= Bounded displacement sets of hyperbolic isometries are compact
In the positive Hermitian determinant-one model of three-dimensional <hyperbolic space>, the base point is $I$ and a matrix $g\in SL(2,\mathbb C)$ sends it to $gg^*$. The distance obeys $2\cosh d(I,gg^*)=\operatorname{tr}(gg^*)$. Bounded displacement therefore bounds every matrix entry, and the determinant-one condition gives a closed bounded subset of a finite-dimensional real vector space. Passing to $PSL(2,\mathbb C)$ preserves compactness. Conjugating moves any chosen base point to $I$. A discrete subgroup meets each such compact set in finitely many elements, proving escape of distinct elements and proper discontinuity, including finite stabilizers.
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