A Uniform Bound for Free Pairs of Bicyclic Units and for Free Points at Algebraic Integers

Gonçalves and del Río asked two questions in the problem session of Oberwolfach Report 55/2007 (Problem 28); the first is also the first half of Question 5.4 of their paper in J. Group Theory 11 (2008). A nonzero complex number z is a free point if the matrices (1 1; 0 1) and (1 0; z 1) generate a free group, and m(z) is the least positive integer k for which kz is a free point. For a finite non-Hamiltonian group G, m(G) is the least positive integer m such that b1 and b2^m generate a free group whenever b1, b2 are bicyclic units of ZG that generate a group which is not nilpotent. Is m(z) bounded over the nonzero algebraic integers z, and is m(G) bounded over all such groups G? We observe that both are bounded by 4. A nonzero algebraic integer has a Galois conjugate of modulus at least 1, freeness of a point is invariant under Galois conjugation, and every complex number of modulus at least 4 is a free point (Sanov, Brenner); hence kz is a free point for every integer k with |k| ≥ 4, and m(z) ≤ 4, with equality for z = ±1. For bicyclic units 1 + a and 1 + b, the element ab acts on ZG by an integer matrix, so if it is not nilpotent it has an eigenvalue λ with |λ| ≥ 1, and the two-dimensional reduction of Gonçalves and del Río turns (1 + a, (1 + b)^m) into (1 1; 0 1), (1 0; mλ 1); hence m(G) ≤ 4, also for the variant M(G) that allows units of different types. The result is an elementary combination of known facts. Exact computations confirm m(S3) = 1, M(S3) = 2 and m(S4) = 2 and give M(S4) = 2; exhaustive computations with numerically located roots give m(S5) = M(S5) = 2. Whether m(G) ≤ 2 for all G, the second part of Question 5.4, remains open. No answer was found in the literature accessible to us; a search that finds nothing is not a proof of novelty, and four later sources could not be read. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-1738-011 (Gonçalves–del Río, Problem 28, Oberwolfach Report 55/2007, Mini-Workshop: Arithmetik von Gruppenringen).