Virtual Algebraic Fibrations of Kähler Groups

Abstract

This paper stems from the observation (arising from work of T. Delzant) that "most" Kähler groups virtually algebraically fiber, i.e. admit a finite index subgroup that maps onto 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 Kähler surfaces. The class of Kähler 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 Q-solvable (vRFRS) elements are commensurable to surface groups, and we show that their Green-Lazarsfeld sets (virtually) coincide with those of surface groups.

BNS Invariants and Algebraic Fibrations of Group Extensions

“…There is one noteworthy class of surface bundles over a surface -namely those who admit a Kähler structure, e.g. Kodaira fibrations -where the BNS invariant is fully understood, thanks to the work of Delzant ([De10], see also [FV19]). To dovetail that result with Theorem 1, it is useful to introduce the following notation.…”

“…The first is the case of surface bundles with monodromy of types I and II in Johnson's trichotomy ( [Jo93]): their fundamental groups contain F 2 × F 2 as subgroup. The second case appears in [FV19] where the authors show that a Kodaira fibration that has virtual Albanese dimension 2 has fundamental group that is noncoherent. Theorem 1 allows us to proceed like in that paper to show the following: Corollary 3.…”

Virtually RFRS Mapping Tori and Coherence

“…This corollary entails that if X be a surface bundle over a surface with base and fiber of genus at least 2, either π 1 (X) is not coherent, or the monodromy along any simple curve on the base gives a non-nonpositively curved graph manifold. This further narrows the class of potential nontrivial examples of coherent fundamental groups of aspherical Kähler surfaces with positive irregularity, already reduced to the case of Kodaira fibrations of virtual Albanese dimension 1 with successive work of [Ka98,Ka13,Py16,FV19a].…”

“…Recently, some attention has been devoted to the study of coherence of some classes of nontrivial extensions (see [FV19a,FV19b,KrWa19]), including free-by-free groups and surface group-by-free groups. An overarching theme of these papers is that in presence of excessive homology (which in the case of Eq.…”

Incoherence of free-by-free and surface-by-free groups

“…This completes a result of [FV19], and it excludes the existence of nontrivial examples of coherent fundamental groups of aspherical Kähler surfaces with positive irregularity. (This question was reduced to the case of Kodaira fibrations of virtual Albanese dimension 1 by successive work of [Kap98,Kap13,Py16,FVar]. )…”