Item Details

On Rank Gradient and P-Gradient of Finitely Generated Groups

Pappas, Nathaniel
Format
Thesis/Dissertation; Online
Author
Pappas, Nathaniel
Advisor
Ershov, Mikhail
Abstract
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.
Date Received
20140425
Published
University of Virginia, Department of Mathematics, PHD (Doctor of Philosophy), 2014
Published Date
2014-04-04
Degree
PHD (Doctor of Philosophy)
Collection
Libra ETD Repository
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