Branched covers of elliptic curves and Kähler groups with exotic finiteness properties

Llosa Isenrich, Claudio

Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 335-363.

Nous construisons des groupes kähleriens ayant des propriétés de finitude arbitraires en considérant des applications holomorphes de produits de surfaces de Riemann vers une courbe elliptique : pour tout $r\ge 3$, nous obtenons une grande classe de groupes kähleriens qui ont un espace classifiant avec un $(r-1)$-squelette fini, mais n’ont aucun espace classifiant avec un nombre fini de $r$-cellules. Nous décrivons des invariants qui distinguent beaucoup de ces groupes. Notre construction est inspirée par les exemples de Dimca, Papadima et Suciu.

We construct Kähler groups with arbitrary finiteness properties by mapping products of closed Riemann surfaces holomorphically onto an elliptic curve: for each $r\ge 3$, we obtain large classes of Kähler groups that have classifying spaces with finite $(r-1)$-skeleton but do not have classifying spaces with finitely many $r$-cells. We describe invariants which distinguish many of these groups. Our construction is inspired by examples of Dimca, Papadima and Suciu.

Classification : 32J27, 20J05, 20F65
Keywords: Kähler groups, Homological finiteness properties, Branched covers
Mot clés : Groupes kähleriens, propriétés de finitude homologiques, Revêtements ramifiés

[1] Bestvina, Mladen; Brady, Noel Morse theory and finiteness properties of groups, Invent. Math., Volume 129 (1997) no. 3, pp. 445-470 | DOI | MR | Zbl

[2] Biswas, Indranil; Mj, Mahan; Pancholi, Dishant Homotopical Height, Int. J. Math., Volume 25 (2014) no. 13, 1450123, 43 pages (Art. ID 1450123, 43 p.) | MR | Zbl

[3] Bridson, Martin R.; Howie, James Normalisers in limit groups, Math. Ann., Volume 337 (2007) no. 2, pp. 385-394 | MR | Zbl

[4] Bridson, Martin R.; Howie, James; Miller, Charles F. III; Short, Hamish The subgroups of direct products of surface groups, Geom. Dedicata, Volume 92 (2002), pp. 95-103 | DOI | MR | Zbl

[5] Bridson, Martin R.; Howie, James; Miller, Charles F. III; Short, Hamish On the finite presentation of subdirect products and the nature of residually free groups, Am. J. Math., Volume 135 (2013) no. 4, pp. 891-933 | DOI | MR | Zbl

[6] Brown, Kenneth S. Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, 1982, x+306 pages | Zbl

[7] Dimca, Alexandru; Papadima, Stefan; Suciu, Alexander I. Quasi-Kähler Bestvina–Brady groups, J. Algebr. Geom., Volume 17 (2008) no. 1, pp. 185-197 | Zbl

[8] Dimca, Alexandru; Papadima, Stefan; Suciu, Alexander I. Non-finiteness properties of fundamental groups of smooth projective varieties, J. Reine Angew. Math., Volume 629 (2009), pp. 89-105 | MR | Zbl

[9] Farb, Benson; Margalit, Dan A primer on Mapping Class groups, Princeton Mathematics Series, 49, Princeton University Press, 2012 | MR | Zbl

[10] Jaco, William On certain subgroups of the fundamental group of a closed surface, Proc. Camb. Philos. Soc., Volume 67 (1970), pp. 17-18 | DOI | MR | Zbl

[11] Llosa Isenrich, Claudio Finite presentations for Kähler groups with arbitrary finiteness properties, J. Algebra, Volume 476 (2017), pp. 344-367 | DOI | MR | Zbl

[12] Suciu, Alexander I. Characteristic varieties and Betti numbers of free abelian covers, Int. Math. Res. Not., Volume 2014 (2014) no. 4, pp. 1063-1124 | DOI | MR | Zbl