Properly discontinuous group action

An action is properly discontinuous when every compact set meets only finitely many of its translates. For an isometric action on a proper metric space, this gives the finiteness needed to extract a constant group element from a sequence with bounded source and target.

For a discrete group action on a locally compact space, a standard proper-discontinuity condition is that only finitely many group elements translate a given compact set to meet itself.

For a group of isometries acting properly discontinuously on the hyperbolic plane, and a point $p$ with trivial point stabilizer, the closed Dirichlet domain isIt 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.

Consider determinant-one matrices $\left(\begin{array}{cc}u& v\\ \overline{v}& \overline{u}\end{array}\right)$ over Gaussian integers with $u+v\equiv 1(\mathrm{mod}2\mathbb{Z}[i])$, modulo their common sign. They act on the Poincare disc model by fractional linear maps. Their origin-centered Dirichlet domain isFor a nonidentity matrix set $N=|v{|}^2\ge 1$ and $uv=m+in$. The group orbit point is $(m+in)/(N+1)$ and its bisector inequality is $2(mx+ny)\le N(1+x^2+y^2)$. Since $m^2+n^2=N(N+1)$, integrality gives $|m|,|n|\le N$, so the four inequalities defining $D$ imply all of them. They are necessary because the matrices with $u=1\pm i$, $v=\pm i$ realize the four combinations $m,n=\pm 1$. The four sides are arcs of the unit-radius circles centered at $\pm 1\pm i$, and the ideal vertices are $1,i,-1,-i$.