Cusp Density and Commensurability of Non-arithmetic Hyperbolic Coxeter Orbifolds

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Introduction
Let H n be the real hyperbolic space of dimension n ≥ 2 with its isometry group Isom H n . The quotient space O n = H n / of H n by a discrete subgroup ⊂ Isom H n of finite covolume is a hyperbolic n-orbifold. By Selberg's lemma, each orbifold is finitely covered by a manifold.
In low dimensions, there are different ways to construct hyperbolic orbifolds and manifolds. In this work, we consider only non-compact space forms of dimension three. The arithmetic constructions in the orientable context are related to Bianchi groups, that is, to Kleinian groups of the form PSL(2, O d ) ⊂ PSL (2, C) where O d is the ring of integers in the field Q( √ −d). A topological way is to look at knot and link complements in S 3 that carry a hyperbolic structure. For n = 3, we are interested in cusped hyperbolic Coxeter n-orbifolds arising as quotients by hyperbolic Coxeter groups, that is, by discrete groups generated by finitely many reflections in hyperplanes of H n . A fundamental polyhedron for a hyperbolic Coxeter group is a so-called Coxeter polyhedron P given by a convex polyhedron all of whose dihedral angles are integral submultiples of π . We assume that P as convex hull of finitely many ordinary or ideal points has at least one vertex on the ideal boundary ∂H n . These orbifolds form a very natural and important family of cusped hyperbolic space forms that include orbifolds of small volume in various dimensions up to n = 18 (see [12,13]).
In contrast to higher dimensions, there are infinitely many distinct Coxeter 3orbifolds, and some of them are intimately related to Bianchi orbifolds or knot and link complements as described above (see [1,Sect. 7], [16,Sect. 3], and Remark 4.3, for example). In order to obtain a survey about the variety of cusped hyperbolic orbifolds, we study them up to commensurability. Two hyperbolic n-orbifolds are commensurable if they have a common finite-sheeted cover, which means that their fundamental groups are commensurable (in the wide sense). Notice that properties such as arithmeticity and cocompactness are stable with respect to commensurability. As an example, the arithmetic 1-cusped Gieseking manifold M G , arising by side identifications of an ideal regular tetrahedron S ∞ reg , has a double cover homeomeorphic to the Figure Eight knot complement, and the fundamental group of M G is commensurable to the Coxeter group associated to S ∞ reg as well as to the Bianchi group PSL (2, O 3 ). In the case of arithmetic hyperbolic 3-orbifolds, there is a well developed and very satisfactory theory about the commensurability of Kleinian groups (see [15,18]). In the case of non-arithmetic hyperbolic 3-manifolds, there is a general algorithm for deciding about their commensurability in terms of horosphere packings and canonical cell decompositions (see [6]).
In this work, we study commensurability of infinitely many distinct non-arithmetic 1-cusped hyperbolic Coxeter 3-orbifolds. More precisely, we consider three infinite sequences (R m ), (S m ), and (T m ) describing simultaneously certain Coxeter prisms (see Fig. 3), their reflection groups and the related Coxeter orbifolds, and defined via their Coxeter graphs as below (for details, see Sect. 2.2). These Coxeter orbifolds are 1-cusped and, for m ≥ 7, non-arithmetic.
The aim of this work is to prove the following result. (a) two distinct elements X k and X l belonging to the same sequence are incommensurable; (b) each element R k is incommensurable with any element X l not belonging to the sequence (R m ); (c) the elements S k and T l are incommensurable for k ≥ l.
For the proof of our Theorem, we first exploit some new commensurability conditions for pairs of hyperbolic Coxeter groups such as those given by Fig. 1. These necessary conditions rely upon the Vinberg space and the Vinberg form related to an arbitrary hyperbolic Coxeter group, and they were recently established by the first author [3,4]. This investigation leads to first yet incomplete conclusions. A complete proof of our Theorem is based on the study of the cusp density δ(X m ) of the orbifold X m . The quantity δ(X m ) is given by the ratio of the volume of the maximal (embedded) cusp in X m to the total volume of X m and forms a commensurability invariant in the context of non-arithmetic 1-cusped hyperbolic orbifolds (it is, however, not a complete invariant; see [6,Sect. 1]). For each of the sequences (R m ), (S m ), and (T m ), we derive explicit formulas for δ(X m ) and prove and exploit the strict monotonicity of their cusp density as a function of m. These monotonicity properties are not of uniform nature but help us in a crucial way to provide a coherent and complete proof of the above theorem.
