On virtual properties of Kähler groups

Stefan Friedl, Stefano Vidussi

Published 2017-04-24Version 1

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 one. We show that the existence of (virtual) algebraic fibrations has implications in the study of coherence and of higher BNSR invariants of the fundamental group of aspherical K\"ahler surfaces. The class of K\"ahler groups of virtual Albanese dimension one contains groups commensurable to surface groups. It is not hard to give further (albeit unsophisticated) examples; however, groups of this class exhibit strong similarities with surface groups. In fact, we show that its only virtually residually finite $\Bbb{Q}$-solvable (vRFRS) elements are commensurable to surface groups, and we show that their Green-Lazarsfeld sets (virtually) coincide with those of surface groups.