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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    $\begingroup$ Here's an approach to proving it. It's true if the group is cocompact, so suppose it is not. Take a maximal $\Gamma$-invariant set of horoballs, call its union $H$. Then $D_2 \setminus H$ has compact closure, so is covered by finitely many translates of $D_1$. We should also be able to cover each of the finitely many horoball components of $D_2$ with finitely many translates of $D_1$, but I don't yet see how to show this cleanly. $\endgroup$ – Grant Lakeland Jun 12 '12 at 18:06
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    $\begingroup$ @Grant: one more observation makes your proof work: by convexity of a Dirichlet domain, its intersection with some horoball neighborhood of each cusp is fundamental domain for the cusp group acting on that horoball whose boundary is two geodesic rays ending at the cusp. $\endgroup$ – Lee Mosher Jun 12 '12 at 19:47
  • $\begingroup$ @Lee: OK, great! I guess this shows any two convex, finite-sided fundamental domains are comparable. $\endgroup$ – Grant Lakeland Jun 12 '12 at 20:18
  • $\begingroup$ By the way, it should not really be a surprise that fundamental domains can be incomparable in the context of group actions which are not cocompact. Many things about noncocompact actions are more subtle than for cocompact actions. $\endgroup$ – Lee Mosher Jun 12 '12 at 22:17
  • $\begingroup$ @Grant and Lee: Thanks, this is very nice! Since the geometry of $D_1, D_2$ is completely irrelevant away from the cusp neighbourhoods, I guess it would suffice for $D_1$ and $D_2$ to be arbitrary finite-sided polygons (not necessarily convex)? $\endgroup$ – David Loeffler Jun 13 '12 at 9:56

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 $B_{\xi'}$.

  • $\begingroup$ Thank you! Am I right in thinking that this actually shows that if $D$ is a finite-sided fundamental domain for $\Gamma$, and $E$ is any hyperbolic polygon with finitely many points at infinity each of which are parabolic points of $\Gamma$, then $E$ only intersects finitely many $\Gamma$-translates of $D$? $\endgroup$ – David Loeffler Jun 13 '12 at 16:35
  • $\begingroup$ That sounds right. $\endgroup$ – Lee Mosher Jun 13 '12 at 21:20

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