  This paper stems from the observation (arising from work of T. Delzant) that
"most" K\"ahler groups virtually algebraically fiber, i.e. admit a finite index
subgroup that maps onto $\Bbb{Z}$ with finitely generated kernel. For the
remaining ones, the Albanese dimension of all finite index subgroups is at most
one, i.e. they have virtual Albanese dimension $va(G) \leq 1$. We show that the
existence of algebraic fibrations has implications in the study of coherence
and higher BNSR invariants of the fundamental group of aspherical K\"ahler
surfaces. The class of K\"ahler groups with $va(G) \leq 1$ includes virtual
surface groups. Further examples exist; nonetheless they exhibit a strong
relation with surface groups. In fact, we show that the Green--Lazarsfeld sets
of groups with $va(G) = 1$ (virtually) coincide with those of surface groups,
and furthermore that the only virtually RFRS groups with $va(G) = 1$ are
virtually surface groups.