This work is structured as follows. In Sect. 2, we review the basic concepts of hyperbolic Coxeter groups, Coxeter polyhedra and their graphs, and present Vinberg's arithmeticity criterion (see Sect. 2.2). For prisms in H 3 giving rise to the Coxeter realisations (R m ), (S m ), and (T m ) and the related cusped orbifolds, we recapitulate a volume formula in terms of the Lobachevsky function. In this way, the cusp density as presented in Sect. 2.1 takes a more concrete analytic form. In Sect. 3, we introduce the notion of Vinberg's quadratic space and use it to formulate the commensurability conditions for a pair of hyperbolic Coxeter groups in Theorem 3.1. Its impact for subfamilies of groups belonging to the sequences (R m ), (S m ), and (T m ) form our first conclusions presented at the end of the section. In Sect. 4, we treat the cusp density δ(X m ) from a polyhedral point of view and look at the cusp density function for the maximal cusp in a corresponding prism R (α, β) ⊂ H 3 defined by two angular parameters α, β with 0 < α + β < π/2 (see Fig. 3). A key technical result to prove strict monotonicity of δ(X m ) is Proposition 4.2 that describes the cusp volume in R (α, β) in terms of the sign of cos α − √ 2 sin β. In Remark 4.3, we consider the case cos α = √ 2 sin β for α = π/m, m ∈ N ≥3 , and discuss briefly the close connection of the prism R (π/m, β) with Thurston's polyhedral model for the m-chain link complement S 3 \ C m . Finally, and based on Schläfli's differential expression for hyperbolic volume, we are able to provide a complete and self-contained proof of our theorem.

Commensurability of Hyperbolic Orbifolds
Let < Isom H n be a hyperbolic lattice, that is, is a discrete group of isometries acting on H n with a fundamental polyhedron P ⊂ H n of finite volume. The latter property describes as being cofinite. The quotient O n = H n / is a hyperbolic norbifold whose volume is given by the volume of P, also denoted by covol n ( ). Two such orbifolds O n 1 and O n 2 are commensurable if they have a common finite sheeted cover. Equivalently, their fundamental groups 1 , 2 ⊂ Isom H n are commensurable in the sense that there is an element γ ∈ Isom H n such that 1 ∩γ 2 γ −1 has finite index in both 1 and γ 2 γ −1 . The commensurability property for groups in Isom H n yields an equivalence relation preserving characteristics such as discreteness, cofiniteness and arithmeticity. In this context, a fundamental result of Margulis (see [22,Chap. 6], for example) states that a hyperbolic lattice ⊂ Isom H n , n ≥ 3, is non-arithmetic if and only if its commensurator is a hyperbolic lattice, and containing as a subgroup of finite index. In particular, Comm( ) is the (unique) maximal group commensurable with a non-arithmetic hyperbolic lattice .

Cusp Density of a Non-Compact Hyperbolic Orbifold
In the sequel, we study commensurability of different infinite families of cusped nonarithmetic hyperbolic 3-orbifolds. A cusp C of an orbifold O n = H n / is a connected subset of O n that lifts to a set of horoballs with disjoint interiors in H n . The set C gives rise to an ideal vertex q ∈ ∂H n of a fundamental polyhedron for , and C is of the form B q / q where B q ⊂ H n is a horoball internally tangent to q and where q < is the stabiliser of q. By Bieberbach's theory, q is a crystallographic group acting discretely and cocompactly by Euclidean isometries on the horosphere ∂ B q containing a translation lattice of rank 2.
Suppose that a hyperbolic orbifold O n has precisely one cusp C, and that C is maximal, that is, there is no cusp of O n containing C. This means that C is tangent to itself at one or more points. The ratio is called the cusp density of O n (and given by C). The numerator vol n (C) of δ(C) can be computed in terms of the volume of a Euclidean fundamental polyhedron P q for the group q as follows. Pass to the upper half space model for H n in R n + where infinitesimal arc length is given by ds = dx/x n . Suppose without loss of generality that q = ∞ and that the bounding horosphere ∂ B ∞ is the hyperplane {x n = 1} at distance 1 from the ground space R n−1 . Then, the volume of the cusp C is given by (see also [11,Sect. 3 The following result is an easy consequence of the above concepts and facts and will play a crucial role (see [18,Prop. 1], [6,Sect. 2]).

