Are any two Dirichlet domains for a Fuchsian group "comparable"?

Question

Let $\Gamma$ be a [EDIT: finitely generated] Fuchsian group of the first kind (i.e. a discrete subgroup of $PSL_2(\mathbf{R})$ acting on the upper half-plane admitting a fundamental domain of finite hyperbolic area). Let's say two fundamental domains $D$ are comparable if each one is contained in a finite union of $\Gamma$-translates of the other.

I was quite shocked to learn that two different fundamental domains needn't be comparable: you can take the standard fundamental domain for $PSL_2(\mathbf{Z})$ and give it an infinite sequence of longer and longer triangular "teeth" sticking out sideways, with corresponding indentations on the other side -- then this won't be comparable with the usual domain.

There's a standard "nice" class of fundamental domains, though. For any $x_0$ that's not an elliptic point, there is the Dirichlet domain with centre $x_0$, given by the set of points closer to $x_0$ than to any other $\Gamma$-translate of $x_0$.

Is it true that any two Dirichlet domains are comparable?

Answer 1

Here are some details of the solution patched together in the comments. There is an implicit assumption that $\Gamma$ is finitely generated, else it does not have a finite sided fundamental domain. Pick a $\Gamma$-equivariant system of pairwise disjoint horoball neighborhoods $B_\xi$ of the cusps $\xi \in \partial \mathbb{H}^2$.

Suppose that $D$ is a finite-sided fundamental domain for $\Gamma$ (bounded by geodesic paths), for example a Dirichlet domain or a Ford domain. Consider a cusp $\xi$ on which $D$ accumulates. Since $D$ is finite sided, the only way it can accumulate on $\xi$ is for there to exist a concentric horoball $B' \subset B_\xi$ such that $D \cap B'$ is the region of $B'$ between two rays (taking this concentric horoball is necessary to avoid the parts where $D$ mucks around inside $B_\xi$ doing unpleasant things close to the boundary of $B_\xi$). After equivariantly shrinking the horoballs, we can assume that $D$ hits each horoball in this standard manner.

Now suppose that $D_1,D_2$ are two such fundamental domains. After further equivariant shrinking of the horoballs, we can assume that both $D_1$ and $D_2$ hit each horoball in the standard manner. It's now easy to prove that $D_1,D_2$ are comparable: their portions outside the horoballs each have compact closure and so are comparable; and if we pick a representative cusp $\xi$ of each $\Gamma$-orbit of cusps, their portions inside $B_{\xi'}$ for cusps $\xi'$ in the $\Gamma$-orbit of $\xi$ translate to a finite union of standard intersections with $B_{\xi'}$.