On FC-central extensions of groups of intermediate growth

TL;DR

This paper investigates FC-central extensions of groups with intermediate growth, demonstrating that such extensions preserve sub-exponential volume growth and introducing new examples with unique properties like infinite centers and oscillating growth functions.

Contribution

It shows that FC-central extensions retain sub-exponential growth and introduces novel groups with infinite centers and oscillating growth, expanding understanding of intermediate growth groups.

Findings

FC-central extensions preserve sub-exponential volume growth

Constructed extensions with centers isomorphic to ^{}

Created groups with prescribed oscillating intermediate growth functions

Abstract

It is shown that FC-central extensions retain sub-exponential volume growth. A large collection of FC-central extensions of the first Grigorchuk group is provided by the constructions in the works of Erschler and Kassabov-Pak. We show that in these examples subgroup separability is preserved. We introduce two new collections of extensions of the Grigorchuk group. One collection gives first examples of intermediate growth groups with centers isomorphic to $\mathbb{Z}^{\infty}$; and the other provides groups with prescribed oscillating intermediate growth functions.