On volumes of arithmetic quotients of PO (n , 1)^° , n odd

Abstract

We determine the minimal volume of arithmetic hyperbolic orientable ndimensional orbifolds (compact and non-compact) for every odd dimension n ≥ 5. Combined with the previously known results it solves the minimal volume problem for arithmetic hyperbolic n-orbifolds in all dimensions.

Scissors Congruence, the Golden Ratio and Volumes in Hyperbolic 5-Space

“…The notion of graph of a pseudo-Coxeter orthoscheme is defined in an analogous way by a graph with edge weigths q p corresponding to (non-right) dihedral angles of the form pπ/q for integers p ≥ 1, q ≥ 3. Of interest will be, among others, the pseudo-Coxeter orthoschemes in H 5 given by (1) In the sequel, by considering dissections of certain Coxeter and quasi-Coxeter polyhedra, new polyhedra will arise with dihedral angles of mixed type, that is, some angles are commensurable with π , some are not. In order to keep the notations as concise as possible, we describe these polyhedra by graphs with rational weights in the first case and with letters such as α, λ, ω representing angular parameters in the second case (see the examples (5), (6) …”

“…We point out that the field Q( √ 5), containing the golden ratio τ , is the ground field associated with the (unique) arithmetic hyperbolic 5-orbifold of minimal volume (cf. [4] and [1]) while μ 5 is the maximal volume among all hyperbolic 5-simplex volumes (cf. [6] and [13]).…”

Hyperbolic Orbifolds of Minimal Volume

“…Recently, in a series of articles, Martin together with Gehring [13,14], Marshall [24] and with Maclachlan and Reid [12], settled the minimal volume problem in a complete and unified way. The works of Martin and his coauthors, put together, prove that the space H 3 / [3,5,3] is the unique three-orbifold of minimal volume. The proof consists of several different parts, beginning with the fact that the rotation subgroup of [3,5,3] is a two-generator, discrete, non-elementary subgroup of P SL(2, C) of restricted order elliptics.…”

“…For n = 2, the above problem was solved by Siegel [27] who showed that the arithmetic Coxeter triangle group [3,7], defined over the field Q[cos(2π/7)] (resp. [3, ∞], defined over Q) provides the unique compact (resp.…”

“…In the arithmetic case, Chinburg and Friedman [6] proved that the Coxeter tetrahedral group [3,5,3] provides, in a unique way, the smallest representative in the class of orientable arithmetic three-orbifolds. The group [3,5,3] In the non-compact case, Meyerhoff [25] proved that the singly cusped quotient of H 3 by the orientation preserving subgroup of the Coxeter simplex [3,3,6] is of minimal volume among all oriented non-compact hyperbolic 3-orbifolds. This orbifold can also be represented as quotient space by the arithmetic group P SL (2, O 3 ), where O 3 is the ring of integers of the field k 2 = Q[ √ −3].…”