Peripheral separability and cusps of arithmetic hyperbolic orbifolds.

D. B. McReynolds

For X = R, C, or H, it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat or almost flat orbifolds modelled on n-dimensional Euclidean space R^n, the (2n+1)-dimensional Heisenberg group, or the (4n+3)-dimensional quaternionic Heisenberg group. We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X--hyperbolic (n+1)-orbifold.

Using our classification theorem, we prove that for n>1, there are infinite families of almost flat manifolds modelled on Heisenberg group which cannot be diffeomorphic to a cusp cross-section of any arithmetic complex hyperbolic (n+1)-orbifold. A similar result is obtained in the quaternionic setting. As a consequence of Corlette's arithmeticity theorem in the quaternionic setting, we have infinite families of almost flat manifolds modelled on the quaternionic Heisenberg group which are not diffeomorphic to a cusp cross-section of any finite volume quaternionic hyperbolic (n+1)-orbifold.

A principal tool in the proof of the characterization of cusp cross-sections of arithmetic X-hyperbolic orbifolds is a subgroup separability result which may be of independent interest. Specifically, for a connected k-algebraic group G and Borel subgroup B of G, we prove that every subgroup of B(O_k) is separable in G(O_k). We deduce several corollaries from this result pertaining to separability of subgroups in arithmetic lattices of algebraic groups and lattices in the isometry group of hyperbolic space.