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

TL;DR

This paper explores the relationship between a conjecture on the mapping class group's representations and the existence of certain surface-by-surface groups that do not virtually algebraically fiber, highlighting open questions in geometric group theory.

Contribution

It establishes an equivalence between a conjecture on Prym representations and the existence of specific surface-by-surface groups lacking virtual algebraic fibering.

Findings

The Putman--Wieland conjecture relates to the nonexistence of finite orbits in higher Prym representations.

Existence of free-by-free and free-by-surface groups that do not algebraically fiber.

The question of whether such non-fibered surface-by-surface groups exist remains open.

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).