A surface-by-surface group is an extension of a non-trivial orientable closed surface group by another such group. It is an open question as to whether every such group contains a free abelian subgroup of rank 2. We show that, for given base an fibre genera, all but finitely many isomorphism classes of surface-by-surface group contain such an abelian subgroup. This can be rephrased in terms of atoroidal surface bundles over surfaces, or in terms of purely loxodromic surface subgroups of the mapping class groups.
First draft: June 2007. Revised: October 2009. Geom. Funct. Anal. 19 (2009) 943-988.