Quantifying lawlessness in finitely generated groups

2021/12/16 by Bradford, Henry
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2112.08875

Abstract

We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" A_Γ : ℕ → ℕ. We show that A_Γ 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 A_Γ grows more slowly that f. We produce torsion lawless groups for which A_Γ is at least linear using Golod-Shafarevich theory, and give some upper bounds on A_Γ for Grigorchuk's group and Thompson's group F. We note some connections between A_Γ and quantitative versions of residual finiteness. Finally, we also describe a function M_Γ quantifying the property of Γ having no mixed identities, and give bounds for nonabelian free groups. By contrast with A_Γ, there are no groups for which M_Γ is bounded: we prove a universal lower bound on M_Γ(n) of the order of log (n).

Related

• Groups of arbitrary lawlessness growth
• Short Laws for Finite Groups and Residual Finiteness Growth
• On the spectrum of residual finiteness growth functions
• Quantifying local embeddings into finite groups
• The free group does not have the finite cover property