On the Second Cohomology of Kähler Groups

Given a group G and a class of manifolds C (e.g. symplectic, contact, Kähler, etc.), it is an old problem to find a manifold M G ∈ C whose fundamental group is G. This article refines it: for a group G and a positive integer r find M G ∈ C such that π 1 (M G ) = G and π i (M G ) = 0 for 1 < i < r. We thus provide a unified point of view systematizing known and new results in this direction for various different classes of manifolds. The largest r for which such an M G ∈ C can be found is called the homotopical height ht C (G). Homotopical height provides a dimensional obstruction to finding a K(G, 1) space within the given class C, leading to a hierarchy of these classes in terms of "softness" or "hardness"à la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes SP and CA of closed symplectic and complex manifolds exhibit intermediate "softness" in the sense that every finitely presented group G can be realized as the fundamental group of a manifold in SP and a manifold in CA. For these classes, ht C (G) provides a numerical invariant for finitely presented groups. We give explicit computations of these invariants for some standard finitely presented groups. We use the notion of homotopical height within the "hard" category of Kähler groups to obtain partial answers to questions of Toledo regarding second cohomology and second group cohomology of Kähler groups. We also modify and generalize a construction due to Dimca, Papadima and Suciu to give a potentially large class of projective groups violating property FP.1450123-1 I. Biswas, M. Mj & D. Pancholi

show abstract

“…Therefore, b 1 (G 3 ) > 0. Hence the Albanese variety of H 3 is non-trivial and so (see [KKM11] for instance) b 2 (G 3 ) > 0.…”

Section: Finiteness Propertiesmentioning

confidence: 99%

Homotopical height

Biswas

Pancholi

2014

Int. J. Math.

View full text Add to dashboard Cite

show abstract

“…In this text, we study actions of fundamental groups of compact Kähler manifolds (referred to as Kähler groups) on finite or infinite dimensional real hyperbolic spaces. For an introduction to the study of Kähler groups, the reader can consult [1] as well as [5,13,31,32,43,45] for more recent developments.…”

mentioning

confidence: 99%

Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat

Delzant

2011

Compositio Math.

View full text Add to dashboard Cite

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL 2 (R). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL 2 (R) on these spaces, and give an application to the study of the Cremona group.

show abstract

“…This is impossible by the results of Carlson, Hernández, Klingler and Toledo. See [CaT89], [CaH91], [Kl10]. Now we prove the second part.…”

Section: Harmonic Maps From Compact Sasakian Manifoldsmentioning

confidence: 76%

On the fundamental groups of compact Sasakian manifolds

Chen

2013

Mathematical Research Letters

View full text Add to dashboard Cite

Abstract. We study the fundamental groups of compact Sasakian manifolds, which we call Sasaki groups. It is shown that the fundamental group of any compact Hodge manifold is Sasaki. In particular, all finite groups are Sasaki. On the other hand, we show that there exist many restrictions on Sasaki groups. We also study the Abel-Jacobi map of a compact Sasakian manifold and its applications to Sasaki groups.

show abstract

On the Second Cohomology of Kähler Groups

Cited by 7 publications

References 19 publications

Homotopical height

Homotopical height

Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat

On the fundamental groups of compact Sasakian manifolds

Contact Info

Product

Resources

About