FINITENESS OF MINIMAL MODULAR SYMBOLS FOR SLn

Let K/Q be a number field with ring of integers O . Let Γ ⊂ SLn(O) be a finite index subgroup, and let ν be the virtual cohomological dimension of Γ. That is, if Γ ⊂ Γ is any finite index torsion-free subgroup, then H (Γ,M) = 0 for i > ν and any ZΓ-module M . Let M be the free abelian group generated by the symbols [v1, . . . , vn], where the vi are nonzero points in K , modulo the following relations: (1) If τ is a permutation on n letters, then [v1, . . . , vn] = sgn(τ)[τ(v1), . . . , τ(vn)], where sgn(τ) is the sign of τ . (2) If q ∈ K, then [qv1, v2, . . . , vn] = [v1, . . . , vn]. (3) If the vi are linearly dependent, then [v1, . . . , vn] = 0. (4) If v0, . . . , vn are nonzero points in K , then ∑