Proposition 2.1
The cusp density is a commensurability invariant for non-arithmetic 1-cusped hyperbolic orbifolds.

Hyperbolic Coxeter Groups and Coxeter Orbifolds
Interpret hyperbolic space in the hyperboloid model H n as a subset of R n+1 equipped with the Lorentzian form q(x) = −x 2 0 +x 2 1 +. . .+x 2 n as usual. The group of isometries Isom H n is given by the group O + (n, 1) of positive Lorentzian matrices.
For N ≥ n + 1, let ⊂ Isom H n be a hyperbolic lattice generated by finitely many reflections s i in hyperplanes H i = e ⊥ i , 1 ≤ i ≤ N , of H n . As a consequence, the vectors e 1 , . . . , e N contain a Lorentzian basis of R n+1 which we suppose to be of Lorentzian norm 1. Consider the convex polyhedron of closed half-spaces H − i ⊂ H n with outer normal vectors e i . The polyhedron P is a Coxeter polyhedron, that is, all the dihedral angles of P are of the form π/m for an integer m ≥ 2. In this way, the group is a hyperbolic Coxeter group and a geometric representation of an abstract Coxeter group in the group O + (n, 1). The associated orbit space of is called a hyperbolic Coxeter n-orbifold. The theory of hyperbolic Coxeter groups and orbifolds has been developed essentially by Vinberg (see [5,20,21] for classification results and further references).
Associated to P and is the Gram matrix G = G(P) of signature (n, 1) formed by the Lorentzian products g ik = e i , e k n,1 . The coefficients of G off the diagonal have the following geometric meaning. (2,5), (3,5), (4,5) k l ∞ ∞

Fig. 2 The four non-arithmetic Coxeter pyramids with exactly one ideal vertex in H 3
In [21, pp. 226-227], Vinberg describes an efficient arithmeticity criterion for a hyperbolic Coxeter group which we only reproduce in the non-cocompact case. To this end, consider 2G(P), and its cycles (of length l) of the form with distinct indices i j in 2 G(P). Then, is arithmetic with field of definition Q if and only if all the cycles of 2 G(P) are rational integers. In this context, define the field K ( ) of all cycles of G(P) and call it the Vinberg field of . For n ≥ 3, the field K ( ) is the smallest field of definition for , and it is moreover an algebraic number field coinciding with the adjoint trace field of . As a consequence, the Vinberg field is a commensurability invariant for (see [4,Sect. 3]).
Often, we visualise a hyperbolic Coxeter group (and its Coxeter polyhedron P) in terms of its Coxeter graph ( ). Each node i of ( ) corresponds to a generator s i (and therefore to the vector e i and the hyperplane H i ). Two nodes i, k are not joined by an edge if the corresponding hyperplanes H i and H k are perpendicular. They are joined by a simple edge if the corresponding hyperplanes intersect under the angle π/3. The edge carries the weight m ik ≥ 4, ∞, or is replaced by a dotted edge (sometimes with weight l ik ), if the hyperplanes H i , H k intersect under the angle π/m ik , are parallel, or at the positive hyperbolic distance l ik , respectively. In contrast to Examples 2.2 and 2.3, there are infinite sequences of Coxeter prisms in H 3 that give rise to non-arithmetic 1-cusped Coxeter orbifolds. They are at the heart of this work and can be characterised as follows. From a combinatorial-metrical point of view, they arise by polar truncation of an orthoscheme R(α, β) ⊂ H 3 with 0 < α + β < π/2. The tetrahedron R(α, β) is an orthogonal tetrahedron of infinite volume bounded by the hyperbolic planes H 1 , . . . , H 4 opposite to the vertices p 1 , . . . , p 4 , say. The planes form one ideal vertex q = p 1 = H 2 ∩ H 3 ∩ H 4 characterised by a Euclidean triangle with angles π/2, β, and β = π/2 − β, and one ultra-ideal vertex The prism R (α, β) ⊂ H 3 with 0 < α + β < π/2 and its Vinberg graph + with positive Lorentzian norm) that we cut off by its polar hyperplane H 4 = {x ∈ H 3 | x, v 3,1 = 0}. Associated to p 4 is the hyperbolic triangle R(α, β) ∩ H 4 with corresponding vertices p 1 , p 2 , and p 3 , and with angles π/2, α, and β. This triangle is at distance l = d H ( p 2 , p 2 ) from the opposite triangle in R(α, β). The truncation by means of the hyperbolic plane H 4 leads to a (simplicial) prism R (α, β) of finite volume that can be described by the Vinberg graph according to Fig. 3.
