A p-group with positive Rank Gradient

2011/05/09 by Jan‐Christoph Schlage‐Puchta, Jan-Christoph Schlage-Puchta, Schlage-Puchta, Jan-Christoph
Mathematics · #20F05 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #msc:20F05

paper · pdf · doi:10.48550/arxiv.1105.1631

arxiv created 2011/05/09 · openalex publication_date 2011/05/09 · arxiv updated 2011/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct for d≥ 2 and ε>0 a d-generated p-group Γ, which in an asymptotic sense behaves almost like a d-generated free pro-p-group. We show that a subgroup of index pⁿ needs (d-ε)pⁿ generators, and that the subgroup growth of Γ satisfies s_pⁿ(Γ)>s_pⁿ(F_d^p)^1-ε, where F_d^p is the d-generated free pro-p-group. To do so we introduce a new invariant for finitely generated groups and study some of its basic properties.

Citations

• A characterisation of large finitely presented groups

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• On profinite groups with positive rank gradient
• Groups with positive rank gradient and their actions
• Arbitrary p-Gradient Values
• Ascending chains of finitely generated subgroups
• Pro-p groups with constant generating number on open subgroups