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Marco Boggi

Publications and source records attributed to Marco Boggi.

24 records · Page 2Linked to original sources
arXiv2013

On the procongruence completion of the Teichmüller modular group

For $2g-2+n>0$, the Teichmüller modular group $Γ_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is 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 $Π_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\hatΠ_{g,n}$ its profinite completion. There is then a natural faithful representation $Γ_{g,n}\hookrightarrow Out(\hatΠ_{g,n})$. The procongruence completion $\checkΓ_{g,n}$ of the Teichmüller group is defined to be the closure of the Teichmüller group $Γ_{g,n}$ inside the profinite group $Out(\hatΠ_{g,n})$. In this paper, we begin a systematic study of the procongruence completion $\checkΓ_{g,n}$. The set of profinite Dehn twists of $\checkΓ_{g,n}$ is the closure, inside this group, of the set of Dehn twists of $\GG_{g,n}$. The main technical result of the paper is a parametrization of the set of profinite Dehn twists of $\checkΓ_{g,n}$ and the subsequent description of their centralizers. This is the basis for the Grothendieck-Teichmüller Lego with procongruence Teichmüller groups as building blocks. As an application, we prove that some Galois representations associated to hyperbolic curves over number fields and their moduli spaces are faithful.

math.AG↗
arXiv2013

Galois coverings of moduli spaces of curves and loci of curves with symmetry

Let $\ccM_{g,[n]}$, for $2g-2+n>0$, be the stack of genus $g$, stable algebraic curves, endowed with $n$ unordered marked points. Looijenga introduced the notion of Prym level structures in order to construct smooth projective Galois coverings of the stack $\ccM_{g}$. In §2 of this paper, we introduce the notion of Looijenga level structure which generalizes Looijenga construction and provides a tower of Galois coverings of $\ccM_{g,[n]}$ equivalent to the tower of all geometric level structures over $\ccM_{g,[n]}$. In §3, Looijenga level structures are interpreted geometrically in terms of moduli of curves with symmetry. A byproduct of this characterization is a simple criterion for their smoothness. As a consequence of this criterion, it is shown that Looijenga level structures are smooth under mild hypotheses. The second part of the paper, from §4, deals with the problem of describing the D-M boundary of level structures. In §6, a description is given of the nerve of the D-M boundary of abelian level structures. In §7, it is shown how this construction can be used to "approximate" the nerve of Looijenga level structures. These results are then applied to elaborate a new approach to the congruence subgroup problem for the Teichmüller modular group.

math.AG↗
arXiv2007

Monodromy of stable curves of compact type: rigidity and extension

Let ${\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, smooth curves. For a family $C\to S$ of such curves over a connected base and a geometric point $ξ$ on $S$, the associated monodromy representation is the induced homomorphism $π_1(S,ξ)\toπ_1({\cal M}_{g,n},[C_ξ])$ on algebraic fundamental groups. It is well known that, if $S$ is irreducible, reduced and locally of finite type over a field $k$ of characteristic zero, the fibre $C_ξ$ and the corresponding monodromy representation determine the relative isomorphism class of the family. In the first part of the paper, it is shown that suitable quotients of this representation suffice. These results are then applied to show that the monodromy representation associated to a family $C\to S$ of $n$-pointed, genus $g$, stable curves of compact type, i.e. the induced homomorphism $π_1(S,ξ)\toπ_1(\widetilde{\cal M}_{g,n},[C_ξ])$ (where, $\widetilde{\cal M}_{g,n}$ denotes the moduli stack of $n$-pointed, genus $g$, stable curves of compact type), characterizes trivial and isotrivial families. Let $U$ be an open subscheme of a normal, irreducible, locally noetherian scheme $S$ over a field $k$ of characteristic zero and let $C\ra U$ be a family of stable curves of compact type. In the second part of the paper, a monodromy criterion is given for extending $C\to U$ to a family of stable curves of compact type over $S$.

math.AG↗
arXiv2007

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 $Γ_{g,n}$, for $2g-2+n>0$, be the Teichmüller 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 $Γ_{g,n}$ 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 $Γ_{g,n}\ra Sp_{2g}(Z)$. 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.

math.AG↗