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Virtual algebraic fibrations of surface-by-surface groups and orbits of the mapping class group

Robert Kropholler, Stefano Vidussi, Genevieve Walsh

Abstract

We show that a conjecture of Putman--Wieland, which posits the nonexistence of finite orbits for higher Prym representations of the mapping class group, is equivalent to the existence of surface-by-surface and surface-by-free groups which do not virtually algebraically fiber. While the question about the existence of such groups remains open, we will show that there exist free-by-free and free-by-surface groups which do not algebraically fiber (hence fail to be virtually RFRS).

Virtual algebraic fibrations of surface-by-surface groups and orbits of the mapping class group

Abstract

We show that a conjecture of Putman--Wieland, which posits the nonexistence of finite orbits for higher Prym representations of the mapping class group, is equivalent to the existence of surface-by-surface and surface-by-free groups which do not virtually algebraically fiber. While the question about the existence of such groups remains open, we will show that there exist free-by-free and free-by-surface groups which do not algebraically fiber (hence fail to be virtually RFRS).

Paper Structure

This paper contains 7 sections, 14 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Coinvariants and homology
  4. Virtual excessive homology and the Putman--Wieland conjecture
  5. The Putman-Wieland Conjecture
  6. Connections with virtually excessive homology
  7. Extensions with no virtual excessive homology

Key Result

Theorem 1.1

For every $g \geq 2$ the Putman--Wieland conjecture NFO($g,0,0$) holds if and only if there exists a surface-by-surface or a surface-by-free group $G$ with fiber of genus $g$ and no virtual excessive homology or, equivalently, that is not virtually algebraically fibered.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.0
  • Proposition 1.0
  • Theorem 1.1
  • Definition
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark
  • ...and 18 more