Ends of groups and compact Kähler manifolds

Asking that a manifold admit a Kähler structure places swingeing restrictions on its topology. The best-known illustration of this comes from Hodge theory: one knows that when a complex structure on a manifold has a compatible symplectic structure, the decomposition of forms into Hodge types descends to cohomology, with far-reaching consequences. However, this turns out to be just the tip of the iceberg, and even the restrictions the Kähler condition places on that most basic of topological invariants, the fundamental group, are by no means well understood. Nonetheless, considerable progress has been made in the past fifteen years, and the aim of this talk is to describe one of the early landmarks in the theory, essentially due to Gromov: Kähler groups are, in a strong sense, indecomposable. The talk is in three sections. § 1 gives geometric and homological definitions of the number e(Γ) of ends of a group Γ. We prove Hopf’s theorem that e(Γ) ∈ {0, 1, 2,∞}, and state Stallings’s famous theorem characterizing groups with infinitely many ends. § 2 lists the basic known examples of Kähler groups, and describes some obvious obstructions for a group to be Kähler. We then prove the ‘fundamental theorem of Kähler groups’: these groups fall into two classes, fibred and nonfibred. Finally, § 3 is devoted to Gromov’s result that a Kähler group has ≤ 1 end. We give a quick recap on L-cohomology, explain how ends are related to L-cohomology, and then outline Gromov’s proof. (If my outline seems rather sketchy, just consider that Gromov dispatched the theorem in seven lines. . . )