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

TL;DR

This paper constructs explicit free product decompositions for congruence subgroups _0(n) with controlled generators, confirming a conjecture and providing geometric and algebraic bounds related to Farey sequences and hyperbolic polygons.

Contribution

It proves a conjecture of Kulkarni by providing explicit free product decompositions of _0(n) with specific generator properties and establishes bounds on the minimal largest denominator in associated cusp sets.

Findings

_0(n) admits a free product decomposition into cyclic factors with controlled components.

Bounds on the minimal largest denominator in cusp sets are established and characterized.

The geometric structure of the hyperbolic convex hull relates to the algebraic decomposition of _0(n).

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.