Kähler groups: Surfaces, Dehn functions and Constructions

The goal of this project is to make fundamental progress in understanding the topology of compact Kähler manifolds by applying methods from Geometric Group Theory. It will take place at the intersection of several disciplines of modern mathematics, including Algebraic Geometry, Differential Geometry and Group Theory. Kähler manifolds are geometric spaces at the interface of Algebraic Geometry and Differential Geometry. In Algebraic Geometry the goal is to understand the geometry of zero sets of polynomials. For instance, we can describe the circle in the Euclidean plane by means of the polynomial equation x*x+y*y=1. Differential Geometry is concerned with the study of smooth geometric spaces, so-called manifolds. A natural class of spaces are those which are both, manifolds as well as zero sets of polynomials. Examples are the circle, described above, and smooth surfaces. Roughly speaking we call such spaces Kähler manifolds. Another key area required for this project is Group Theory. It can be interpreted as the study of symmetries of objects. For example, the reflectional and rotational symmetries of a cube form what we call a group. Given a geometric space we can assign to it its fundamental group. This is the set of all closed loops in the space with two loops equal if one can be deformed into the other. A Kähler group is the fundamental group of a compact Kähler manifold. This project is composed of three main objectives in the study of Kähler groups. Objective A is to obtain new results on the interplay between Kähler groups and subgroups of direct products of surface groups (short SPSGs). In particular, we plan to classify Kähler groups among all groups contained in interesting subclasses of SPSGs and to explore applications of our results to Kodaira fibrations. Objective B concerns the Dehn functions of Kähler groups and residually free groups. Our pioneering work with Romain Tessera has shown the existence of a Kähler group whose Dehn function is not linear, quadratic or exponential. Our goal is to obtain more precise results in this area, by studying the concrete class of Kähler SPSGs. A central step will be the study of Dehn functions of residually free groups. Objective C concerns the construction problem for Kähler groups. In particular, we plan to conduct the first construction of explicit examples by means of the Bogomolov--Katzarkov method and to study their properties.