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Department of Mathematics, University of Fribourg, CH-1700 Fribourg, Switzerland
Received: 10 July 2017 / Accepted: 15 August 2017 / Published: 22 August 2017
We study lattices with a non-compact fundamental domain of small volume in hyperbolic space . First, we identify the arithmetic lattices in of minimal covolume for even n up to 18. Then, we discuss the related problem in higher odd dimensions and provide solutions for and in terms of the rotation subgroup of certain Coxeter pyramid groups found by Tumarkin. The results depend on the work of Belolipetsky and Emery, as well as on the Euler characteristic computation for hyperbolic Coxeter polyhedra with few facets by means of the program CoxIter developed by Guglielmetti. This work complements the survey about hyperbolic orbifolds of minimal volume.
Let and consider a hyperbolic lattice, that is, a discrete group whose orbit space or orbifold is of finite volume. By a celebrated result of Kazhdan and Margulis, the set of all volumes has a positive minimal element . In the work , we provided a survey about the values and the volume minimising n-orbifolds for dimensions n satisfying by taking into account (non-)compactness, orientability, arithmeticity, and dimension parity.
In this work, we consider only the volumes of non-compact or cusped hyperbolic n-orbifolds and study the corresponding volume spectrum
with minimal element . The set contains the proper subset of volumes of orientable quotient spaces of by arithmetic lattices in with corresponding minimal element . By deep results of Belolipetsky and Emery (see [2,3,4,5]), the values are explicitly known for . Our aim is to describe the hyperbolic lattices whose covolumes equal for .
In this context, hyperbolic lattices generated by finitely many reflections in hyperplanes of , called hyperbolic Coxeter groups, are of particular interest (see Section 2.3). In fact, for , the smallest covolume hyperbolic Coxeter n-simplex groups (generated by reflections) are all arithmetic and yield the unique non-compact volume minimisers (up to a factor of two, in the exceptional case ; for references, see , Section 3). In this way, the values and could be entirely specified. However, in , cofinite Coxeter simplex groups do not exist for and, apart from Borcherds’ example  for , nothing is known about the existence of cofinite hyperbolic Coxeter groups for .
In the sequel of our commensurability classification of Coxeter pyramid groups with generators existing up to dimension 17 (see ), Guglielmetti  developed the software program CoxIter testing various properties such as arithmeticity and providing invariants such as the Euler characteristic of a hyperbolic Coxeter group. In Section 2.6, we give several instructive examples. By a result of Emery , the covolume of the single Coxeter pyramid group with Coxeter graph given by Figure 1 yields the minimal value among all for (see also Section 3.1.2).
Based on these facts, we are able to identify the orientable cusped arithmetic hyperbolic n-orbifolds as orbit spaces by the action of certain hyperbolic Coxeter groups for the even dimensions n with and for the odd dimensions and (the proof for is based on a combinatorial argument due to S. Tschantz ). The results are presented in Proposition 1, Proposition 2 and Proposition 3 of Section 3.1. The work ends with a couple of remarks about the mysterious case of dimension and the non-arithmetic case.
2. Hyperbolic Lattices with Parabolic Elements
2.1. Hyperbolic n-Space
Denote by one of the standard geometric spaces given by either the Euclidean space , the standard sphere , or hyperbolic space . View each space in the context of a linear space equipped with a suitable bilinear form . In particular, is interpreted as a connected component of the two-sheeted hyperboloid in according to:
In this picture, the isometry group is isomorphic to the group of positive Lorentzian matrices leaving invariant the product and (cf.  Chapter 3). By passing to the upper half space model of in , the line element and the volume element are given by:
Furthermore, a horizontal hyperplane , where , carries a Euclidean metric up to a distortion factor . Such a subset is called a horosphere based at ∞ and bounds the horoball above it.
