On the existence of two-generator finite solvable groups with short derived series (Q1917723)

Let \(\ell,m, \mu\) be integers \(\geq 2\) with \((\ell,m) = (\mu,m) = 1\) and the triple \((\ell,m,\mu)\neq(2,3,2)\). The authors prove that the Fuchsian triangle group \((\ell,m, \mu \ell)\) has a solvable quotient \(G_n\) for each \(n\), of derived length \(\leq 3\). The construction supercedes, and is motivated by, earlier results on large solvable groups of automorphisms of compact Riemann surfaces.