= Dirichlet domain
{c}
{title2=$D_G(p)$}
= Dirichlet region
{c}
{synonym}
For a <group> of <isometries> acting properly discontinuously on the <hyperbolic plane>, and a point $p$ with trivial <point stabilizer>, the closed <Dirichlet domain> is
$$
D_G(p)=\{x:\rho(x,p)\le\rho(x,gp)\text{ for every }g\in G\}.
$$
It is the intersection of the closed half-planes bounded by <perpendicular bisectors> between $p$ and its other <group orbit> points. The <group orbit> is locally finite and the <hyperbolic plane> is a <proper metric space>, so every point has a nearest <group orbit> point. This shows that the translates of $D_G(p)$ cover the plane. Distinct interiors are disjoint, since being strictly closer to two different <group orbit> points is impossible. Boundary ties are allowed in the closed domain. Thus it is a geometric fundamental region.
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