Dirichlet fundamental domains and complex-projective varieties

TL;DR

This paper demonstrates that any finitely-presented group can be realized as the fundamental group of a specific type of complex-projective variety with controlled singularities.

Contribution

It constructs 2-dimensional irreducible complex-projective varieties with prescribed fundamental groups and limited singularities, advancing the understanding of the relationship between algebraic groups and geometric structures.

Findings

Any finitely-presented group G can be realized as the fundamental group of such a variety.

The constructed varieties have only normal crossing and Whitney umbrella singularities.

This bridges group theory and complex algebraic geometry in a new way.

Abstract

We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.