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 $g{g}^{\ast }$. The distance obeys $2\cosh d(I,g{g}^{\ast })=\mathrm{tr}(g{g}^{\ast })$. 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.