Near classification of compact hyperbolic Coxeter \(d\)-polytopes with \(d+4\) facets and related dimension bounds (Q6568832)

Let \(\mathbb{H}^d\) be the d-dimensional real hyperbolic space. A hyperbolic Coxeter polytope is a domain in \(\mathbb{H}^d\) bounded by a collection of geodesic hyperplanes, such that each intersecting pair of hyperplanes meets at dihedral angle \(\frac{\pi}{m}\) for some integer \(m \geq 2\).\N\NHyperbolic Coxeter polytopes are precisely the fundamental domains of discrete hyperbolic reflection groups.\N\NThe Euclidean and spherical Coxeter polytopes were classified by Coxeter in 1934. But until now no complete classification is known in the hyperbolic case. The known classifications are the compact Coxeter polytopes with \(d + 1\), \(d+2\) and \(d+3\) facets.\N\NThe author completes the classification for compact Coxeter polytopes with \(d+4\) facets. For \(d = 4\), there exist \(348\) polytopes and at dimension \(d= 5\) there are \(51\) polytopes.\N\NFurthermore, new upper bounds on the dimension \(d\) of compact hyperbolic Coxeter polytopes with \(d+k\) facets are given when \(k \leq 10\) -- and no better bounds have previously been published for \(k \geq 5\). As a consequence, a compact hyperbolic Coxeter \(29\)-polytope has at least \(40\) facets.\N\NAn important tool frequently used are the Gale diagrams on simple polytopes defined by the dual construction regarding the facets, which include all compact Coxeter polytopes. The Gale diagram of a simple \(d\)-polytope with \(d + k\) facets consists of a set of \(d + k\) points on the sphere \(S^{k-2} \subset \mathbb{R}^{k-1}\).\N\NThen, for \(k = 4\) this on \(S^2\) can be replaced by affine Gale diagrams that encode the same information; they themselves can be reduced to points on the Euclidean plane.