Elementary amenable groups of cohomological dimension 3

Jonathan A. Hillman

Abstract

We show that torsion-free elementary amenable groups of Hirsch length $$\leq3$$ are solvable, of derived length $$\leq3$$. This class includes all solvable groups of cohomological dimension $$3$$. We show also that groups in the latter subclass are either polycyclic, semidirect products $$BS(1,n)\rtimes\mathbb{Z}$$, or properly ascending HNN extensions with base $$\mathbb{Z}^2$$ or $$\pi_1(Kb)$$.

Keywords: cohomological dimension, elementary amenable, finitely presentable, Hirsch length, solvable, torsion-free.

: Primary 20J05.

This paper is available as a pdf (252kB) file. It is also on the arXiv: arxiv.org/abs/2102.02947 [math.GR].

 Thursday, June 17, 2021