Elementary amenable groups of cohomological dimension 3

Jonathan A. Hillman

Type: Article

Publication Date: 2023-08-02

Citations: 0

DOI: https://doi.org/10.1515/jgth-2022-0049

Abstract

Abstract We show that torsion-free elementary amenable groups of Hirsch length at most 3 are solvable, of derived length at most 3. This class includes all solvable groups of cohomological dimension 3. We show also that groups in the latter subclass are either polycyclic, semidirect products <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>BS</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo rspace="0.055em" stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="0.222em">⋊</m:mo> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> </m:math> \mathrm{BS}(1,n)\rtimes\mathbb{Z} , or properly ascending HNN extensions with base <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> \mathbb{Z}^{2} or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>π</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>Kb</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \pi_{1}(\mathrm{Kb}) .

Locations

Journal of Group Theory - View
arXiv (Cornell University) - View - PDF

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+	Almost finitely presented soluble groups	1978	Robert Bieri Ralph Strebel
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