On volumes of hyperbolic Coxeter polytopes and quadratic forms

Abstract

Abstract. In this paper, we compute the covolume of the group of units of the quadratic form f n d (x) = x 2 1 + x 2 2 + · · · + x 2 n − dx 2 n+1 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f n 3 . We apply our covolume formula to compute the volumes of these hyperbolic Coxeter polytopes.

On Minimal Covolume Hyperbolic Lattices

“…By a result of Mcleod [33], the group PO( f 3 ; Z) is reflective for n ≤ 13. As an extension of their work for PO(n, 1; Z) to the group PO( f d ; Z), Ratcliffe and Tschantz determined the covolumes of the groups PO( f 3 ; Z) and, for each n ≡ 3 (mod 4), they computed furthermore the commensurability ratio κ n ∈ Q of ∆ n and PO( f 3 ; Z) showing that κ n = 1 (see [34], (35)).…”

“…Denote by P ⊂ Isom H 11 the Coxeter polyhedron of Γ. In particular, by the result ( [34], Table 1), one gets the value covol 11 (PO( f 3 ; Z)) = 13 × 31…”

“…where L(s, D) is Dirichlet's L-function according to ([34], (12)), as well as κ 11 = 1 4 (2 5 − 1) (2 6 + 1) = 2015 4 (see [34], Section 7). This implies that vol 11 (P) = 2 · covol 11 (PO( f 3 ; Z)) and that covol 11 (∆ 11 ) = 2 2015 vol 11 (P).…”

CoxIter – Computing invariants of hyperbolic Coxeter groups

“…Output of CoxIter for the reflection group corresponding to the automorphism group of the quadratic form −3x0 + x vertices (1,20,24) and find 11 other triples, corresponding to the 11 other vertices of the second kind. The 12 triples of vertices are the following: (1,20,24), (1,6,12), (5,16,20), (5,10,21), (9,16,19), (9,4,14), (13,2,8), (13,19,24), (17,2,10), (17,4,12), (22,8,21), (22,6,14). Now, vertices of the first kind are connected among themselves by lines labelled with an ∞, while vertices of the second kind are connected among themselves by broken edges.…”