Kähler groups and subdirect products of surface groups
| Autor: | Claudio Llosa Isenrich |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Surface (mathematics)
Pure mathematics Geometric topology (object) branched covers surface groups Kähler groups Algebraic geometry 20J05 32J27 20F65 (Primary) 32Q15 20J05 (Secondary) Mathematics - Geometric Topology Mathematics - Algebraic Geometry 32J27 homological finiteness properties 32Q15 Geometry and Topology 20F65 Mathematics - Group Theory Group theory Mathematics |
| Zdroj: | Geom. Topol. 24, no. 2 (2020), 971-1017 Geometry and Topology |
| Popis: | We present a construction that produces infinite classes of K\"ahler groups that arise as fundamental groups of fibres of maps to higher dimensional tori. Following the work of Delzant and Gromov, there is great interest in knowing which subgroups of direct products of surface groups are K\"ahler. We apply our construction to obtain new classes of irreducible, coabelian K\"ahler subgroups of direct products of $r$ surface groups. These cover the full range of possible finiteness properties of irreducible subgroups of direct products of $r$ surface groups: For any $r\geq 3$ and $2\leq k \leq r-1$, our classes of subgroups contain K\"ahler groups that have a classifying space with finite $k$-skeleton while not having a classifying space with finitely many $(k+1)$-cells. We also address the converse question of finding constraints on K\"ahler subdirect products of surface groups and, more generally, on homomorphisms from K\"ahler groups to direct products of surface groups. We show that if a K\"ahler subdirect product of $r$ surface groups admits a classifying space with finite $k$-skeleton for $k>\frac{r}{2}$, then it is virtually the kernel of an epimorphism from a direct product of surface groups onto a free abelian group of even rank. Comment: 30 pages, V4: Some results were strengthened (in particular, Theorems 1.2 and 1.5 now include some additional consequences). Minor corrections and improvements to the exposition. Final accepted version, to appear in Geometry & Topology |
| Databáze: | OpenAIRE |
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