The minimal derived lengths of smooth finite solvable quotients of co-compact Fuchsian groups

This paper produces a follow-up to a recent Journal of Algebra paper by M.D.E. Conder and D.W. Young, extending the main theorem of the latter paper to arbitrary co-compact Fuchsian groups. In particular, it shows that every non-perfect co-compact Fuchsian group has a smooth finite soluble quotient of derived length c ≤ 3 and classifies those for which $$c = 1$$ or $$c=3$$ . In this way it also extends and generalizes certain theorem by Chih-Han Sah published in Acta Mathematica of 1969 and improves main theorem of R. Zomorrodian published in Illinois J. Math. of 2007.