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Prym Representations of the Handlebody Group

Philipp Bader

Abstract

Let $S$ be an oriented, closed surface of genus $g.$ The mapping class group of $S$ is the group of orientation preserving homeomorphisms of $S$ modulo isotopy. In 1997, Looijenga introduced the Prym representations, which are virtual representations of the mapping class group that depend on a finite, abelian group. Let $V$ be a genus $g$ handlebody with boundary $S$. The handlebody group is the subgroup of those mapping classes of $S$ that extend over $V.$ The twist group is the subgroup of the handlebody group generated by twists about meridians. Here, we restrict the Prym representations to the handlebody group and further to the twist group. We determine the image of the representations in the cyclic case.

Prym Representations of the Handlebody Group

Abstract

Let be an oriented, closed surface of genus The mapping class group of is the group of orientation preserving homeomorphisms of modulo isotopy. In 1997, Looijenga introduced the Prym representations, which are virtual representations of the mapping class group that depend on a finite, abelian group. Let be a genus handlebody with boundary . The handlebody group is the subgroup of those mapping classes of that extend over The twist group is the subgroup of the handlebody group generated by twists about meridians. Here, we restrict the Prym representations to the handlebody group and further to the twist group. We determine the image of the representations in the cyclic case.
Paper Structure (9 sections, 31 theorems, 97 equations, 12 figures)

This paper contains 9 sections, 31 theorems, 97 equations, 12 figures.

Table of Contents

  1. Preliminaries
  2. Mapping class group, handlebody group and twist group
  3. Definition of Prym representations
  4. Statements of the results
  5. Image of the Handlebody Group
  6. The lower right block $D$
  7. The upper right block $B$
  8. The genus 2 case
  9. Image of the Twist Group

Key Result

Theorem 1

For every $d \in \mathbb{N},$ and every genus $g \ge 2,$ there is a finite index subgroup $\Gamma$ of the punctured handlebody group $\mathcal{H}_V(S,x_0)$ and a representation whose image is the subgroup Here, $\zeta_d$ is a $d^{\text{th}}$ root of unity and $D^*$ is the adjoint matrix. Equivalently, there is a finite index subgroup of the (non-punctured) handlebody group $\mathcal{H}_V(S)$ tha

Figures (12)

  • Figure 1: The curves $E_{\pm i}$
  • Figure 2: The covering space $\widetilde{S}$ and the curves $e_{\pm i}$
  • Figure 3: The lift of $\alpha$
  • Figure 4: The subsurfaces $S'$ and $T$
  • Figure 5: The subsurfaces $S_i$ and $M$
  • ...and 7 more figures

Theorems & Definitions (53)

  • Theorem : Main
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Remark
  • ...and 43 more