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We show that a conjecture of Putman--Wieland, which posits the nonexistenceof finite orbits for higher Prym representations of the mapping class group, isequivalent to the existence of surface-by-surface and surface-by-free groupswhich do not virtually algebraically fiber. While the question about theexistence of such groups remains open, we will show that there existfree-by-free and free-by-surface groups which do not algebraically fiber (hencefail to be virtually RFRS), and study some properties of the normal subgroupsof those groups, as well as those of potential counterexamples to thePutman--Wieland conjecture.

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