arXiv subjects

Marco Boggi

Publications and source records attributed to Marco Boggi.

At least 19 recordsLinked to original sources

arXiv2026

Automorphisms of profinite mapping class groups

For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, such that $χ(S)=2-2g-n<0$, let $\mathrm{P}Γ(S)$ be the pure mapping class group of $S$ and $\mathrm{P}\widehatΓ(S)$ and $\mathrm{P}\checkΓ(S)$ be, respectively, its profinite and its congruence completions, the latter being identified with the image of the natural representation $\mathrm{P}\widehatΓ(S)\to\operatorname{Out}({\widehatπ}_1(S))$ (where ${\widehatπ}_1(S)$ is the profinite completion of the fundamental group of $S$). We determine the automorphism groups of procongruence completions under a natural rigidity condition, and show that the profinite Grothendieck-Teichmüller group embeds into the outer automorphism group of the profinite completion. Let $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))$ and $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\checkΓ(S))$ be the groups of outer automorphisms which preserve the conjugacy class of a procyclic subgroup generated by a nonseparating Dehn twist (a condition trivially satisfied for $g=0$). Our main result gives that, for $χ(S)<g-2$ and $(g,n)\neq (1,2)$, there is a natural isomorphism: \[\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\checkΓ(S))\congΣ_n\times\widehat{\operatorname{GT}},\] where $Σ_{n}$ is the symmetric group on $n$ letters and $\widehat{\operatorname{GT}}$ denotes the profinite Grothendieck-Teichmüller group. We also prove that, for $χ(S)<g-2$, there is a natural faithful representation $\widehat{\operatorname{GT}}\hookrightarrow\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))$.

math.GT↗

arXiv2026

Lannes' $T$-functor and mod-$p$ cohomology of profinite groups

The Lannes-Quillen theorem relates the mod-$p$ cohomology of a finite group $G$ with the mod-$p$ cohomology of centralizers of abelian elementary $p$-subgroups of $G$, for $p>0$ a prime number. This theorem was extended to profinite groups whose mod-$p$ cohomology algebra is finitely generated by Henn. In a weaker form, the Lannes-Quillen theorem was then extended by Symonds to arbitrary profinite groups. Building on Symonds' result, we formulate and prove a full version of this theorem for all profinite groups. For this purpose, we develop a theory of products for families of discrete torsion modules, parameterized by a profinite space, which is dual, in a very precise sense, to the theory of coproducts for families of profinite modules, parameterized by a profinite space, developed by Haran, Melnikov and Ribes. In the last section, we give applications to the problem of conjugacy separability of $p$-torsion elements and finite $p$-subgroups.

math.GR↗

arXiv2026

A congruence subgroup property for symmetric mapping class groups

We prove the congruence subgroup property for the centralizer of a finite subgroup $G$ in the mapping class group of a hyperbolic oriented and connected surface of finite topological type $S$ such that the genus of the quotient surface $S/G$ is at most $2$. As an application, we show that torsion elements in the mapping class group of a surface of genus $\leq 2$ are conjugacy distinguished.

math.GT↗

arXiv2026

Automorphisms of the procongruence pants complex

We show that every automorphism of the congruence completion of the extended mapping class group that preserves the set of conjugacy classes of procyclic groups generated by Dehn twists is inner, and that its automorphism group is naturally isomorphic to the automorphism group of the procongruence pants complex. In the genus-zero case, we prove the stronger result that all automorphisms of the profinite completion of the extended mapping class group are inner.

math.GT↗

arXiv2025

A generating set for the Johnson kernel

For a connected orientable hyperbolic surface $S$ without boundary and of finite topological type, the Johnson kernel ${\mathcal K}(S)$ is the subgroup of the mapping class group of $S$ generated by Dehn twists about separating simple closed curves on $S$. We prove that ${\mathcal K}(S)$ is generated by the Dehn twists about separating simple closed curves on $S$ bounding either: a closed subsurface of genus $1$ or $2$; a closed subsurface of genus $1$ minus one point; a closed disc minus two points.

