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!
 A: 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. 
A: In general, given a group of isometries $G$
acting on a metric space $(X,d)$,
and some point $c\in X$
such that $\forall g\in G$,
$g(c)\neq c$,
the Dirichlet domain for the action of $G$
on $X$,
centered at $c$,
is
$$\mathscr{D}_c(G):=\{x\in X\mid\forall g\in G, d(x,c)\leq d(g(x),c)\}.$$
When computing this,
it is useful to characterize it more constructively,
looking at how each isometry might affect it.
So we often look at $\mathscr{D}_c(G)$
as an intersection of "half-spaces"
contributed by elements of $G$
(all we're doing is replacing the universal quantifier in the definition with a list of conditions indexed over $G$).
That is,
$$\mathscr{D}_c(G)=\underset{g\in G}\bigcap\{x\in X\mid d(x,c)\leq d(g(x),c)\}.$$
Since we want to get a precise characterization of the boundary,
notice that for some $g\in G$,
the boundary of it's corresponding "half-space'' is
$$m(g):=\{x\in X\mid d(x,c)=d(g(x),c)\},$$
which (since $g$ is an isometry)
can also be written
$$m(g)=\{x\in X\mid d(x,c)=d(x,g^{-1}(c))\}.$$
In words:
the boundary of the half-space contributed by $g$
is the set of points equidistant to $c$
and $g^{-1}(c)$ (don't forget that inverse).
If $X$
has enough geometric structure for "right angles" to make sense,
and has the property that there is a unique geodesic connecting every pair of distinct points (this is all true in your hyperbolic plane),
then $m(g)$
is the perpendicular bisector of the geodesic from $c$
to $g^{-1}(c)$.
(I use the notation $m(g)$
because,
in Mexico,
we call it a "mediatriz.")
So, to wrap up and answer your question,
the boundary of $\mathscr{D}_c(G)$
can be pieced together out of the geodesic pieces
$m(g)\cap\mathscr{D}_c(G)$,
over $g\in G$.
Naturally, this will be empty for many choices of $g$.
If your Fuchsian group is finitely generated,
it will be empty for all but some finite set of generators.
If not,
everything else still goes through and you could still use these tools to talk about its boundary, though it may be more complicated.
[If you want more detail than that, I recommend section 7.4 of the book Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots (Bonahon, 2009).]
A: 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
