Arithmetic and geometry of the Hecke groups

TL;DR

This paper investigates the arithmetic and geometric properties of Hecke groups, deriving conditions for subgroups with specific characteristics related to genus, cusps, and conjugacy classes, with special cases for odd q.

Contribution

It provides explicit criteria linking subgroup properties of Hecke groups to their geometric and arithmetic invariants, extending understanding of their structure.

Findings

Derived conditions for subgroup existence based on genus, cusps, and conjugacy classes.

Established relations between subgroup indices and geometric invariants.

Simplified criteria for odd q cases where certain conditions are redundant.

Abstract

We study the arithmetic and geometry properties of the Hecke group $G_q$. In particular, we prove that $G_q$ has a subgroup $X $ of index $d$, genus $g$ with $v_{\infty} $ cusps, and $\tau_2$ (resp. $v_{r_i}$) conjugacy classes of elements that are conjugates of $S$ (resp. $R^{q/r_i}$) if and only if (i) $ 2g-2 + \tau_2/2 +\sum_{i=1}^k v_{r_i}(1-1/r_i) + v_{\infty} = d(1/2-1/q)$, and (ii) $ m _0= 4g-4 +\tau_2 + 2 v_{\infty} + \sum _{i=1}^k v_{r_i}(2-q/r_i)\ge 0$ is a multiple of $q-2$, (iii) $m \ge 0$. In the case $q$ is odd, (ii) is a consequence of (i).