Right-angled Coxeter polytopes, hyperbolic six-manifolds, and a problem of Siegel

Abstract

By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic −1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.The n-dimensional manifold Siegel problem. Determine the minimum possible volume obtained by an orientable hyperbolic n-manifold.All hyperbolic manifolds in this paper will be complete Riemannian manifolds of constant sectional curvature −1. The "full" Siegel problem refers to the problem above for orbifolds rather than manifolds. Nevertheless, the manifold Siegel problem is one with a long and venerable history. This paper describes our solution when n = 6. But first, an overview and some background. The Euler characteristic χ creates a big difference between even and odd dimensions. When n is even, the Gauss-Bonnet theorem gives vol(M ) = κ n χ(M ), with κ n = (−2π) n 2 /(n − 1)!! for the volume of an n-dimensional hyperbolic manifold M . As χ(M ) ∈ Z, the most obvious place to look for solutions to the problem is when |χ| = 1. A compact orientable M satisfies |χ(M )| ∈ 2Z, so the minimum volume is most likely achieved by a non-compact manifold. When n is odd, χ(M ) = 0, and a different approach must be found. For these reasons progress in even dimensions has been more rapid.

Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers

“…is a solution to (7), where P ∞ (t) denotes the reciprocal polynomial of P ∞ (t), that is, P ∞ (t) = t deg P∞(t) P ∞ (t −1 ). Since Q ∞ (t) is a product of cyclotomic polynomials Φ k (t) with k ≥ 2, one has Q ∞ (t) = Q ∞ (t) = t deg Q∞(t) Q ∞ (t −1 ).…”

Hyperbolic groups containing subgroups of type $\mathscr{F}_{3}$ not $\mathscr{F}_{4}$

“…To deduce Theorem 1 from Theorems 3 and 5, we will use a certain hyperbolic manifold M 8 = H 8 /Γ built from the dual of the 8-dimensional Euclidean Gosset polytope studied in [19], together with a continuous map u : M 8 → S 1 . The manifold M 8 and the corresponding map to the circle were constructed in [26].…”

Virtually special embeddings of integral Lorentzian lattices

“…We apply the lattice embeddings together with the explicit relationship between the unimodular lattices I n,1 and RACGs given by [ERT12] to construct many examples of C-special hyperbolic manifold groups in dimension 3 and 4. The following theorem extends and improves the results in [Chu19] (see also [Chu18]) and [DMP18, Theorem 2.6].…”