We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the ‘rank vs. Heegaard genus’ conjecture on hyperbolic 3-manifolds is incompatible with the ‘fixed price problem’ in topological dynamics.
Let be a compact lie group. We introduce the set for every smooth manifold . It consists of equivalence classes of pair where is a map which defines a homotopy equivalence from to . Two pairs , for , are equivalent if there is a homotopy equivalence such that is homotopic to .Properties of the set and related to the representation of on the tangent spaces of and at the fixed points. For the case and is the manifold defined by a “linear” action on complex...
The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B 4,4,4ℍ3. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B 4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation...