Partial Answers to Kourovka Problems 11.18 and 17.80: Infinite Groups G(a,b) and Their Two-Dimensional Images

R. Brandl asked in the Kourovka Notebook (Problem 11.18, 1990) whether the group G(a,b) = ⟨x, y | x = [x,_a y], y = [y,_b x]⟩ is finite; he noted that G(1,b) = 1 and G(2,2) = 1 and that nothing is known about G(2,3). O. Macedońska asked (Problem 17.80, 2010) whether M_n = G(n,n) is infinite for every n > 2 (UnsolvedMath records KOU-11.18 and KOU-17.80). We give partial answers: G(a,b) is infinite whenever m divides both a and b for some m ∈ {18, 20, 26, 28, 30, 42, 44}, so M_n is infinite whenever one of these numbers divides n (for m = 18 this is in print); every homomorphism of G(a,b) into SL(2,K) or PSL(2,K) factors through G(d,d) with d = gcd(a,b); the case G(2,3) and the general questions stay open. That M_18 is infinite follows from a result of Quek, Wong and Wong (2014), proved there by a computation credited to the referee, as A. Abdollahi's zbMATH review points out; we know that paper only from the review, and we prove the result again. For each m we give an explicit epimorphism of G(m,m) onto PSL(2,q) with q ∈ {8, 13, 29, 41, 43} and integer vectors that define homomorphisms onto Z of its kernel or of the preimage of a split torus; these subgroups have abelianizations of free rank exactly 7, 2, 14, 4, 4, 22, 12 for m = 18, 20, 26, 28, 30, 42, 44. The certificates are machine-readable and are checked by independent programs; those for m = 20, 28, 30, 42, 44 are new in this note. Infiniteness passes to G(a,b) because G(a,b) is a quotient of G(a′,b′) whenever a | a′ and b | b′. The second result holds over every field: if X, Y ∈ SL(2,K) satisfy X = [X,_a Y] and Y = [Y,_b X], then tr X = tr Y, a conjugation in GL(2,K) swaps X and Y, and the relations hold with gcd(a,b) in place of a and b. Hence G(2,3), and every G(a,b) with coprime a, b, has no nontrivial homomorphism into GL(2,K) or PGL(2,K) for any field K. Tested evidence: G(2,3) has no nontrivial homomorphism into A_n, n ≤ 11. For the cases other than m = 18, and for the second result, no source was found in zbMATH, OpenAlex, arXiv and Crossref; two related papers of Quek, Wong and Wong were not accessible to us; we claim no priority. 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: KOU-11.18 (Kourovka Notebook Problem 11.18, R. Brandl; with Problem 17.80, O. Macedońska).