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Cubulable Kähler groups

Thomas Delzant and Pierre Py

Geometry & Topology 23 (2019) 2125–2164

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.

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Kähler manifolds, cubical complexes
Mathematical Subject Classification 2010
Primary: 20F65, 32Q15
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
Thomas Delzant
Université de Strasbourg
Pierre Py
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria
Ciudad de México
Université de Strasbourg