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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.

If a group has a properly discontinuous group action by isometries that is also a cocompact group action on a nonempty proper metric space that is a geodesic metric space, then the group is finitely generated and every orbit map from the word metric for any finite generating set of a group is a quasi-isometry. Finite stabilizers are permitted.

For a nonempty finite set $\{p_1,\dots ,p_m\}$ in finite-dimensional hyperbolic space, the continuous function $R(x)={\max }_{j}d(x,p_j)$ attains its minimum. Indeed, a bounded candidate radius confines all possible improving centers to a closed bounded ball about $p_1$. Such a ball is compact in the Poincare ball model, since its points remain away from the ideal boundary. Minimizing $R$ on it supplies a closed ball of smallest enclosing radius. More generally this argument works for any nonempty bounded set in a proper metric space when the supremum-distance function is finite.

The Milnor–Švarc lemma applies to an isometric properly discontinuous group action that is also a cocompact group action on a nonempty proper metric space that is a geodesic metric space. It concludes that the group is finitely generated and, for a basepoint $x$ and any finite generating set of a group $S$, the orbit mapHere $d_S$ is the word metric, and a quasi-isometry means a map satisfying two-sided coarse distance bounds with a uniformly coarsely dense image. A free action is not required: finite stabilizers are compatible with proper discontinuity. The properness and cocompactness hypotheses ensure that only finitely many group elements move a chosen basepoint by the bounded amount used to generate the group. Replacing a finite generating set by another does not change the conclusion, by equivalence of finite word metrics.

The bounded sequence $(x_n)$ lies in a compact set because $X$ is a proper metric space. After taking a subsequence, $x_n\to x$. Since $d(x_n,{\varphi }_n{x}_n)\to 0$, both $x_n$ and ${\varphi }_n{x}_n$ eventually lie in one compact neighborhood $L$ of $x$. A properly discontinuous group action has only finitely many $g\in \Gamma$ with $gL\cap L\ne \varnothing$, so some conjugate $\psi$ occurs as ${\varphi }_n$ along an infinite subsequence. Continuity then givesThus $\psi$ fixes $x$. Since $\psi =\gamma \varphi {\gamma }^{-1}$ for some $\gamma$, the original element $\varphi$ fixes ${\gamma }^{-1}x$.