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We study the commutator subgroup of integral orthogonal groups belonging toindefinite quadratic forms. We show that the index of this commutator is 2 formany groups that occur in the construction of moduli spaces in algebraicgeometry, in particular the moduli of K3 surfaces. We give applications tomodular forms and to computing the fundamental groups of some moduli spaces.

Abelianisation of orthogonal groups and the fundamental group of modular varieties