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Quotients of the mapping class group by power subgroups

Javier Aramayona, Louis Funar

Abstract

We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup $\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that $\operatorname{Mod}_g^1 /\operatorname{Mod}_g^1[p]$ contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher-rank lattice, for all but finitely many explicit values of $p$. Next, we prove that $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$ contains a Kähler subgroup of finite index, for every $p\ge 2$ coprime with six. Finally, we observe that the existence of finite-index subgroups of $\operatorname{Mod}_g^0$ with infinite abelianization is equivalent to the analogous problem for $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$.

Quotients of the mapping class group by power subgroups

Abstract

We study the quotient of the mapping class group of a surface of genus with punctures, by the subgroup generated by the -th powers of Dehn twists. Our first main result is that contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher-rank lattice, for all but finitely many explicit values of . Next, we prove that contains a Kähler subgroup of finite index, for every coprime with six. Finally, we observe that the existence of finite-index subgroups of with infinite abelianization is equivalent to the analogous problem for .

Paper Structure

This paper contains 12 sections, 21 theorems, 63 equations.

Table of Contents

  1. Introduction and statements
  2. Non-lattice properties.
  3. Virtually Kähler quotients
  4. Abelianization of finite-index subgroups
  5. Preliminaries
  6. Power subgroups
  7. The Birman short exact sequence
  8. Algebraic version of the Birman exact sequence
  9. Normal subgroups via covers
  10. A Birman exact sequence for quotient subgroups
  11. Virtually Kähler groups
  12. Abelianization of finite index subgroups

Key Result

Theorem 1

Let $g,p \ge 2$, where $p \ge 4$ if $g=2$, and $p\notin \{2,3,4,6,8,12\}$ if $g\ge 3$. Then has a descending normal series $Q_1 \trianglerighteq Q_2\trianglerighteq \ldots$ such that $Q_{i+1}$ has infinite index in $Q_i$ for all $i\ge 1$.

Theorems & Definitions (41)

  • Theorem 1
  • Corollary 2
  • Proposition 3
  • Corollary 4
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Remark 1
  • Conjecture 1
  • Theorem 8
  • ...and 31 more