Cusps and fundamental domains for congruence subgroups

2025/03/16 by Nie, Zhaohu
#11F06 #20H05 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2503.12352

Abstract

We characterize the cusp classes and their widths for the congruence subgroups Γ(N), Γ₁(N) and Γ₀(N). We relate the cusp classes of Γ₀(N) with those produced by the connected fundamental domain in the previous work of Nie and Parent. By further studying the interesting functions M and W on \mathbb Z/N, we establish an identity relating the widths.

Related

• Connected fundamental domains for congruence subgroups
• Fundamental domains for congruence subgroups of SL2 in positive characteristic
• Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre
• The congruence subgroup problem
• Cusps, Congruence Groups and Monstrous Dessins