Kissing spheres and group combinatorics in complex hyperbolic geometry

We discuss some connections between sphere packing (kissing spheres) in Euclidean spaces $\mathbb R^n$ (or spheres $S^n={\mathbb R^n} \cup{\infty}$) and possible “block-buildings” in the complex (or quaternionic) hyperbolic geometry and CR-geometry at its infinity. This approach grows from several successful applications of our method of block-building of real hyperbolic manifolds/orbifolds, their deformations, unusual ergodic actions of their fundamental groups (hyperbolic lattices) and related cobordisms.