The growth of residually soluble groups

2025/11/10 by Eberhard, Sean, Maini, Elena
Mathematics · #20E26 #20F16 #20F65 #20F69 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory

paper · doi:10.48550/arxiv.2511.07018

openalex publication_date 2025/11/10 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

Building on work of Wilson, we show that if G is a finitely generated residually soluble group whose growth function γ satisfies (log γ(n))/ n^1/4 → 0 as n → ∞ then G is virtually nilpotent. This shows that Grigorchuk's Gap Conjecture holds for all exponents β< 1/4 within the class of residually soluble groups (improving Wilson's exponent 1/6). We also discuss stronger versions of the Gap Conjecture.

Citations

• Explicit universal minimal constants for polynomial growth of groups
• Growth of periodic Grigorchuk groups
• On the Gap Conjecture concerning group growth
• A finitary version of Gromov's polynomial growth theorem

Related

• On the Gap Conjecture concerning group growth
• Full residual finiteness growths of nilpotent groups
• Residual Finiteness Growth in Minimax Groups
• On the spectrum of residual finiteness growth functions
• Arbitrarily Large Residual Finiteness Growth