This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups $G$ virtually algebraically fiber, that is, admit a finite index subgroup that maps onto $\mathbb{Z}$ 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 $va(G)\leqslant 1$. 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 Kähler surfaces. The class of Kähler groups with $va(G)=1$ 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.

中文翻译：

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KÄHLER 群的虚拟代数纤维化

本文源于对“大多数”Kähler 集团的观察（源自 Delzant 的工作）$G$实际上是代数纤维，也就是说，允许一个有限索引子群映射到$\mathbb{Z}$具有有限生成的内核。对于其余的，所有有限索引子群的 Albanese 维数至多为 1，即它们有虚拟的阿尔巴尼亚维$va(G)\leqslant 1$. 我们表明代数纤维化的存在对研究非球面 Kähler 表面的基本组的相干性和更高的 BNSR 不变量具有重要意义。Kähler 群的类$va(G)=1$包括虚拟表面组。还有更多的例子；尽管如此，它们与表面基团表现出很强的关系。事实上，我们证明了 Green-Lazarsfeld 组具有$va(G)=1$（实际上）与表面基团的那些一致，此外，只有几乎 RFRS 基团具有$va(G)=1$实际上是表面基团。