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Cubulable Kähler groups

Thomas Delzant and Pierre Py

Geometry & Topology 23 (2019) 2125–2164
Abstract

We prove that a Kähler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite-index subgroup isomorphic to a direct product of surface groups, possibly with a free abelian factor. Similarly, we prove that a closed aspherical Kähler manifold with a cubulable fundamental group has a finite cover which is biholomorphic to a topologically trivial principal torus bundle over a product of Riemann surfaces. Along the way, we prove a factorization result for essential actions of Kähler groups on irreducible, locally finite CAT(0) cubical complexes, under the assumption that there is no fixed point in the visual boundary.

Keywords
Kähler manifolds, cubical complexes
Mathematical Subject Classification 2010
Primary: 20F65, 32Q15
References
Publication
Received: 20 February 2018
Revised: 23 October 2018
Accepted: 2 December 2018
Published: 17 June 2019
Proposed: Jean-Pierre Otal
Seconded: Martin Bridson, Benson Farb
Authors
Thomas Delzant
IRMA
Université de Strasbourg
CNRS
Strasbourg
France
Pierre Py
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria
Ciudad de México
Mexico
IRMA
Université de Strasbourg
CNRS
Strasbourg
France