Nilpotent groups with balanced presentations. II

2021/07/21 by J. A. Hillman, Hillman, J. A.
Engineering · Mathematics · #20F18 #20J05 #57N13 #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Rings, Modules, and Algebras #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2107.09985

openalex publication_date 2021/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If G is a nilpotent group with a balanced presentation and G\not≅ℤ³ then β₁(G;ℚ)≤2 \citeHi22. We show that if such a group G has an abelian normal subgroup A such that G/A≅ℤ² then G is torsion-free and has Hirsch length h(G)≤4. On the other hand, if β₁(G;ℚ)=1 and G has an abelian normal subgroup A such that G/A≅ℤ then G≅ℤ/mℤ\rtimes_nℤ, for some m,n\not=0 such that m divides a power of n-1.

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