Cubulable Kähler groups

Delzant, Thomas¹ ; Py, Pierre²

¹ IRMA, Université de Strasbourg, CNRS, Strasbourg, France
² Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México, Mexico, IRMA, Université de Strasbourg, CNRS, Strasbourg, France

Geometry & topology, Tome 23 (2019) no. 4, pp. 2125-2164.

Résumé

We prove that a Kähler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a $CAT\left(0\right)$ 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 $CAT\left(0\right)$ cubical complexes, under the assumption that there is no fixed point in the visual boundary.

Détail

DOI : 10.2140/gt.2019.23.2125

Classification : 20F65, 32Q15
Keywords: Kähler manifolds, cubical complexes

Bibliographie

[1] I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045

[2] J Amorós, M Burger, K Corlette, D Kotschick, D Toledo, Fundamental groups of compact Kähler manifolds, 44, Amer. Math. Soc. (1996) | DOI

[3] W Ballmann, J Świątkowski, On groups acting on nonpositively curved cubical complexes, Enseign. Math. 45 (1999) 51

[4] J Behrstock, R Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012) 339 | DOI

[5] N Bergeron, F Haglund, D T Wise, Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. 83 (2011) 431 | DOI

[6] N Bergeron, D T Wise, A boundary criterion for cubulation, Amer. J. Math. 134 (2012) 843 | DOI

[7] M R Bridson, On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc. 127 (1999) 2143 | DOI

[8] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) | DOI

[9] M R Bridson, J Howie, Subgroups of direct products of elementarily free groups, Geom. Funct. Anal. 17 (2007) 385 | DOI

[10] M R Bridson, J Howie, C F Miller Iii, H Short, The subgroups of direct products of surface groups, Geom. Dedicata 92 (2002) 95 | DOI

[11] M R Bridson, J Howie, C F Miller Iii, H Short, Subgroups of direct products of limit groups, Ann. of Math. 170 (2009) 1447 | DOI

[12] K S Brown, Cohomology of groups, 87, Springer (1994) | DOI

[13] M Burger, Fundamental groups of Kähler manifolds and geometric group theory, from: "Séminaire Bourbaki 2009/2010", Astérisque 339, Soc. Mat. de France (2011) 305

[14] P E Caprace, M Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21 (2011) 851 | DOI

[15] J A Carlson, D Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces, Inst. Hautes Études Sci. Publ. Math. 69 (1989) 173 | DOI

[16] F Catanese, Differentiable and deformation type of algebraic surfaces, real and symplectic structures, from: "Symplectic –manifolds and algebraic surfaces" (editors F Catanese, G Tian), Lecture Notes in Math. 1938, Springer (2008) 55 | DOI

[17] I Chatterji, T Fernós, A Iozzi, The median class and superrigidity of actions on CAT(0) cube complexes, J. Topol. 9 (2016) 349 | DOI

[18] M W Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. 117 (1983) 293 | DOI

[19] M W Davis, T Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991) 347 | DOI

[20] M Davis, T Januszkiewicz, R Scott, Nonpositive curvature of blow-ups, Selecta Math. 4 (1998) 491 | DOI

[21] O Debarre, Tores et variétés abéliennes complexes, 6, Soc. Mat. de France (1999)

[22] T Delzant, Trees, valuations and the Green–Lazarsfeld set, Geom. Funct. Anal. 18 (2008) 1236 | DOI

[23] T Delzant, L’invariant de Bieri–Neumann–Strebel des groupes fondamentaux des variétés kählériennes, Math. Ann. 348 (2010) 119 | DOI

[24] T Delzant, M Gromov, Cuts in Kähler groups, from: "Infinite groups: geometric, combinatorial and dynamical aspects" (editors L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk), Progr. Math. 248, Birkhäuser (2005) 31 | DOI

[25] J.-P. Demailly, Complex analytic and differential geometry, book project (2012)

[26] A Dimca, Ş Papadima, A I Suciu, Non-finiteness properties of fundamental groups of smooth projective varieties, J. Reine Angew. Math. 629 (2009) 89 | DOI

[27] C Druţu, M Kapovich, Geometric group theory, 63, Amer. Math. Soc. (2018)

[28] R Geoghegan, Topological methods in group theory, 243, Springer (2008) | DOI

[29] M Gromov, Hyperbolic groups, from: "Essays in group theory" (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 | DOI

[30] M Gromov, Kähler hyperbolicity and L2–Hodge theory, J. Differential Geom. 33 (1991) 263 | DOI

[31] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, 152, Birkhäuser (1999) | DOI

[32] 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 | DOI

[33] M F Hagen, D T Wise, Cubulating hyperbolic free-by-cyclic groups: the general case, Geom. Funct. Anal. 25 (2015) 134 | DOI

[34] F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551 | DOI

[35] C H Houghton, Ends of locally compact groups and their coset spaces, J. Austral. Math. Soc. 17 (1974) 274 | DOI

[36] 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 | DOI

[37] I Kapovich, The nonamenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups, Enseign. Math. 48 (2002) 359

[38] M Kapovich, Energy of harmonic functions and Gromov’s proof of Stallings’ theorem, Georgian Math. J. 21 (2014) 281 | DOI

[39] A Kar, M Sageev, Ping pong on CAT(0) cube complexes, Comment. Math. Helv. 91 (2016) 543 | DOI

[40] N J Korevaar, R M Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993) 561 | DOI

[41] P H Kropholler, M A Roller, Relative ends and duality groups, J. Pure Appl. Algebra 61 (1989) 197 | DOI

[42] P Li, On the structure of complete Kähler manifolds with nonnegative curvature near infinity, Invent. Math. 99 (1990) 579 | DOI

[43] P Li, L F Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992) 359 | DOI

[44] C Llosa Isenrich, Branched covers of elliptic curves and Kähler groups with exotic finiteness properties, Ann. Inst. Fourier (Grenoble) 69 (2019) 335 | DOI

[45] T Napier, M Ramachandran, Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems, Geom. Funct. Anal. 5 (1995) 809 | DOI

[46] T Napier, M Ramachandran, Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces, Geom. Funct. Anal. 11 (2001) 382 | DOI

[47] T Napier, M Ramachandran, Filtered ends, proper holomorphic mappings of Kähler manifolds to Riemann surfaces, and Kähler groups, Geom. Funct. Anal. 17 (2008) 1621 | DOI

[48] T Napier, M Ramachandran, L2 Castelnuovo–de Franchis, the cup product lemma, and filtered ends of Kähler manifolds, J. Topol. Anal. 1 (2009) 29 | DOI

[49] P Py, Coxeter groups and Kähler groups, Math. Proc. Cambridge Philos. Soc. 155 (2013) 557 | DOI

[50] M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. 71 (1995) 585 | DOI

[51] M Sageev, CAT(0) cube complexes and groups, from: "Geometric group theory" (editors M Bestvina, M Sageev, K Vogtmann), IAS/Park City Math. Ser. 21, Amer. Math. Soc. (2014) 7

[52] P Scott, Ends of pairs of groups, J. Pure Appl. Algebra 11 (1977/78) 179 | DOI

[53] C Simpson, Lefschetz theorems for the integral leaves of a holomorphic one-form, Compositio Math. 87 (1993) 99

[54] Y T Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980) 73 | DOI

[55] D T Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004) 150 | DOI

[56] D T Wise, From riches to raags : 3–manifolds, right-angled Artin groups, and cubical geometry, 117, Amer. Math. Soc. (2012) | DOI