Kähler groups, quasi-projective groups and 3-manifold groups

Abstract

We prove two results relating 3‐manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3‐manifold. If N has non‐empty, toroidal boundary and π1(N) is a Kähler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and π1(N) is a quasi‐projective group, then all the prime components of N are graph manifolds.

Kählerian three-manifold groups

“…This generalizes both the theorem of Dimca-Suciu [4] and a very recent result of Friedl and Suciu [6], who considered compact three-manifolds with non-empty toroidal boundary. The problem of determining all Kählerian three-manifold groups was suggested by [6]. Discussing infinite groups only, as we do here, is an insignificant restriction, since all finite groups are in fact Kähler by a classical result of Serre.…”

Quasiprojective Three-Manifold Groups and Complexification of Three-Manifolds

“…As a consequence of our results we deduce the restrictions on quasiprojective 3-manifold groups obtained by the authors of [DPS,FrSu,Ko2] and the restrictions on good complexifications of 3-manifolds deduced in [To] (this is done in in Section 3.1.1). We also indicate, in Remark 3.12, how to deduce the classification of (closed) 3-manifold Kähler groups [DiSu,Ko1,BMS] using the techniques of Theorem 1.1, thus providing a unified treatment of known results.…”

“…A group is called quasiprojective (respectively, Kähler) if it is the fundamental group of a smooth complex quasiprojective variety (respectively, compact Kähler manifold). Kähler and quasiprojective 3manifold groups have attracted much attention of late [DiSu,Ko1,BMS,DPS,FrSu,Ko2]. In this paper we characterize quasiprojective 3-manifold groups.…”

Characteristic varieties of graph manifolds and quasi-projectivity of fundamental groups of algebraic links

“…Theorem 3.1 (Papadima [20], Friedl-Suciu [11], Biswas-Mj [6]). The local group of an algebraic link is quasi-projective if and only if it is the group of a quasihomogeneous singularity.…”