Quasi-projectivity, Artin-Tits Groups, and Pencil Maps

TL;DR

This paper investigates the fundamental groups of complex quasi-projective varieties, focusing on Artin-Tits groups, using Alexander invariants to determine their geometric origin and exploring their finiteness properties.

Contribution

It provides a method to identify when Artin-Tits groups are fundamental groups of complex varieties and analyzes their finiteness properties with new examples.

Findings

Solved the problem for large families of Artin-Tits groups

Identified examples of hyperplane complements with specific finiteness properties

Developed Alexander-based invariants for group classification

Abstract

We consider the problem of deciding if a group is the fundamental group of a smooth connected complex quasi-projective (or projective) variety using Alexander-based invariants. In particular, we solve the problem for large families of Artin-Tits groups. We also study finiteness properties of such groups and exhibit examples of hyperplane complements whose fundamental groups satisfy $\text{F}_{k-1}$ but not $\text{F}_k$ for any $k$.