Branched covers of elliptic curves and Kähler groups with exotic finiteness properties  [ Revêtements ramifiés de courbes elliptiques et groupes kähleriens avec propriétés de finitude exotiques ]
Annales de l'Institut Fourier, à paraître, 29 p.

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 r3, 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 r3, 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.

Reçu le : 2016-10-04
Révisé le : 2017-07-11
Accepté le : 2017-12-13
Publié le : 2019-03-08
Classification:  32J27,  20J05,  20F65
Mots clés: Groupes kähleriens, propriétés de finitude homologiques, Revêtements ramifiés
@unpublished{AIF_0__0_0_A9_0,
     author = {Llosa Isenrich, Claudio},
     title = {Branched covers of elliptic curves and K\"ahler groups with exotic finiteness properties},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Llosa Isenrich, Claudio. Branched covers of elliptic curves and Kähler groups with exotic finiteness properties. Annales de l'Institut Fourier, à paraître, 29 p.

[1] Bestvina, Mladen; Brady, Noel Morse theory and finiteness properties of groups, Invent. Math., Tome 129 (1997) no. 3, pp. 445-470 | Zbl 0888.20021

[2] Biswas, Indranil; Mj, Mahan; Pancholi, Dishant Homotopical Height, Int. J. Math., Tome 25 (2014) no. 13, 1450123, 43 pages (Art. ID 1450123, 43 p.) | Zbl 1308.32024

[3] Bridson, Martin R.; Howie, James Normalisers in limit groups, Math. Ann., Tome 337 (2007) no. 2, pp. 385-394 | Zbl 1139.20037

[4] Bridson, Martin R.; Howie, James; Miller, Charles F. Iii; Short, Hamish The subgroups of direct products of surface groups, Geom. Dedicata, Tome 92 (2002), pp. 95-103 | Zbl 1048.20009

[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., Tome 135 (2013) no. 4, pp. 891-933 | Zbl 1290.20024

[6] Brown, Kenneth S. Cohomology of groups, Springer, Graduate Texts in Mathematics, Tome 87 (1982), x+306 pages | Zbl 0584.20036

[7] Dimca, Alexandru; Papadima, Stefan; Suciu, Alexander I. Quasi-Kähler Bestvina–Brady groups, J. Algebr. Geom., Tome 17 (2008) no. 1, pp. 185-197 | Zbl 1176.20037

[8] Dimca, Alexandru; Papadima, Stefan; Suciu, Alexander I. Non-finiteness properties of fundamental groups of smooth projective varieties, J. Reine Angew. Math., Tome 629 (2009), pp. 89-105 | Zbl 1170.14017

[9] Farb, Benson; Margalit, Dan A primer on Mapping Class groups, Princeton University Press, Princeton Mathematics Series, Tome 49 (2012) | Zbl 1245.57002

[10] Jaco, William On certain subgroups of the fundamental group of a closed surface, Proc. Camb. Philos. Soc., Tome 67 (1970), p. 17-18 | Zbl 0184.48901

[11] Llosa Isenrich, Claudio Finite presentations for Kähler groups with arbitrary finiteness properties, J. Algebra, Tome 476 (2017), pp. 344-367 | Zbl 06688024

[12] Suciu, Alexander I. Characteristic varieties and Betti numbers of free abelian covers, Int. Math. Res. Not., Tome 2014 (2014) no. 4, pp. 1063-1124 | Zbl 1352.57004