2.2. Cusped Hyperbolic Orbifolds of Finite Volume
Consider a discrete subgroup with fundamental domain , and suppose that is of finite covolume or cofinite for short. This means that the volume of the orbifold , as given by the volume of D, is finite, and we call a hyperbolic lattice, for short. In the sequel we are particularly interested in lattices giving rise to non-compact or cusped orbifolds . Such a group contains at least one non-trivial subgroup of parabolic type whose elements stabilise a point , say (see , Section 3.1). Associated to the fixed point q at infinity is a cusp neighborhood which is an embedded subset given by the quotient of a -precisely invariant horoball in . The action of on the boundary horosphere is by Euclidean isometries and with a compact fundamental domain so that can be interpreted as a crystallographic subgroup of . By Bieberbach’s theory, any crystallographic group in contains a finite index translational lattice of rank . For , it is known (see [13,14] and also Section 3) that small volume cusped hyperbolic n-orbifolds are intimately related to dense lattice packings in .
2.3. Hyperbolic Coxeter Polyhedra and Discrete Reflection Groups with Few Generators
Consider a geometric polyhedron , that is, P is an n-dimensional convex polyhedron of finite volume in , bounded by hyperplanes with unit normal vector directed away from, say, P. Denote by , the Gram matrix of P. In [15,16], Vinberg developed a very satisfactory theory to conclude the existence and describe arithmetic, combinatorial, and metrical properties of P in terms of . There are explicit criteria by means of certain submatrices of allowing one to decide whether P is compact or of finite volume. In this work, we are dealing mainly with non-compact hyperbolic polyhedra of finite volume, being the convex hull of finitely many ordinary points in and at least one point in the boundary at infinity of . In fact, the fundamental group of a finite volume cusped hyperbolic n-orbifold admits a fundamental domain whose closure is a hyperbolic polyhedron and any of its parabolic subgroups stabilises a vertex at infinity of the fundamental polyhedron.
A geometric polyhedron is a Coxeter polyhedron if all of its dihedral angles formed by pairs of intersecting hyperplanes in the boundary of P, and with associated ridge , are of the form for an integer . Coxeter polyhedra arise in a natural way as building blocks in the context of regular polyhedra and as closures of fundamental regions of discrete reflection groups. Denote by the group generated by the N reflections with respect to the hyperplanes H bounding the Coxeter polyhedron P; the group is called the geometric Coxeter group associated to P. The group is a discrete subgroup in which admits a particularly simple presentation with relations satisfying and
for distinct generators and , with hyperplanes H and intersecting in along a ridge where P has dihedral angle .
Most conveniently, geometric Coxeter polyhedra of simple combinatorics (and Coxeter groups with few generators) are described by their Coxeter graph . Each node in corresponds to a hyperplane H (and the reflection ) and is joined to another node by an edge with weight m if the corresponding dihedral angle, formed by their hyperplanes at the ridge F in P, equals . Usually, edges with a weight 2 are omitted and edges with weight 3 (resp. 4) are drawn as simple (resp. double edges). In the hyperbolic case, and for parallel hyperplanes intersecting in , the nodes are connected by a bold edge; for hyperplanes disjoint in and of hyperbolic distance l, the nodes are connected by a dotted edge (and sometimes marked with the weight l).
In contrast to the spherical and Euclidean cases, Coxeter polyhedra in are classified only very partially. There is a complete list for hyperbolic Coxeter simplices, characterised by , and they exist for , only. Hyperbolic Coxeter polyhedra with are classified and they exist for . Notice that examples of compact Coxeter polyhedra in are known just for . In higher dimensions, there are only single examples in for , and 21 whose discovery is due to Kaplinskaja, Vinberg, and Borcherds, respectively. Notice that Coxeter polyhedra in do not exist for . For a survey, we refer to information and references collected on the webpage of Felikson and Tumarkin .
As for even dimensions above 17, there are only the two Coxeter polyhedra and explicitly known (for more details, see Example 7). The polyhedron exists in and is non-compact and bounded by 37 hyperplanes only forming dihedral angles of and . The polyhedron was discovered by Kaplinskaja and Vinberg  and is associated with the maximal reflection group in the group of integral positive linear transformations leaving invariant the unimodular quadratic form for . Observe that the quadratic forms are reflective in the above sense, providing finite volume non-compact hyperbolic Coxeter polyhedra , for . More precisely, the group is the automorphism group of the odd unimodular Lorentzian lattice with quadratic form whose maximal reflection subgroup is of finite index equal to the order of the symmetry group of its (fundamental) Coxeter polyhedron . The order of is different from 1 for and equal to 2 (resp. 4) for (resp. ), whereas the symmetric group appears according and . The results can be found in [18,19] and ( part II, Chapter 6, §2).
