A $π_1$ obstruction to having finite index monodromy and an unusual subgroup of infinite index in $\textrm{Mod}(Σ_g)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929273462325248 |
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| author | Banerjee, Ishan |
| author_facet | Banerjee, Ishan |
| contents | Let $X$ be an algebraic surface with $\mathcal{L}$ an ample line bundle on $X$.
Let $Γ(X, \mathcal{L})$ be the \emph{geometric monodromy} group associated to family of nonsingular curves in $X$ that are zero loci of sections of $\mathcal{L}$. We provide obstructions to $Γ(X, \mathcal{L})$ being finite index in the mapping class group. We also show that for any $k \ge 0$, the image of monodromy is finite index in appropriate subgroups of the quotient of the mapping class group by the $k$th term of the Johnson filtration assuming that $\mathcal{L}$ is sufficiently ample. This enables us to construct several subgroups of the mapping class group with unusual properties, in some cases providing the first examples of subgroups with those properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A $π_1$ obstruction to having finite index monodromy and an unusual subgroup of infinite index in $\textrm{Mod}(Σ_g)$ Banerjee, Ishan Geometric Topology Algebraic Geometry Let $X$ be an algebraic surface with $\mathcal{L}$ an ample line bundle on $X$. Let $Γ(X, \mathcal{L})$ be the \emph{geometric monodromy} group associated to family of nonsingular curves in $X$ that are zero loci of sections of $\mathcal{L}$. We provide obstructions to $Γ(X, \mathcal{L})$ being finite index in the mapping class group. We also show that for any $k \ge 0$, the image of monodromy is finite index in appropriate subgroups of the quotient of the mapping class group by the $k$th term of the Johnson filtration assuming that $\mathcal{L}$ is sufficiently ample. This enables us to construct several subgroups of the mapping class group with unusual properties, in some cases providing the first examples of subgroups with those properties. |
| title | A $π_1$ obstruction to having finite index monodromy and an unusual subgroup of infinite index in $\textrm{Mod}(Σ_g)$ |
| topic | Geometric Topology Algebraic Geometry |
| url | https://arxiv.org/abs/2403.07280 |