Coding of billiards in hyperbolic 3-space

The paper’s main theorem states the exact coding rules (a)–(c) and uniqueness for billiards in an ideal hyperbolic polyhedron with dihedral angles π/λ (λ∈N), and sketches necessity via unfolding and a projection argument, and sufficiency via a half-space lemma and an “algorithmic” unfolding; these ideas match the intended result, but key steps are asserted without adequate justification. In particular, it is claimed that reflecting Π across faces yields a tessellation of B^3, which generally requires Coxeter-type hypotheses and is not proved here, yet is used later in the continuity/conjugacy arguments . The necessity proof for (b) relies on a projection “angle > π” contradiction that is not rigorous as written , and the sufficiency proof only partially addresses the required divergence of successive planes (Lemma 3.1) before breaking to a second case with triples, without fully closing the argument in the text we see . The model’s solution gives a clean Coxeter-group-based construction and uniqueness via endpoints at infinity, but it assumes, without verification, that face reflections generate a discrete reflection group with Π as fundamental chamber via Poincaré’s polyhedron theorem. That requires additional local-to-global compatibility beyond “all dihedral angles are π/λ,” which the paper does not assume or prove. Thus, while both aim for the same result and are largely on the right track, each leaves nontrivial gaps.