[Math] Canonical fundamental domain for a discrete subgroup Γ of SL₂(R) acting on hyperbolic plane

hyperbolic-geometry

Let a discrete subgroup $\Gamma$ of $SL_2(\mathbb R)$ act on the hyperbolic plane by Möbius transformations. Is there a "best" or "most canonical" fundamental domain for this action?

Some (mostly unhelpful) observations:

  • For the action of $SL_2(\mathbb Z)$ the usually taken fundamental domain is this one. Any particular reason?

  • Unfortunately the hyperbolic triangle with vertices $0, 1, \infty$ also won't do for $SL_2(\mathbb Z)$, as it is not (quite) a fundamental domain.

  • If we are considering a finite-index sugroup of $SL_2(\mathbb Z)$, then we can form the fundamental domain as a finite union of the fundamental domain of $SL_2(\mathbb Z)$. This is not very nice-looking, but it can be determined effectively given a description of $\Gamma$.

  • Let the action be specified for $\Gamma$. Then, for any point $z_0 \in \mathbb H$, there is the standard polygon around it. This was pointed out to me by sigfpe. Thanks a lot, sigfpe. This is a very nice and very canonical construction, once you have a starting point $z_0$. But this is very theoretical. We do not know what are the vertices, in a computable way. We just know that they exist. And then again, what would be a canonical choice for the point $z_0$? If any point inside the hyperbolic plane is canonical, it is $i$. But the two examples considered above for $SL_2(\mathbb Z)$ does not arise as a fundamental polygon or standard polygon for $i$. In fact $i$ is on the edge of both of these fundamental domains.

Best Answer

As I alluded to in a comment to one of your previous questions, many people have thought about how to explicitly compute fundamental domains, homology, etc., for congruence subgroups of $SL_2(\mathbb Z)$: this leads to the theory of modular symbols, which is not only a theory, but serves as the foundation for all computational work on modular forms too. (See papers of Mazur and Manin from the 1970s, and for the computational aspects, more recent writings of Cremona, William Stein, and others.)

On a related note, let me remark that it it actually quite feasible, and reasonable, to generate a fundamental domain for a finite index subgroup by translating the fundamental domain for $SL_2(\mathbb Z)$.
For a carefully worked example, see here (especially the discussion on page 6).

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