Groups of arbitrary lawlessness growth
Pith reviewed 2026-05-22 23:06 UTC · model grok-4.3
The pith
For any unbounded nondecreasing function f meeting mild assumptions, there exists a finitely generated lawless group where the lawlessness function A_Γ matches f exactly.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For any unbounded nondecreasing function f : ℕ → ℕ satisfying mild assumptions, there exists a finitely generated lawless group Γ such that A_Γ is equivalent to f. This generalizes the first author's prior theorem on slow-growing A_Γ and Petschick's result on super-tower growth.
What carries the argument
The function A_Γ(n), the smallest M_n such that every nontrivial reduced word of length at most n is falsified by some k-tuple of word length at most M_n.
If this is right
- All previously constructed growth rates for A_Γ become special cases of a single existence theorem.
- Lawless groups exist with A_Γ growing slower than any prescribed function or faster than any tower of exponentials.
- The set of realizable growth rates for A_Γ includes every function satisfying the mild assumptions.
Where Pith is reading between the lines
- The result implies that lawlessness growth can be made to match functions such as iterated logarithms or other slow-growing unbounded sequences if they meet the assumptions.
- Similar control over A_Γ may extend to other group-theoretic invariants that count the shortest witnesses to non-identity evaluations.
Load-bearing premise
The target function f must satisfy the mild assumptions stated in the abstract for the inductive group construction to succeed.
What would settle it
An explicit unbounded nondecreasing f that meets the mild assumptions but for which no finitely generated lawless group Γ has A_Γ equivalent to f.
read the original abstract
For a finitely generated lawless group $\Gamma$ and $n \in \mathbb{N}$, let $\mathcal{A}_{\Gamma} (n)$ 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 $k \geq 2$, there exists $\overline{g} \in \Gamma^k$ of word-length at most $M_n$ with $w(\overline{g}) \neq e$. For any unbounded nondecreasing function $f : \mathbb{N} \rightarrow \mathbb{N}$ satisfying some mild assumptions, we construct $\Gamma$ such that the function $\mathcal{A}_{\Gamma}$ is equivalent to $f$. Our result generalizes both a Theorem of the first named author, who constructed groups for which $\mathcal{A}_{\Gamma}$ is unbounded but grows more slowly than any prescribed function $f$, and a result of Petschick, who constructed lawless groups for which $\mathcal{A}_{\Gamma}$ grows faster than any tower of exponential functions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves an existence result: for any unbounded nondecreasing function f:ℕ→ℕ satisfying mild assumptions, there exists a finitely generated group Γ such that the lawlessness function A_Γ is asymptotically equivalent to f. The argument synthesizes an inductive or amalgam-based construction generalizing the first author's prior result on slow growth of A_Γ and Petschick's construction for super-tower growth.
Significance. If the construction holds, the result establishes that A_Γ can realize essentially arbitrary growth rates (subject only to the mild assumptions on f), completing the picture of possible lawlessness growth in finitely generated groups. The synthesis of prior techniques into a single flexible construction is a clear strength.
minor comments (2)
- [§1] The mild assumptions on f are referenced but not stated explicitly in the introduction or abstract; listing them (e.g., as a numbered list in §1) would clarify the precise scope of the theorem.
- [Theorem 1.1] The definition of asymptotic equivalence between A_Γ and f should be recalled or referenced at the first use in the main theorem statement to avoid ambiguity for readers unfamiliar with the prior papers.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript and for recommending minor revision. We are pleased that the referee recognizes the result as completing the picture of possible lawlessness growth rates in finitely generated groups.
Circularity Check
No significant circularity; construction generalizes independent prior results
full rationale
The paper's central claim is an existence result obtained by explicit construction of groups Γ realizing any prescribed growth function A_Γ (under mild assumptions on f). This directly extends two earlier constructions—one by the first author and one by Petschick—without the new argument reducing to a redefinition, a fitted parameter renamed as prediction, or a self-citation chain that itself lacks independent verification. The mild assumptions on f are stated as sufficient for the inductive/amalgam-based construction to succeed, and the derivation chain remains self-contained against external benchmarks. No load-bearing step collapses to its own inputs by the paper's equations or citations.
discussion (0)
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