Even unimodular Lorentzian lattices and hyperbolic volume

Abstract

We compute the hyperbolic covolume of the automorphism group of each even unimodular Lorentzian lattice. The result is obtained as a consequence of a previous work with Belolipetsky, which uses Prasad's volume to compute the volumes of the smallest hyperbolic arithmetic orbifolds.Comment: minor modifications. To appear in J. Reine Angew. Mat

On commensurable hyperbolic Coxeter groups

“…In fact, the group * is closely related to the even unimodular group P SO(II 17,1 ) which, by a result of Emery [4,Theorem 1], is the fundamental group of the (unique up to isometry) hyperbolic n-space form of minimal volume among all orientable arithmetic hyperbolic norbifolds for n ≥ 2. The Coxeter graph of * is given in Fig.…”

“…In this way, in dimension 3, a natural bisection shows that the Coxeter pyramid groups [∞, 3, 3, ∞] resp. [∞, 4, 4, ∞] are subgroups of index 2 in the Coxeter simplex groups [3,4,4] resp. [4,4,4], and the latter group is related to the last but one by an index 3 subgroup relation arising by a tetrahedral trisection.…”

“…[∞, 4, 4, ∞] are subgroups of index 2 in the Coxeter simplex groups [3,4,4] resp. [4,4,4], and the latter group is related to the last but one by an index 3 subgroup relation arising by a tetrahedral trisection. As an example in dimension 4, by [15, Consider the new unit vector f = (e 5 − e 4 )/ √ 2 which is Lorentz-orthogonal to the bisecting hyperplane H f of H 4 and H 5 .…”

“…In the following we make use of the description of P 1 and P 2 according to the proofs of Lemma 1 and Proposition 1. , q 2 , q 3 be the vertices of F such that q i q i+1 ⊥ q i+1 q i+2 for i = 1, 2. In fact, F is a simply asymptotic orthoscheme [4,4,3] in H 3 whose edge lengths satisfy tanh q 1 q 2 = 1/2 and tanh q 1 q 3 = 1/ √ 2 (see [18]). In a similar way, the glue facet F = F(P 2 ) for P 2 is a simply asymptotic orthoscheme [6,3,3] in H 3 whose corresponding edge lengths satisfy tanh q 1 q 2 = 1/2 and tanh q 1 q 3 = 1/ √ 3.…”

“…is of finite index in [4,4,3], and the intersection K 2 := K ∩ [6,3,3] is of finite index in [6,3,3] . Hence, the groups K 1 resp.…”