Here, the nodes i and 4 correspond to the planes H i and H 4 , and two nodes are not joined if the associated planes are Lorentz-orthogonal. For the weight l = l αβ of the dotted edge corresponding to the length of the common perpendicular of H 4 and H 4 , an easy computation exploiting the vanishing of the determinant of the Gram matrix of R (α, β) yields the expression tanh l αβ = tan α tan β. (2.7) For the volume of R (α, β) ⊂ H 3 , there is a closed formula in terms of α, β, and the Lobachevsky function JI(ω) = − ω 0 log |2 sin t| dt as follows (see [10]).
Observe that the Lobachevsky function JI(ω) is odd, π -periodic and satisfies a certain distribution relation. As an example, which allows one to deduce JI(π/6) = 3JI(π/3)/2 ≈ 0.50747 for its maximum value. For computations, the series representation  with Bernoulli coefficients B 1 = 1/6, B 2 = 1/30, . . ., converges rapidly for small ω (see [17,App.]). The formula (2.8) can be derived by integrating Schläfli's differential formula which expresses the infinitesimal volume change of a non-Euclidean polyhedron in terms of the variation of its dihedral angles (see [10], for example). In particular, when keeping the angle parameter β = β 0 constant, the volume differential of R (α, β 0 ) is given by which leads to (2.8) by using vol 3 (R (β 0 , β 0 )) = 0 as integration constant. Among the prisms R (α, β) ⊂ H 3 with 0 < α + β < π/2, there are three distinguished infinite families, indexed by an integer m, of Coxeter prisms R m , S m , and T m with Coxeter graphs given in Fig. 4 (see also Fig. 1 in the Introduction). By means of Vinberg's arithmeticity criterion, the 1-cusped Coxeter orbifolds associated to the Coxeter groups given by R m , S m , and T m are non-arithmetic at least for m ≥ 7. In the sequel and for convenience, we shall use the same symbol X m for the Coxeter prism as well as for the associated reflection group and its quotient space.
Our aim is to prove first that the members belonging to a fixed sequence, and secondly, that most pairs from different sequences are incommensurable hyperbolic Coxeter groups. To do this we follow two different paths. The first one is algebraic and based on the study of Vinberg spaces and the relevant results of the first author [3,4]. We shall see that this approach has limitations. The second path is geometric and based on certain analytic properties of the cusp density function such as strict monotonicity. It leads to a coherent and complete proof of our theorem.

The Vinberg Space and Commensurability
Let m ≥ 7, and consider the three sequences (X m ) of non-arithmetic Coxeter prism groups in Isom H 3 depicted in Fig. 4. The subsequent machinery is due to Vinberg, and the new results about commensurability based on it are due to the first author (see [3,4] and the references therein).
Associated to each group X m of the sequence (X m ) is the Vinberg field K (X m ) generated by all the cycles g i 1 i 2 ...i l of the Gram matrix G(X m ) = (g ik ) of the prism X m (see Sect. 2.2). Following the description as given in Fig. 3, denote by e 1 , . . . , e 4 and e 5 the outer normal unit vectors in (R 4 , · , · 3,1 ) of the hyperplanes H 1 , . . . , H 4 and H 4 =: H 5 bounding X m .
(3.1) Recall that two quadratic forms q 1 and q 2 , defined on vector spaces V 1 and V 2 of dimension m over a field K , respectively, are similar if and only if there exists a scalar λ ∈ K * such that (V 1 , q 1 ) and (V 2 , λq 2 ) are isometric spaces. Representing the quadratic forms q 1 , q 2 by means of their bilinear forms with matrices Q 1 , Q 2 ∈ Mat(m, K ), the isometry of (V 1 , q 1 ) to (V 2 , λq 2 ) then means that there is a matrix S ∈ GL(m, K ) such that Q 1 = S t (λQ 2 )S.
