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Homological representations of low genus mapping class groups

Trent Lucas

Abstract

Given a finite group $G$ acting on a surface $S$, the centralizer of G in the mapping class group $\textrm{Mod}(S)$ has a natural representation given by its action on the homology $H_1(S; \mathbb{Q})$. We consider the question of whether this representation has arithmetic image. Several authors have given positive and negative answers to this question. We give a complete answer when S has genus at most 3.

Homological representations of low genus mapping class groups

Abstract

Given a finite group acting on a surface , the centralizer of G in the mapping class group has a natural representation given by its action on the homology . We consider the question of whether this representation has arithmetic image. Several authors have given positive and negative answers to this question. We give a complete answer when S has genus at most 3.
Paper Structure (13 sections, 8 theorems, 58 equations, 8 figures, 2 tables)

This paper contains 13 sections, 8 theorems, 58 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Equivariant mapping classes and homology
  3. Homology as a representation
  4. The action of mapping classes
  5. Dividing up the cases
  6. First reductions
  7. Prior work
  8. Determining the target algebraic group
  9. Direct proofs of arithmeticity
  10. Lifting twists
  11. Computing the action of lifted twists
  12. Checking finite index
  13. Results

Key Result

Theorem 1.2

If $S$ has genus $g \leq 3$, then for every action of a finite group $G$ on $S$ and for every irreducible $\mathbb{Q}$-representation $V$ of $G$, the image of the map $\Phi_V:\mathop{\mathrm{Mod}}\nolimits(S)^G \rightarrow \mathop{\mathrm{Aut}}\nolimits_{\mathbb{Q}[G]}(H_1(S;\mathbb{Q})_V, \hat{i})$

Figures (8)

  • Figure 1: The action of $C_2$ in Case 2.b.
  • Figure 2: The action of $C_2$ in Case 3.b.
  • Figure 3: Verifying that the desired twists lie in $\mathop{\mathrm{Mod}}\nolimits(S)^{C_2}$.
  • Figure 4: The action of $C_3$ in Case 3.e.
  • Figure 5: Building the polygon $P$, with $h=1$ and $n=3$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 1.2
  • Theorem 2.1: Chevalley-Weil, Gaschütz, Koberda-Silberstein
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 3 more