Optimal independent generating system for the congruence subgroups ${\Gamma }_0(p)$ and ${\Gamma }_0(p^2)$

Nhat Minh Doan, Sang-hyun Kim, Mong Lung Lang +1

Let $n$ be a prime or its square. We prove that the congruence subgroup ${\Gamma }_0(n)$ admits a free product decomposition into cyclic factors in such a way that the $(2,1)$-component of…