Algebraic entropy of elementary amenable groups

TL;DR

This paper investigates the algebraic entropy of elementary amenable groups, establishing conditions under which such groups exhibit nilpotent subgroups of finite index and analyzing the distribution of entropy values.

Contribution

It proves that finitely generated elementary amenable groups with zero entropy contain a finite index nilpotent subgroup and shows that zero is an accumulation point of their entropy set.

Findings

Groups of zero entropy contain finite index nilpotent subgroups

Elementary amenable groups of exponential growth have uniform exponential growth

Zero entropy is an accumulation point of entropy values

Abstract

We prove that any finitely generated elementary amenable group of zero (algebraic) entropy contains a nilpotent subgroup of finite index or, equivalently, any finitely generated elementary amenable group of exponential growth is of uniformly exponential growth. We also show that 0 is an accumulation point of the set of entropies of elementary amenable groups.