Counting congruence subroups

D. Goldfeld, A. Lubotzky, L. Pyber

Let $\Gamma$ denote the modular group $SL(2,\mathbb{Z})$ and $C_n(\Gamma )$ the number of congruence subgroups of $\Gamma$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log C_n…