Torsion homology of arithmetic lattices and $$K_2$$ K 2 of imaginary fields

Abstract

Let $$X = G/K$$ be a symmetric space of noncompact type. A result of Gelander provides exponential upper bounds in terms of the volume for the torsion homology of the noncompact arithmetic locally symmetric spaces $$\Gamma \backslash X$$ . We show that under suitable assumptions on $$X$$ this result can be extended to the case of nonuniform arithmetic lattices $$\Gamma \subset G$$ that may contain torsion. Using recent work of Calegari and Venkatesh we deduce from this upper bounds (in terms of the discriminant) for $$K_2$$ of the ring of integers of totally imaginary number fields $$F$$ . More generally, we obtain such bounds for rings of $$S$$ -integers in $$F$$ .