Pseudo-Anosov subgroups of surface bundles over tori

In 2002, Farb and Mosher introduced the notion of convex cocompactness in the mapping class group to capture coarse geometric information of the associated surface group extensions. Convex cocompact subgroups are necessarily finitely generated and purely pseudo-Anosov, but it is an open question whether the converse is true. Several partial results are known in certain settings, however. For example, work of Dowdall, Kent, Leininger, Russell, and Schleimer give a positive answer for subgroups of fibered 3-manifold groups (aka surface-by-cyclic extensions) naturally embedded in punctured mapping class groups via the Birman exact sequence. We present a generalization of this in the setting of surface-by-abelian extensions.