math.GT↗

arXiv2024

Finite subgroups of the profinite completion of good groups

Let $G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$ induces a bijective correspondence between conjugacy classes of finite $p$-subgroups of $G$ and those of its profinite completion $\hat{G}$. Moreover, we prove that the centralizers and normalizers in $\hat{G}$ of finite $p$-subgroups of $G$ are the closures of the respective centralizers and normalizers in $G$. With somewhat more restrictive hypotheses, we prove the same results for finite solvable subgroups of $G$. In the last section, we give a few applications of this theorem to hyperelliptic mapping class groups and virtually compact special toral relatively hyperbolic groups (these include fundamental groups of $3$-orbifolds and of uniform standard arithmetic hyperbolic orbifolds).

math.GR↗

arXiv2024

Characterizing closed curves on Riemann surfaces via homology groups of coverings

Let $S$ be a hyperbolic oriented Riemann surface of finite type. The main purpose of this paper is to show that non-trivial geometric intersection between closed curves on $S$ is detected by some symplectic submodules they naturally determine in the homology groups of the compactifications of unramified $p$-coverings of $S$, for $p\geq 2$ a fixed prime. In particular, this gives a characterization of simple closed curves on $S$ in terms of homology groups of $p$-coverings. We then define a $p$-adic Reidemeister pairing on the fundamental group of $S$ and show that the free homotopy classes of two loops have trivial geometric intersection if and only if they are orthogonal with respect to this pairing. As an application, we give a geometric argument to prove that oriented surface groups are conjugacy $p$-separable (a combinatorial proof of this fact was recentely given by Paris).

math.AT↗

arXiv2024

Notes on hyperelliptic mapping class groups

Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus $2$ case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.

math.GT↗

arXiv2024

Generating the homology of covers of surfaces

Putman and Wieland conjectured that if $\tildeΣ \rightarrow Σ$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of $H_1(\tildeΣ;\mathbb{Q})$ under the action of lifts to $\tildeΣ$ of mapping classes on $Σ$ are infinite. We prove that this holds if $H_1(\tildeΣ;\mathbb{Q})$ is generated by the homology classes of lifts of simple closed curves on $Σ$. We also prove that the subspace of $H_1(\tildeΣ;\mathbb{Q})$ spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that $H_1(\tildeΣ;\mathbb{Q})$ is generated by the homology classes of lifts of loops on $Σ$ lying on subsurfaces homeomorphic to 3-holed spheres.

math.GT↗

arXiv2023

Automorphisms of procongruence curve and pants complexes

In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.

math.GT↗

arXiv2023

A remark on the homology of finite coverings of a surface

Let $p: S\to S_g$ be a finite covering of an orientable closed surface of genus $g$. We prove that, for $g\geq 3$, the rational homology group $H_1(S;{\mathbb Q})$ is generated by cycles supported on simple closed curves $γ\subset S$ such that $p(γ)$ is contained in a $3$-punctured, genus $0$ subsurface of $S_g$. In particular, this answers positively, for $g\geq 3$ and rational coefficients, a question by Autumn Kent.

math.GT↗

arXiv2022

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 $\operatorname{Mod}(S_g,B)^ι$, which is a subgroup of $\operatorname{Mod}(S_g\smallsetminus B)$ associated to a given hyperelliptic involution $ι$ 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 $g\geq 2$, there are virtual linear representations of the hyperelliptic mapping class group with nontrivial finite orbits, associated to $G$-coverings of $(S_g,ι)$ ramified over the locus of Weierstrass points.

math.AT↗

arXiv2022

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.

math.AG↗

arXiv2018

The congruence subgroup property for the hyperelliptic modular group: the open surface case

Let ${\cal M}_{g,n}$ and ${\cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $Γ_{g,n}$ and $H_{g,n}$, the so called Teichm{ü}ller modular group and hyperelliptic modular group. A choice of base point on ${\cal H}_{g,n}$ defines a monomorphism $H_{g,n}\hookrightarrowΓ_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmüller group $Γ_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. As a subgroup of $Γ_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookrightarrow\operatorname{Out}(π_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^λ$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of $π_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}\to\operatorname{Out}(π_1(S_{g,n})/K)$ is contained in $H^λ$. The main result of the paper is an affirmative answer to this question for $n\geq 1$.

