2021/12/16 by H. F. Bradford, Henry Bradford, Bradford, Henry · 1 citation
Computer Science · Mathematics · #Bounded function #Combinatorics #FOS: Mathematics #Finite Group Theory Research #Finitely-generated abelian group #Geometric and Algebraic Topology #Group Theory (math.GR) #Lawlessness #Mathematical analysis #Mathematics #Order (exchange) #Upper and lower bounds #math.GR #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2112.08875
published in arXiv (Cornell University) (Cornell University) · Theorem 1.2 has been significantly strengthened; other minor corrections
openalex publication_date 2021/12/16 · arxiv created 2022/01/08 · arxiv updated 2022/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" AΓ : ℕ → ℕ. We show that AΓ is bounded iff Γ has a nonabelian free subgroup. By contrast we construct, for any nondecreasing unbounded function f: ℕ → ℕ, an elementary amenable lawless groups for which AΓ grows more slowly that f. We produce torsion lawless groups for which AΓ is at least linear using Golod-Shafarevich theory, and give some upper bounds on AΓ for Grigorchuk's group and Thompson's group F. We note some connections between AΓ and quantitative versions of residual finiteness. Finally, we also describe a function MΓ quantifying the property of Γ having no mixed identities, and give bounds for nonabelian free groups. By contrast with AΓ, there are no groups for which MΓ is bounded: we prove a universal lower bound on MΓ(n) of the order of log (n).