"uller group associated with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$, i.e. the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $K_{g,n}$ be the normal subgroup of $\Gamma_{g,n}$ generated by Dehn twists along separating circles on $S_{g,n}$. As a first application of the above theory, a characterization of $K_{g,n}$ is given for all $n\geq 0$ (for $n=0,1$, this was done by Johnson). Let then ${\cal T}_{g,n}$ be the Torelli group, i.e. the kernel of the natural representation \Gamma_{g,n}\ra Sp_{2g}(Z). The abelianization of ${\cal T}_{g,n}$ is determined for all $g\geq 1$ and $n\geq 1$, thus completing classical results by Johnson and Mess.Comment: 25 pages; minor corrections in some proofs; typos corrected; theorem numbering change