Abelianisation of orthogonal groups and the fundamental group of modular varieties

TL;DR

This paper investigates the structure of orthogonal groups related to indefinite quadratic forms, revealing their commutator subgroup properties and implications for the fundamental groups of moduli spaces in algebraic geometry.

Contribution

It establishes that the index of the commutator subgroup is 2 for many orthogonal groups relevant to moduli spaces, with applications to modular forms and fundamental group computations.

Findings

The commutator subgroup index is 2 for many orthogonal groups.

Applications to the fundamental groups of moduli spaces of K3 surfaces.

Insights into modular forms related to these orthogonal groups.

Abstract

We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces.