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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?

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

Let $\Gamma$ be a 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?