In the case of odd dimensions n ≥ 3, the theorem above combined with the Theorem of Hasse-Minkowski produces the following commensurabilitry condition (see [4,Lem. 3.16]). Proposition 3.2 (Ratio Test) For n ≥ 3 odd, let 1 and 2 be two commensurable hyperbolic Coxeter groups acting on H n with Vinberg field K and Vinberg forms q 1 and q 2 , respectively. Then, det(q 1 ) ≡ det(q 2 ) mod (K * ) 2 .
Let us illustrate the above theorem and examine as far as possible the (in-)commensurability of the non-arithmetic groups X m = R m , S m , and T m for m ≥ 7. In order to establish their Gram matrices G(X m ) = (g ik ) and compute the Vinberg fields, we determine the weights l mp = l π/m,π/ p , p = 3, 4, 6, according to (2.7) and obtain the following results.
cosh l m4 = cos(π/m) √ cos(2π/m) , cosh l m3 = cos(π/m) √ 2 cos(2π/m) − 1 , 3) The extension degree of K m equals [K m : Q] = ϕ(m)/2, where ϕ(k) denotes the Euler totient function counting the positive integers smaller than or equal to k that are relatively prime to k. Recall that ϕ(k) is not injective since, for example, ϕ(2k) = ϕ(k) for odd k. Next, we determine for each X m the Vinberg form by following Vinberg's construction. To this end, we construct the outer normal unit vectors e 1 , . . . , e 5 and choose a basis v 1 , . . . , v 4 for the Vinberg space V (X m ) in the set of vectors defined by (3.1). Their Gram matrix Q(X m ) := ( v i , v k 3,1 ) 1≤i,k≤4 yields the Vinberg form q(V (X m )). For comparison by means of the Ratio Test above, it suffices to compute the determinant of Q(X m ) modulo K 2 m . We summarise the computations as follows. Consider the Coxeter prism R m as given by the Coxeter graph (R m ) depicted in Fig. 4 and with weight l m4 according to (3.2). We put R m in H 3 in such a way that its outer normal unit vectors are given by

First Conclusions
(A) For a fixed sequence (X m ), m ≥ 7, of non-arithmetic Coxeter prism groups given by one of the Coxeter graphs according to Fig. 4, two groups X m and X m with ϕ(m) = ϕ(m ) (and hence different Vinberg fields) are incommensurable. However, if K (X m ) = K (X m ) =: K , the Ratio Test does not allow us to conclude about their incommensurability since the determinants of the Vinberg forms q(X m ) and q(X m ) are equal modulo K 2 . (B) Let k, l ≥ 7. For a group R k and a group X l not belonging to (R m ), the Ratio Test proves their incommensurability. (C) Let k, l ≥ 7. A group S k and a group T l are incommensurable if ϕ(k) = ϕ(l).
In the case K (S k ) = K (T l ) =: K , the Ratio Test does not allow us to conclude incommensurability since the determinants of the Vinberg forms q(S k ) and q(T l ) are equal modulo K 2 .

Cusp Density and Commensurability
In the sequel, we provide a complete proof, based on the cusp density invariant, of the theorem as stated in the Introduction for the infinite sequences (R m ), (S m ), and (T m ) given by Fig. 1. To this end, we generalise the context as follows.
Consider the two-parameter family R (α, β) ⊂ H 3 with 0 < α+β < π/2 of prisms in H 3 as depicted in Fig. 3. Each prism results from polar truncation of an orthoscheme R(α, β) = 1≤i≤4 H − i with ideal vertex q = p 1 and ultra-ideal vertex p 4 . For i ≤ 3, denote by p i the intersection of H 4 with the geodesic defined by the vertices p i and p 4 . By construction, the vertices p 1 , p 2 , and p 3 describe the hyperbolic triangle [ p 1 p 2 p 3 ] opposite to the triangular base [ p 1 p 2 p 3 ] of the prism R (α, β), and it has angles π/2, α, and β while being orthogonal to H 1 , H 2 , and H 3 .
The quantity l αβ is given by (2.7) and appears as coefficient in Schläfli's differential according to (2.11).
