## Geometry & Topology

### Cubulable Kähler groups

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.gt/1563242526

Digital Object Identifier
doi:10.2140/gt.2019.23.2125

Mathematical Reviews number (MathSciNet)
MR3988093

Zentralblatt MATH identifier
07094914

#### Citation

Delzant, Thomas; Py, Pierre. Cubulable Kähler groups. Geom. Topol. 23 (2019), no. 4, 2125--2164. doi:10.2140/gt.2019.23.2125. https://projecteuclid.org/euclid.gt/1563242526

#### References

• 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 https://www-fourier.ujf-grenoble.fr/~demailly/documents.html {\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)