Symmetries of Hyperbolic 4-Manifolds

In this paper, for each finite group G, we construct the first explicit examples of non-compact complete finite-volume arithmetic hyperbolic 4-manifolds M such that Isom M similar or equal to G, or Isom(+) M similar or equal to G. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on the one hand, and the combinatorics of simplicial complexes, on the other hand. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realizing a given finite group G as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group G as their isometry group.