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On the arithmetic dimension of triangle groups

Let $Δ=Δ(a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $π/a,π/b,π/c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $Δ$ to be the number of split real places of the quaternion algebra generated by $Δ$ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic dimension $1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an efficient algorithm to completely enumerate the list of triples of bounded arithmetic dimension.