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A $π_1$ obstruction to having finite index monodromy and an unusual subgroup of infinite index in $\textrm{Mod}(Σ_g)$

Ishan Banerjee

Abstract

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.

A $π_1$ obstruction to having finite index monodromy and an unusual subgroup of infinite index in $\textrm{Mod}(Σ_g)$

Abstract

Let be an algebraic surface with an ample line bundle on . Let be the \emph{geometric monodromy} group associated to family of nonsingular curves in that are zero loci of sections of . We provide obstructions to being finite index in the mapping class group. We also show that for any , the image of monodromy is finite index in appropriate subgroups of the quotient of the mapping class group by the th term of the Johnson filtration assuming that 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.
Paper Structure (9 sections, 21 theorems, 16 equations)

This paper contains 9 sections, 21 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Vanishing cycles and their relation to monodromy
  3. Existing work and a general question
  4. Acknowledgements
  5. Containment
  6. Infinite index
  7. $\Gamma(X, \mathcal{L}^{\otimes r})$ contains framed mapping class groups of subsurfaces
  8. The subgroups $\mathcal{A}(V)$ and Johnson homomorphisms
  9. Surjectivity of monodromy groups into quotients

Key Result

Theorem 1

Let $i: C \to X$ be the inclusion map. Let $i_*: \pi_1 (C) \to \pi_1(X)$ denote the induced map on $\pi_1$. The group $\Gamma (X, \mathcal{L})$ is contained in the group $\mathcal{A}[i_*]$ where $\mathcal{A}[i_*] \subseteq \textrm{Mod}(C)$ is the subgroup generated by Dehn twists about nonseparating

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Proposition 1
  • Proposition 2
  • Definition 1
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 31 more