Divisible homology classes in the special linear group of a number field Original Research Article

The integral homology groups of the infinite special linear group SL(F) over a number field F are in general not finitely generated but they have the following property: for any integer image is the direct sum of a free abelian group of finite rank and a torsion group. The purpose of this paper is to investigate partially the structure of that torsion subgroup. The main theorem asserts that, if image denotes the subgroup of divisible elements in Hi(SL(F); image), then image is an abelian group of finite exponent for any i ≥ 0 (and image is in general non-trivial). The following vanishing result is also proved: if N is a positive integer and ℓ a prime number> N with the property that K2nF contains no ℓ-torsion divisible elements for all n≤ N, then the ℓ-torsion subgroup of image is trivial for all i ≤ 2N.