Fields of definition for triangle groups as Fuchsian groups

TL;DR

The paper resolves when cocompact hyperbolic triangle groups Δ(p,q,r) admit a faithful representation into PSL_2(L) for totally real fields L, proving that exactly eleven such groups embed into PSL_2(K) where $K$ is the invariant trace field $K=\mathbf{Q}(\cos(π/p),\cos(π/q),\cos(π/r))$, and that no others embed into any totally real field. The authors convert the obstruction to a PSL_2(K) model into the splitting problem for the quaternion algebra B/K and recast it as a quantitative lattice-approximation problem in $\mathbf{R}^3/\Lambda$, then employ Fourier-analytic and geometric-number theory techniques to obtain explicit bounds. A central technical result, Theorem 5lemma, produces a uniform lower bound on the distance from $t(1/p,1/q,1/r)$ to the lattice $\Lambda$, enabling an effective finitess argument via a careful analysis of codimension cases and Jacobsthal-function bounds; this is complemented by a lower-dimensional, twisting-based framework to manage edge cases. The work confirms Waterman–Machlachlan's finiteness predictions and settles five related conjectures of McMullen, yielding a complete classification of the Hilbert-series–type triangle groups admitting a model over a totally real field and establishing a robust, effective method for studying trace-field realizations in arithmetic Fuchsian contexts. The results have implications for the arithmetic of triangle groups, Hilbert modular varieties, and the broader study of fields of definition for discrete groups in Lie groups. All of the above are supported by explicit computational verification, including extensive lattice checks and reductions to finitely many cases.

Abstract

The compact hyperbolic triangle group $Δ(p,q,r)$ admits a canonical representation to $\mathrm{PSL}_2(\mathbf{R})$ with discrete image which is unique up to conjugation. The trace field of this representation is \[K = \mathbf{Q}(\cos(π/p), \cos(π/q), \cos(π/r)).\] We prove that there are exactly eleven such groups which are conjugate to subgroups of $\mathrm{PSL}_2(K)$. Moreover, we prove that there are no additional compact hyperbolic triangle groups which are conjugate to subgroups of $\mathrm{PSL}_2(L)$ 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.

Figures (5)

• Figure 1.1.0: A (partial) tiling of hyperbolic space by $(2,3,7)$ triangles, taken from Fricke.
• Figure 2.1.0: The region $|(x,y,z) - \Lambda| \ge 6/5$ in $\mathbf{R}^3/(2 \mathbf{Z})^3$.
• Figure 2.4.2: The region $|(x,y,z) - \Lambda| \ge 6/5$ in $\mathbf{R}^3/(2 \mathbf{Z})^3$ together with the line $(t/3,t/12,t/4)$.
• Figure 6.1.1: The region $\Omega(x)$ for $x = 1/6, 1/3, 1/2$ and the intersection $\Omega$ of all $\Omega(x)$.
• Figure 6.1.2: Lines intersecting $\Omega$.

Theorems & Definitions (67)

• Definition 1.1.1: McMullen, Curt
• Conjecture A
• Conjecture B
• Conjecture C
• Conjecture D
• Conjecture E
• Theorem 2
• Theorem 3
• Definition 1.5.1
• Definition 1.5.2