Abelianisation of orthogonal groups and the fundamental group of modular varieties

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. Many moduli spaces in algebraic geometry can be described via period domains as quotients of a symmetric space by a discrete group, or modular group. We shall be concerned with the case of the symmetric space DL associated with a lattice L of signature (2, n), and discrete subgroups of the orthogonal group O(L) that act on DL. Such groups arise in the study of the moduli of K3 surfaces and of other irreducible symplectic manifolds (see [GHS1], [GHS3] and the references there), and of polarised abelian surfaces. Orthogonal groups of indefinite forms also appear elsewhere in geometry, for instance in the theory of singularities (see [Br], [Eb]). In this paper we study the commutator subgroups and abelianisations of orthogonal modular groups of this kind, especially for lattices of signature (2, n). Notation. For definitions and notation concerning locally symmetric varieties and toroidal compactification we refer to [GHS2]. We write 〈X〉 for the group generated by a subset X of some group. If n is an integer 〈n〉 means the rank-1 lattice generated by an element of square n. For a group G, we write [G,G] for the commutator subgroup (derived subgroup) of G (not G because we want to keep the notation O(L) from [Kn1]) and we use Gab for the abelianisation, i.e. the quotient G/ [G,G], which is also the group Hom(G,C) of characters of G. The commutator subgroup and the abelianisation of a modular group carry important information about modular forms. For example the fact that SL2(Z) ab ∼= Z/12Z reflects the existence of the Dedekind η-function.