turkmath.org

Let G be a discrete group. It is well known that the homology groups of G have both algebraic and topological definitions. Now consider a subgroup H of G. In the literature there are two versions of relative homology groups for the pair (G,H), the theory by Adamson generalizes in a natural way the algebraic definition, while the theory by Takasu generalizes in a natural way the topological definition. In the first part of the talk, we present both theories, we give simple examples that show that these theories does not coincide in general, and we give a sufficient condition on the subgroup Hin order that the two theories coincide. In the second part of the talk, we define invariants of complete, orientable, non-compact hyperbolic 3-manifolds of finite volume, in Adamson relative homology groups. We explain the relation between these invariants and the elements in Takasu relative homology groups given by Zickert, which are used to define and invariant of such hyperbolic 3-manifolds in the extended Bloch group defined by Neumann.