Arbitrary p-Gradient Values

TL;DR

This paper constructs finitely generated groups with any prescribed p-gradient, demonstrating the existence of uncountably many non-commensurable, infinite, torsion, non-amenable, residually-p groups.

Contribution

It introduces a method to realize any positive real number as the p-gradient of a finitely generated group, expanding understanding of group invariants.

Findings

Existence of groups with arbitrary p-gradient values

Uncountably many non-commensurable groups with specified properties

Construction of torsion, non-amenable, residually-p groups

Abstract

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} 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.