Our first aim is to derive a formula for the (polyhedral) cusp density Here, C(α, β) is the maximal cusp inside R (α, β) and results from intersecting the maximal horoball B q associated to q with the prism R (α, β). Notice that B q is tangent to the facet(s) closest to q but disjoint to the remaining one among all facets not containing q in R (α, β). More precisely, the orthogonality properties reigning in R (α, β) imply that the horosphere S q = ∂ B q is either touching H 1 at p 2 as depicted in Fig. 5, or H 4 at p 1 as depicted in Fig. 6. Therefore, the size of C(α, β) depends on the geometric position of the planes H 1 , . . . , H 4 and H 4 which can be quantified in terms of the distance = (α, β) of S q to H 1 and to H 4 , respectively. The prism R (α, β) ⊂ H 3 with 0 < α + β < π/2 such that cos α ≥ √ 2 sin β The following result about horocycle geometry will be useful (see [2,Sect. 4]). Consider a hyperbolic triangle T with one ideal vertex Q, a right angle at the vertex A 1 and the angle ω at the vertex A 2 . Let a = d H (A 1 , A 2 ), and consider the horocyclic segment of Euclidean length h based at Q and passing through A 1 . The situation is depicted in Fig. 7. H (A 1 , A 2 ) according to Fig. 7. Then h = cos ω = tanh a. R (α, β) ⊂ H 3 with 0 < α + β < π/2 be a hyperbolic prism with one ideal vertex q. Then, the volume of the maximal cusp neighborhood C(α, β) of q in R (α, β) is given according to the following dichotomy.

Proposition 4.2 Let
Proof Start from the representation R (α, β) = 1≤i≤5 H − i where the plane H 5 equals the truncating polar plane H 4 associated to the ultra-ideal vertex p 4 of the underlying orthoscheme R(α, β). Since the maximal cusp C(α, β) is either tangent to H 1 at p 2 or to H 4 at p 1 , there are only two possible cases for the relative position of C(α, β), and they are depicted in Figs. 5 and 6, respectively. In both cases, the Euclidean area of the right-angled triangle [s 2 s 3 s 4 ] forming the boundary of C(α, β) is given by (h 2 4 /2) cot β where h 4 denotes the Euclidean length of the segment [s 2 s 3 ] (see Fig. 5). By (2.3), the volume of C(α, β) equals (h 2 4 /4) cot β. Hence, it remains to determine the quantity h 4 in terms of α and β as asserted. Accordingly, we distinguish two cases.
Case (i) Suppose that the horosphere S q centred at q touches the plane H 1 at p 2 . By the orthogonality properties of R (α, β), the dihedral angle α is equal to the angle at p 3 in the triangle [qp 2 p 3 ]. Hence, by Lemma 4.1, the Euclidean length h 4 of the horocyclic segment [s 2 s 3 ] is equal to cos α implying that vol 3 (C(α, β)) = (1/4) cos 2 α cot β.
It remains to show that the above assumption holds if cos α ≤ √ 2 sin β. Since the angle at s 4 in the Euclidean triangle [s 2 s 3 s 4 ] is equal to β, the Euclidean length h 3 of its hypotenuse is given by Observe that h > 1 since α + β < π/2, and that furthermore h = cosh d H ( p 1 , p 2 ) by elementary trigonometry for [ p 1 p 2 p 3 ].
Next, we show that the horosphere S q does not intersect the plane H 4 which, by the orthogonality properties of R (α, β), is equivalent to show that = d H ( p 1 , s 4 ) ≥ 0. For this, we put the Lambert quadrilateral [qp 1 p 2 p 2 ] in the upper half plane model for H 2 as follows. Assume without loss of generality that its ideal vertex q is ∞, and that the horocycle defined by the segment [s 2 s 4 ] and of Euclidean length h is at height 1; see Fig. 8. For the distance = d H ( p 1 , s 4 ), we have = log 1 ρ , where ρ denotes the radius of the geodesic semicircle carrying the edge [ p 1 p 2 ]. The geodesic semicircle carrying the edge [ p 2 p 2 ] is of radius 1 and orthogonal to the former one. Furthermore, the centers of these semicircles are at (Euclidean) distance h given in (4.2). Hence, As a consequence, ≥ 0 if and only if ρ ≤ 1, which in turn is equivalent to This finishes the proof of (i).
