Infinite groups in projective and symplectic geometry

Summary

Introduction

For a long time it was not known if the groups which appear as fundamental groups of projective manifolds are similar to linear groups. First J. P. Serre asked if there are smooth complex projective manifolds with nonresidually finite fundamental groups. D. Toledo [22] found examples of nonresidually finite groups which appear as fundamental groups of complex projective manifolds. Further examples were given by Catanese, Kollàr and Nori [6]. We shall call a group Kähler if it can be realized as a fundamental group of a compact Kähler manifold. We assume (this is a popular belief) that Kähler groups can be realized as fundamental groups of projective manifolds and hence by the Lefschetz hyperplane section theorem they can also be realized as fundamental groups of complex projective surfaces.

The problem, which arises naturally is to find some “natural borders” for the class of groups which appear as fundamental groups of projective (Kähler manifolds) inside the class of finitely presented groups. In this paper we are going to present a construction which shows that fundamental groups of projective surfaces (and hence Kähler groups) are rather densely distributed among all finitely presented groups.