A construction of representations of $3$-manifold groups into $\mathrm{PU}(2,1)$ through Lefschetz fibrations

Ruben Dashyan

doi:10.5802/afst.1751

A construction of representations of

3

-manifold groups into

PU (2, 1)

through Lefschetz fibrations

Ruben Dashyan ¹

¹ Institut de Mathématiques de Jussieu-Paris Rive Gauche, 4 place Jussieu, boîte courrier 247, 75252 Paris Cedex 5, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 4, pp. 769-803.

Abstract
Abstract

We obtain infinitely many (non-conjugate) representations of $3$ -manifold fundamental groups into a lattice in $Isom (ℍ_{ℂ}^{2})$ , the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of the projective plane along an arrangement of hyperplanes constructed by Hirzebruch. The $3$ -manifolds are related to a Lefschetz fibration of the complex hyperbolic orbifold.

Nous obtenons un nombre infini de représentations non-conjuguées de groupes fondamentaux de $3$ -variétés dans un réseau du groupe $Isom (ℍ_{ℂ}^{2})$ des isométries holomorphes de l’espace hyperbolique complexe. Ce réseau est un groupe fondamental orbifold d’un revêtement du plan projectif ramifié le long d’un arrangement de droites, construit par Hirzebruch. Les $3$ -variétés sont liées à une fibration de Lefschetz de l’orbifold hyperbolique complexe.

Received: 2019-10-13
Accepted: 2021-07-22
Published online: 2023-11-13

DOI: 10.5802/afst.1751

Author's affiliations:

Ruben Dashyan ¹

¹ Institut de Mathématiques de Jussieu-Paris Rive Gauche, 4 place Jussieu, boîte courrier 247, 75252 Paris Cedex 5, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Ruben Dashyan},
     title = {A construction of representations of $3$-manifold groups into $\mathrm{PU}(2,1)$ through {Lefschetz} fibrations},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {769--803},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Ruben Dashyan. A construction of representations of $3$-manifold groups into $\mathrm{PU}(2,1)$ through Lefschetz fibrations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 4, pp. 769-803. doi : 10.5802/afst.1751. https://afst.centre-mersenne.org/articles/10.5802/afst.1751/

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