In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic $4$-space, on one hand, and the combinatorics of simplicial complexes, on the other.

32 pages, 10 figures; Int. Math. Res. Notices (2015); SAGE worksheet available at https://doi.org/10.7910/DVN/0YUU6O; a minor mistake in the proof of Proposition 4.4 corrected in Proposition 2.5 / Remark 2.6 of arXiv:1710.07534