The deficiencies of Kähler groups. (Q2914383)

From the introduction: ``The deficiency \(\mathrm{def}(\Gamma)\) of a finitely presentable group \(\Gamma\) is the maximum over all presentations of the difference of the number of generators and the number of relators \dots. In Kähler geometry, the deficiency has also long been known to play a rôle because of the following:NEWLINENEWLINE Theorem~1. (Green-Lazarsfeld, Gromov) If the fundamental group of a compact Kähler manifold \(X\) has deficiency at least 2, then the Albanese image of \(X\) is a curve.NEWLINENEWLINE\dots\ The main purpose of this paper is to prove the following definitive version of Theorem 1:NEWLINENEWLINE Theorem~2. A group of deficiency at least 2 is the fundamental group of a compact Kähler manifold \(X\) if and only if it is isomorphic to the orbifold fundamental group of a curve of genus \(g\geq 2\). The curve is the Albanese image of \(X\) and the isomorphism is induced by the Albanese map of \(X\).NEWLINENEWLINE\dots\ Our proof, like Gromov's, uses \(l^2\)-cohomology, starting from the well known fact that groups of deficiency at least two have positive first \(l^2\)-Betti number. However, unlike in Gromov's approach, we do not use any \(l^2\)-Hodge theory or complex analysis, but instead shift the focus onto the formal properties of the first \(l^2\)-Betti number, replacing analytic techniques by algebraic ones.''NEWLINENEWLINE A corollary of the main result is that a positive integer can arise as the deficiency of a Kähler group if and only if it is odd; it is further shown that, with the possible (but unlikely) exceptions of \(-5\) and \(-7\), all non-positive integers arise as deficiencies of Kähler groups.NEWLINENEWLINE Theorem~2 implies a classification of one-relator Kähler groups with at least 3 generators. Theorem~3 extends this to the 1 and 2 generator case (which was proved independently by Bismas and Mj, using different techniques).NEWLINENEWLINE Theorem~3. An infinite one-relator group is the fundamental group of a compact Kähler manifold \(X\) if and only if it is isomorphic to the orbifold fundamental group of an orbifold of genus \(g\geq 1\) with at most one point with multiplicity greater then 1. Moreover, the isomorphism is induced by the Albanese map of \(X\).NEWLINENEWLINE Similar arguments to those used in the proofs of Theorem~2 and~3 are employed to prove that a non-Abelian limit group is a Kähler group if and only if it is a surface group of genus at least~2. Extensions of the main results to groups of deficiency 1 and for non-Kähler complex surfaces and for Vaisman manifolds are also discussed.