Case (ii) The proof is very similar to the one for (i). Suppose that the horosphere S q centred at q touches the plane H 4 at p 1 according to Fig. 6. We determine first the quantity h = h 3 giving the Euclidean length of [s 2 s 4 ] in terms of the hyperbolic length of the edge [ p 2 p 2 ] in the quadrilateral Q = [p 2 p 2 p 3 p 3 ] opposite to q. The angle at p 2 in Q equals β while the other angles of Q are right ones. Hence, Q is a Lambert quadrilateral giving the identity Finally, it remains to show that the horosphere S q does not intersect the plane H 1 which, by the orthogonality properties of R (α, β), is equivalent to show that = d H (s 2 , p 2 ) ≥ 0. Again, consider the quadrilateral [qp 1 p 2 p 2 ] in the upper half plane model for H 2 and assume that its ideal vertex q is ∞, and that the horocycle defined by the segment [s 2 s 4 ] of Euclidean length h is at height 1. By performing the exchanges Figure 8 gets suitably adapted. As in (4.3), we deduce that with the consequence that = log(1/ρ) ≥ 0 if and only if cos α ≥ √ 2 sin β.

Remark 4.3
Consider the limiting case cos α = √ 2 sin β in Proposition 4.2. The cusp C(α, β) touches both, the plane H 1 at p 2 and H 4 at p 1 in the prism R (α, β) (see Figs. 5 and 6). In the particular instance α = π/k with k ∈ N ≥3 , the prism P k := R (π/k, β) appears as building block for each of the two isometric drums that glued together represent a polyhedral model P of the (orientable) complement S 3 \ C k of the sphere S 3 by the k-link chain C k . This construction is due to and nicely illustrated by Thurston [19,Exam. 6.8.1]. A closer look reveals that each drum can be decomposed into 4k copies of P k so that the polyhedron P associated to S 3 \ C k is an ideal one consisting of 8k prisms of type P k . As a consequence, the volume of S 3 \ C k is given by where β = π/2 − β by convention. For k = 3 and k = 4, the fundamental group of S 3 \ C k is commensurable to PSL(2, O 7 ) and PSL(2, O 3 ), respectively (see [19,Examples 6.8.2 and 6.8.3]). The quotient space of S 3 \ C k by the rotational symmetry group Z k of C k is obtained by generalised Dehn surgery on the Whitehead link W , so that Finally, we remark that the manifold S 3 \ W is commensurable with the 2-cusped Coxeter orbifold given by the Coxeter pyramid group with graph • - Our next aim is to analyse the cusp density δ(α, β 0 ) for fixed β 0 with 0 < α+β 0 < π/2 according to (4.1) and to prove strict monotonicity for the function on a suitable interval [0, α 0 ] with α 0 ∈ (0, π/2). We treat the cases β 0 = π/4, β 0 = π/3, and β 0 = π/6 separately in view of the related sequences (R m ), (S m ), and (T m ) given by Fig. 4. We start with the easiest case.
For the two remaining cases β 0 = π/3 and β 0 = π/4, the monotonicity behavior differs but the proof will be uniform. Proof First, observe that cos α ≤ √ 2 sin β 0 holds for all α in the case (a) with β 0 = π/3 as well as in the case (b) with β 0 = π/4. Hence, by (i) of Proposition 4.2, the cusp volume c(α) is given in both cases by In contrast to the function given by (4.7), the numerator c(α) of δ(α) given by (4.8) is strictly decreasing so that we can not conclude as in the proof of Lemma 4.4. Here, we proceed as follows. Let l(α) = l αβ 0 be the length of the ridge of α which is related to v(α) according v (α) = −l(α)/2. Again, v(α) is a strictly decreasing function. By (2.7), l(α) satisfies the identity l(α) = artanh (tan α tan β 0 ). (4.9) We study the sign of the derivative of δ(α) that can be expressed as More precisely, we investigate whether the sign of the quantity behaves as claimed according to the cases (a) and (b).
We are now ready to provide a uniform proof of the following result announced in the Introduction. This proof is different by nature and allows us to complete the partial conclusions (A) and (C) based on Vinberg's form as stated at the end of Sect. 3.
Theorem For an integer m ≥ 7, consider the three sequences of non-arithmetic 1cusped hyperbolic Coxeter 3-orbifolds induced by (R m ), (S m ), and (T m ) according to Fig. 1. Then: (a) two distinct elements X k and X l belonging to the same sequence are incommensurable; (b) each element R k is incommensurable with any element X l not belonging to the sequence (R m ); (c) the elements S k and T l are incommensurable for k ≥ l.
Data Availability Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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