Optimal independent generating system for the congruence subgroups $\Gamma_0(p)$ and $\Gamma_0(p^2)$

Abstract

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 each cyclic generator is either $n$ or $0$, answering a conjecture of Kulkarni. We can also require that the Frobenius norm of each generator is less than $2n-1$. A crucial observation is that if $P$ denotes the convex hull of the extended Farey sequence of order $\lfloor \sqrt{n} \rfloor$ in the hyperbolic plane $\mathbb{H}^2$, then the projection $\pi: \mathbb{H}^2\to \mathbb{H}^2/\Gamma_0(n)$ is injective on the interior of $P$ and each connected component of $\pi(\mathbb{H}^2)\setminus\pi(P)$ is either an order-three cone of area $\pi/3$ or an ideal triangle. Denoting by $m(\Gamma_0(n))$ the minimum of the largest denominator in the cusp set of $Q$ where $Q$ ranges over all possible special (fundamental) polygons for $\Gamma_0(n)$, we establish the inequality $ \lfloor \sqrt{n} \rfloor \le m(\Gamma_0(n))\le \lfloor \sqrt{4n/3} \rfloor$, and completely characterize the cases in which the bounds are achieved. We also prove analogous results when $n$ is the multiplication of two sufficiently close odd primes.