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This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups
virtually algebraically fiber, that is, admit a finite index subgroup that maps onto
with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, that is, they have virtual Albanese dimension
. 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 Kähler surfaces. The class of Kähler groups with
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
(virtually) coincide with those of surface groups, and furthermore that the only virtually RFRS groups with
are virtually surface groups.
Conversely, Catanese (2017) inquires about what conditions need to be satisfied by a group of that sort in order to be the fundamental group of a Kodaira fibration. In this short note we collect some restrictions on the image of the classifying map
$m \colon \Pi_b \to \Gamma_g$
in terms of the coinvariant homology of
. In particular, we observe that if π is the fundamental group of a Kodaira fibration with relative irregularity g−s, then
$g \leq 1+ 6s$
, and we show that this effectively constrains the possible choices for π, namely that there are group extensions as above that fail to satisfy this bound, hence it cannot be the fundamental group of a Kodaira fibration. A noteworthy consequence of this construction is that it provides examples of symplectic 4-manifolds that fail to admit a Kähler structure for reasons that eschew the usual obstructions.
Mendes Lopes and Pardini showed that minimal general type surfaces of Albanese dimension one have slopes K2/χ dense in the interval [2,8]. This result was completed to cover the admissible interval [2,9] by Roulleau and Urzua, who proved that surfaces with fundamental group equal to that of any curve of genus g ≥ 1 (in particular, having Albanese dimension one) give a set of slopes dense in [6,9]. In this note we provide a second construction that complements that of Mendes Lopes–Pardini, to recast a dense set of slopes in [8,9] for surfaces of Albanese dimension one. These surfaces arise as ramified double coverings of cyclic covers of the Cartwright–Steger surface.
It follows from earlier work of Silver and Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split, and whether it is totally split.
Let M be a closed 4-manifold with a free circle action. If the orbit manifold N3 satisfies an appropriate fibering condition, then we show how to represent a cone in H2(M; ℝ) by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández et al. In the case that M is the product 4-manifold S1 × N, our construction complements our previous results and allows us to determine completely the symplectic cone of such 4-manifolds.
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