On growth and torsion of groups

TL;DR

This paper establishes new bounds on the growth function of the Fabrykowski-Gupta group, providing insights into its algebraic structure and answering a question about free subsemigroups in groups.

Contribution

It offers the first subexponential upper bound and superpolynomial lower bound on the group's growth, addressing a key open problem.

Findings

Growth function has subexponential upper bound

Growth function has superpolynomial lower bound

Negative answer to characterization of groups without free subsemigroups

Abstract

We give a subexponential upper bound and a superpolynomial lower bound on the growth function of the Fabrykowski-Gupta group. As a consequence, we answer negatively a question by Longobardi, Maj and Rhemtulla about characterizing groups containing no free subsemigroups on two generators.