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 \(G(a,b)=\langle x,y\mid x=[x,{}_{a}\,y],\ y=[y,{}_{b}\,x]\rangle\) is finite; he noted that \(G(1,b)=1\) and \(G(2,2)=1\) and that nothing is known about \(G(2,3)\). O. Macedo\'nska 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 \(m\in\{18,20,26,28,30,42,44\}\), 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 \(\mathrm{SL}(2,K)\) or \(\mathrm{PSL}(2,K)\) 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 \(\mathrm{PSL}(2,q)\) with \(q\in\{8,13,29,41,43\}\) and integer vectors that define homomorphisms onto \(\mathbb{Z}\) 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 \(a\mid a'\) and \(b\mid b'\). The second result holds over every field: if \(X,Y\in \mathrm{SL}(2,K)\) satisfy \(X=[X,{}_{a}\,Y]\) and \(Y=[Y,{}_{b}\,X]\), then \(\operatorname{tr} X=\operatorname{tr} Y\), a conjugation in \(\mathrm{GL}(2,K)\) 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 \(\mathrm{GL}(2,K)\) or \(\mathrm{PGL}(2,K)\) for any field \(K\). Tested evidence: \(G(2,3)\) has no nontrivial homomorphism into \(A_n\), \(n\le11\). 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.