Groups of arbitrary lawlessness growth

2025/03/30 by H. F. Bradford, Bradford, Henry, Willis, Jacob
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2503.23582

openalex publication_date 2025/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a finitely generated lawless group Γ and n ∈ ℕ, let A_Γ (n) be the minimal positive integer M_n such that for all nontrivial reduced words w of length at most n in the free group of fixed rank k ≥ 2, there exists g ∈ Γ^k of word-length at most M_n with w(g) ≠ e. For any unbounded nondecreasing function f : ℕ → ℕ satisfying some mild assumptions, we construct Γ such that the function A_Γ is equivalent to f. Our result generalizes both a Theorem of the first named author, who constructed groups for which A_Γ is unbounded but grows more slowly than any prescribed function f, and a result of Petschick, who constructed lawless groups for which A_Γ grows faster than any tower of exponential functions.

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