math.AG↗

arXiv2018

Congruence topologies on the mapping class group

Let $Γ(S)$ be the pure mapping class group of a connected orientable surface $S$ of negative Euler characteristic. For ${\mathscr C}$ a class of finite groups, let $\hatπ_1(S)^{\mathscr C}$ be the pro-${\mathscr C}$ completion of the fundamental group of $S$. The \emph{${\mathscr C}$-congruence completion $\checkΓ(S)^{\mathscr C}$ of $Γ(S)$} is the profinite completion induced by the embedding $Γ(S)\hookrightarrow{\operatorname{Out}}(\hatπ_1(S)^{\mathscr C})$. In this paper, we begin a systematic study of such completions for different ${\mathscr C}$. We show that the combinatorial structure of the profinite groups $\checkΓ(S)^{\mathscr C}$ closely resemble that of $Γ(S)$. A fundamental question is how ${\mathscr C}$-congruence completions compare with pro-${\mathscr C}$ completions. Even though, in general (e.g.\ for ${\mathscr C}$ the class of finite solvable groups), $\checkΓ(S)^{\mathscr C}$ is not even virtually a pro-${\mathscr C}$ group, we show that, for ${\mathbb Z}/2\in{\mathscr C}$, $g(S)\leq 2$ and $S$ open, there is a natural epimorphism from the ${\mathscr C}$-congruence completion $\checkΓ(S)(2)^{\mathscr C}$ of the abelian level of order $2$ to its pro-${\mathscr C}$ completion $\widehatΓ(S)(2)^{\mathscr C}$. In particular, this is an isomorphism for the class of finite groups and for the class of $2$-groups. Moreover, in these two cases, the result also holds for a closed surface.

math.GR↗

arXiv2017

A restricted Magnus property for profinite surface groups

Magnus proved that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle^F=\langle y\rangle^F$, then $x$ is conjugated either to $y$ or $y^{-1}$. More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for ${\mathscr S}$ a class of finite groups, we prove that, if $x$ and $y$ are \emph{algebraically simple} elements of the pro-${\mathscr S}$ completion $\hatΠ^{\mathscr S}$ of an orientable surface group $Π$, such that, for all $n\in{\mathbb N}$, there holds $\langle x^n\rangle^{\hatΠ^{\mathscr S}}=\langle y^n\rangle^{\hatΠ^{\mathscr S}}$, then $x$ is conjugated to $y^s$ for some $s\in(\hat{\mathbb Z}^{\mathscr S})^\ast$. As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions. The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists to profinite Dehn multitwists.

math.GR↗

arXiv2017

Deforming a canonical curve inside a quadric

Let $C\subset{\mathbb P}^{g-1}$ be a canonically embedded nonsingular nonhyperelliptic curve of genus $g$ and let $X\subset{\mathbb P}^{g-1}$ be a quadric containing $C$. Our main result states among other things that the Hilbert scheme of $X$ is at $[C\subset X]$ a local complete intersection of dimension $g^2-1$, and is smooth when $X$ is. It also includes the assertion that the minimal obstruction space for this deformation problem is in fact the full associated $\operatorname{Ext}^1$-group and that in particular the deformations of $C$ in $X$ are obstructed in case $C$ meets the singular locus of $X$. As we will show in a forthcoming paper, this has applications of a topological nature.

math.AG↗

arXiv2016

Continuous cohomology and homology of profinite groups

We develop cohomological and homological theories for a profinite group $G$ with coefficients in the Pontryagin dual categories of pro-discrete and ind-profinite $G$-modules, respectively. The standard results of group (co)homology hold for this theory: we prove versions of the Universal Coefficient Theorem, the Lyndon-Hochschild-Serre spectral sequence and Shapiro's Lemma.

math.GR↗