Dirichlet Domain of a Fuchsian Group Recall that a Fuchsian group is a discrete group $\Gamma \leq PSL(2, \mathbb{R})$ and that a Dirichlet domain for $\Gamma$ is a set $D \subset \mathbb{H}^2$ of the form $$\{z \in \mathbb{H}^2: d(z, p) \leq d(z, \gamma p) \forall \gamma \in \Gamma \}$$

where $p \in \mathbb{H}^2$ is a point not fixed by any element of $\Gamma - e$. 

Every book I read very casually declares that because $D$ is an intersection of hyperbolic half-planes $\{z : d(z,p) \leq d(z, \gamma p)\}$, D is "bounded by segments of the geodesics $\{z :d(z,p) = d(z, \gamma p)\}$."  While this is somewhat clear if this intersection is finite, its not at all clear to me that this is true (or what this even means!) when D cannot be written as a finite intersection of halfplanes. Presumably you want to say something like: "D is the region bounded by a simple closed curve in $\mathbb{H}^2$ which is comprised of geodesic arcs and arcs along the x-axis" but I'm not at all sure if thats true or how to prove it.  

Can someone give me a very explicit and technical statement (ie without using words like "is comrpised of") about the boundary of Dirichlet regions and some indication of how its proved? Thanks!
The key is that the collection if bisectirs defining Dirichlet domain is locally finite (this follows immediately from discreetness if the group). Therefore, the same proof as in the case if finitely many bisectors goes through.
There is some online software to generate data for 3-manifolds which might be worth looking at before going further. Here is a link to the screenshots: http://www.math.uic.edu/t3m/SnapPy/doc/screenshots.html