Virtual algebraic fibrations of K\"ahler groups

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.