Geometry & Topology

Cubulable Kähler groups

Thomas Delzant and Pierre Py

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Geom. Topol., Volume 23, Number 4 (2019), 2125-2164.

Received: 20 February 2018
Revised: 23 October 2018
Accepted: 2 December 2018
First available in Project Euclid: 16 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 32Q15: Kähler manifolds

Kähler manifolds cubical complexes


Delzant, Thomas; Py, Pierre. Cubulable Kähler groups. Geom. Topol. 23 (2019), no. 4, 2125--2164. doi:10.2140/gt.2019.23.2125.

Export citation


  • I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045–1087
  • J Amorós, M Burger, K Corlette, D Kotschick, D Toledo, Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs 44, Amer. Math. Soc., Providence, RI (1996)
  • W Ballmann, J Świ\katkowski, On groups acting on nonpositively curved cubical complexes, Enseign. Math. 45 (1999) 51–81
  • J Behrstock, R Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012) 339–356
  • N Bergeron, F Haglund, D T Wise, Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. 83 (2011) 431–448
  • N Bergeron, D T Wise, A boundary criterion for cubulation, Amer. J. Math. 134 (2012) 843–859
  • M R Bridson, On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc. 127 (1999) 2143–2146
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer (1999)
  • M R Bridson, J Howie, Subgroups of direct products of elementarily free groups, Geom. Funct. Anal. 17 (2007) 385–403
  • M R Bridson, J Howie, C F Miller, III, H Short, The subgroups of direct products of surface groups, Geom. Dedicata 92 (2002) 95–103
  • M R Bridson, J Howie, C F Miller, III, H Short, Subgroups of direct products of limit groups, Ann. of Math. 170 (2009) 1447–1467
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994)
  • M Burger, Fundamental groups of Kähler manifolds and geometric group theory, from “Séminaire Bourbaki 2009/2010”, Astérisque 339, Soc. Mat. de France, Paris (2011) Exposé 1022, 305–321
  • P-E Caprace, M Sageev, Rank rigidity for $\mathrm{CAT}(0)$ cube complexes, Geom. Funct. Anal. 21 (2011) 851–891
  • J A Carlson, D Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces, Inst. Hautes Études Sci. Publ. Math. 69 (1989) 173–201
  • F Catanese, Differentiable and deformation type of algebraic surfaces, real and symplectic structures, from “Symplectic $4$–manifolds and algebraic surfaces” (F Catanese, G Tian, editors), Lecture Notes in Math. 1938, Springer (2008) 55–167
  • I Chatterji, T Fernós, A Iozzi, The median class and superrigidity of actions on $\rm CAT(0)$ cube complexes, J. Topol. 9 (2016) 349–400 With an appendix by P-E Caprace
  • M W Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. 117 (1983) 293–324
  • M W Davis, T Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991) 347–388
  • M Davis, T Januszkiewicz, R Scott, Nonpositive curvature of blow-ups, Selecta Math. 4 (1998) 491–547
  • O Debarre, Tores et variétés abéliennes complexes, Cours Spécialisés 6, Soc. Mat. de France, Paris (1999)
  • T Delzant, Trees, valuations and the Green–Lazarsfeld set, Geom. Funct. Anal. 18 (2008) 1236–1250
  • T Delzant, L'invariant de Bieri–Neumann–Strebel des groupes fondamentaux des variétés kählériennes, Math. Ann. 348 (2010) 119–125
  • T Delzant, M Gromov, Cuts in Kähler groups, from “Infinite groups: geometric, combinatorial and dynamical aspects” (L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk, editors), Progr. Math. 248, Birkhäuser, Basel (2005) 31–55
  • J-P Demailly, Complex analytic and differential geometry, book project (2012) \setbox0\makeatletter\@url {\unhbox0
  • A Dimca, \commaaccentS Papadima, A I Suciu, Non-finiteness properties of fundamental groups of smooth projective varieties, J. Reine Angew. Math. 629 (2009) 89–105
  • C Dru\commaaccenttu, M Kapovich, Geometric group theory, American Mathematical Society Colloquium Publications 63, Amer. Math. Soc., Providence, RI (2018)
  • R Geoghegan, Topological methods in group theory, Graduate Texts in Mathematics 243, Springer (2008)
  • M Gromov, Hyperbolic groups, from “Essays in group theory” (S M Gersten, editor), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75–263
  • M Gromov, Kähler hyperbolicity and $L_2$–Hodge theory, J. Differential Geom. 33 (1991) 263–292
  • M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progr. Math. 152, Birkhäuser, Boston, MA (1999)
  • M Gromov, R Schoen, Harmonic maps into singular spaces and $p$–adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992) 165–246
  • M F Hagen, D T Wise, Cubulating hyperbolic free-by-cyclic groups: the general case, Geom. Funct. Anal. 25 (2015) 134–179
  • F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551–1620
  • C H Houghton, Ends of locally compact groups and their coset spaces, J. Austral. Math. Soc. 17 (1974) 274–284
  • V A Kaimanovich, W Woess, The Dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality, Probab. Theory Related Fields 91 (1992) 445–466
  • I Kapovich, The nonamenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups, Enseign. Math. 48 (2002) 359–375
  • M Kapovich, Energy of harmonic functions and Gromov's proof of Stallings' theorem, Georgian Math. J. 21 (2014) 281–296
  • A Kar, M Sageev, Ping pong on $\rm CAT(0)$ cube complexes, Comment. Math. Helv. 91 (2016) 543–561
  • N J Korevaar, R M Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993) 561–659
  • P H Kropholler, M A Roller, Relative ends and duality groups, J. Pure Appl. Algebra 61 (1989) 197–210
  • P Li, On the structure of complete Kähler manifolds with nonnegative curvature near infinity, Invent. Math. 99 (1990) 579–600
  • P Li, L-F Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992) 359–383
  • C Llosa Isenrich, Branched covers of elliptic curves and Kähler groups with exotic finiteness properties, Ann. Inst. Fourier (Grenoble) 69 (2019) 335–363
  • T Napier, M Ramachandran, Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems, Geom. Funct. Anal. 5 (1995) 809–851
  • T Napier, M Ramachandran, Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces, Geom. Funct. Anal. 11 (2001) 382–406
  • T Napier, M Ramachandran, Filtered ends, proper holomorphic mappings of Kähler manifolds to Riemann surfaces, and Kähler groups, Geom. Funct. Anal. 17 (2008) 1621–1654
  • T Napier, M Ramachandran, $L^2$ Castelnuovo–de Franchis, the cup product lemma, and filtered ends of Kähler manifolds, J. Topol. Anal. 1 (2009) 29–64
  • P Py, Coxeter groups and Kähler groups, Math. Proc. Cambridge Philos. Soc. 155 (2013) 557–566
  • M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. 71 (1995) 585–617
  • M Sageev, $\rm CAT(0)$ cube complexes and groups, from “Geometric group theory” (M Bestvina, M Sageev, K Vogtmann, editors), IAS/Park City Math. Ser. 21, Amer. Math. Soc., Providence, RI (2014) 7–54
  • P Scott, Ends of pairs of groups, J. Pure Appl. Algebra 11 (1977/78) 179–198
  • C Simpson, Lefschetz theorems for the integral leaves of a holomorphic one-form, Compositio Math. 87 (1993) 99–113
  • Y T Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980) 73–111
  • D T Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004) 150–214
  • D T Wise, From riches to raags: $3$–manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics 117, Amer. Math. Soc., Providence, RI (2012)