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.

With some minor revisions, the multiplications of two nearby primes are also considered