On Rank Gradient and p-Gradient of Finitely Generated Groups

Rank gradient and p-gradient are group invariants that assign some real number greater than or equal to -1 to a finitely generated group. Though the invariants originated in the study of topology (3-manifold groups), there is growing interest among group theorists. For most classes of groups for which rank gradient and p-gradient have been computed, the value is zero. The research presented consists of two main parts. First, for any prime number p and any positive real number alpha, we construct a finitely generated group G with p-gradient equal to alpha. This construction is used to show that there exist uncountably many pairwise non-commensurable groups that are finitely generated, infinite, torsion, non-amenable, and residually-p. Second, rank gradient and p-gradient are calculated for free products, free products with amalgamation over an amenable subgroup, and HNN extensions with an amenable associated subgroup using various methods. The notion of cost of a group is used to obtain lower bounds for the rank gradient of amalgamated free products and HNN extensions. For p-gradient, the Kurosh subgroup theorems for amalgamated free products and HNN extensions are used.