Fields of definition for triangle groups as Fuchsian groups

2025/01/03 by Calegari, Frank, Chen, Qiankang
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.01780

Abstract

The compact hyperbolic triangle group Δ(p,q,r) admits a canonical representation to PSL₂(R) with discrete image which is unique up to conjugation. The trace field of this representation is K = Q(cos(π/p), cos(π/q), cos(π/r)). We prove that there are exactly eleven such groups which are conjugate to subgroups of PSL₂(K). Moreover, we prove that there are no additional compact hyperbolic triangle groups which are conjugate to subgroups of PSL₂(L) for any totally real field L. This answers a question first raised by Waterman and Machlachlan, and also resolves (in the positive) five (interrelated) recent conjectures of McMullen.

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