Most introductory textbooks on the modular group begin with an introduction of it as the group generated by the two Möbius transformations:

\begin{gather*} z'=z+1 \\ z'=-\frac{1}{z} \end{gather*}

and immediately after that, they describe the canonical fundamental domain of the modular group as a certain hyperbolic triangle with angles $\pi/3$, $\pi/3$, $0$, which under the action of the modular group tiles up the whole hyperbolic plane. My question refers to the opposite process: given an arbitrary hyperbolic triangle with angles $\pi/n$, $\pi/m$, $\pi/l$ (where $n$, $m$, $l$ are positive integers), how to construct the generators $T$, $S$ of the group of Möbius transformations such that this triangle is its fundamental domain?

I'd also like to see at least one computational example in addition to an explicit formula, with special emphasis on "symmetric" (equilateral) hyperbolic triangles, such as the simplest equilateral hyperbolic triangle — a triangle with angles $\pi/4$, $\pi/4$, $\pi/4$. If someone knows a good source with explicit formula, it will be blessed if he will write it here!


1 Answer 1


My question [is] how to construct the generators $T$, $S$ of the group of Möbius transformations such that this triangle is its fundamental domain?

Via large amounts of hyperbolic trigonometry. See Section 10.6 (Triangle groups) of Beardon's book The geometry of discrete groups.

A long time ago I computed explicit generators (working in the upper half plane) for the (orientation preserving) triangle group $\Delta(p, q, \infty)$. The way this works is we define $z_p = \exp(i\pi/p)$ and $z_q = \exp(i\pi(q - 1)/q)$ on the upper semicircle. We then conjugate the $2\pi/p$ and $2\pi/q$ rotations about $i$ to $z_p$ and $z_q$. We check our work by multiplying these (in the correct order) and obtaining the parabolic element $z \mapsto z + 2(\operatorname{Re}(z_p) - \operatorname{Re}(z_q))$.

As special case we recover the Hecke groups $\Delta(2, q, \infty)$; these are discussed by Beardon.

I've also worked out the generators for $\Delta(2, 3, 7)$. It was not as fun as I hoped it might be — when you do this I strongly suggest you carefully check your work using a computer.

Edit 1: There is a related MathOverflow question How to characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk from a purely analytic point of view? about the group $\Delta(4, 4, 4)$.

Edit 2: Magnus, as part of Theorem 2.11 of his book Noneuclidean tesselations and their groups gives Mobius generators for the group $\Delta(2, p, q)$. You should be able to use the generators for $\Delta(2, 3, 8)$ to find generators for $\Delta(4, 4, 4)$.

  • 2
    $\begingroup$ The question you link was asked by the same user, by the way. $\endgroup$
    – LSpice
    Jan 24, 2022 at 0:18

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