A formula for the normal subgroup growth of Baumslag-Solitar groups

2007/08/18 by J. O. Button, Button, J. O.
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.0708.2494

arxiv created 2007/08/18 · openalex publication_date 2007/08/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an exact formula for the number of normal subgroups of each finite index in the Baumslag-Solitar group BS(p,q) when p and q are coprime. Unlike the formula for all finite index subgroups, this one distinguishes different Baumslag-Solitar groups and is not multiplicative. This allows us to give an example of a finitely generated profinite group which is not virtually pronilpotent but whose zeta function has an Euler product.

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• Subgroup growth of all Baumslag-Solitar groups
• Metric properties of Baumslag-Solitar groups
• A simple proof of a theorem of Karrass and Solitar
• Growth in higher Baumslag-Solitar groups
• A new proof of the growth rate of the solvable Baumslag-Solitar groups