2.4. Arithmetic Hyperbolic Coxeter Groups
The group is a model of an arithmetic group, a notion which will not be explained in detail here (see [4,5], for example). One characterisation is —by using a result of Margulis (see , Theorem 10.3.5, for example) —that a lattice is non-arithmetic if and only if its commensurator group is discrete in (and containing G with finite index). Here, the group is defined by:
However, for hyperbolic Coxeter groups such as , Vinberg developed a very useful criterion for arithmeticity. This criterion simplifies drastically when the group has a non-compact (fundamental) Coxeter polyhedron . Consider the Gram matrix of P and form the matrix with coefficients for . A cycle in H is a product of the type .
Theorem (Vinberg’s Criterion).
Let be a non-compact hyperbolic Coxeter polyhedron with Coxeter group Γ and Gram matrix G. Then, Γ is arithmetic if and only if all the cycles of the matrix are rational integers.
The non-compact hyperbolic Coxeter simplices with graphs given in Table 1 are all arithmetic.
The Coxeter polyhedron given by the graph in Figure 1 is bounded by 19 hyperplanes and has precisely two vertices at infinity. It has the combinatorial type of a pyramid over a product of two (eight-dimensional) simplices. Coxeter polyhedra of this type have been classified by Tumarkin . The polyhedron yields an arithmetic reflection group , as is easily checked by means of Vinberg’s criterion above. The group is the maximal reflection in the automorphism group of the even unimodular Lorentzian lattice . Due to the obvious two-fold symmetry of the graph, one can pass to the -extension of the group , which is arithmetic of half the covolume of .
2.5. The Euler Characteristic and the Covolume of a Hyperbolic Coxeter Group
Let be a Coxeter group with presentation according to (1) and fundamental polyhedron of finite volume. Consider the growth series
where is the word length of with respect to the generating set S of . Denote the set of all subsets T of S such that the group generated by the elements in T is finite. Notice that the groups of type are spherical Coxeter groups with finite growth series. In order to represent their growth polynomials, we use the standard notations , and so on, and denote by the exponents of the Coxeter group (see , Section 9.7). For the list of irreducible finite Coxeter groups, see Table 2.
By a result of Steinberg , is the power series of a rational function and satisfies the following important formula.
where . For the Euler characteristic , one obtains, for any abstract Coxeter group ,
In terms of the volume of P and therefore the quotient space , one deduces the following identity (see ).
The formulas (3) and (5) are very useful when computing the volume of an even-dimensional hyperbolic Coxeter polyhedron. Since the list of irreducible finite Coxeter groups is known and comparatively short (see Table 2), this volume computation can be realised by a computer program.
2.6. The Computer Program CoxIter
By means of the computer program CoxIter designed by Guglielmetti  in 2015 (freely accessible online https://coxiter.rgug.ch/, https://coxiterweb.rafaelguglielmetti.ch/), different invariants of a Coxeter group acting by reflections on can be computed. The input are the dimension n and the Coxeter graph with the number of its nodes and with the edge weights (either 0 or for parallel or disjoint hyperplanes, respectively). Then, the program CoxIter answers questions such as cocompactness, cofiniteness, arithmeticity, Euler characteristic and covolume of , number of vertices at infinity, and the f-vector (with components equal to the number of i-dimensional faces) of its Coxeter polyhedron P.
Consider the two Coxeter pyramids and with Coxeter graphs given by Figure 2 and Figure 3. By the tools mentioned in Section 2.4 and Section 2.5, one can check easily that the associated Coxeter groups and are arithmetic. By means of the webversion of CoxIter, one computes their Euler characteristic as being equal to (see also , Appendix A3) and (see the output given in Figure 4).
Consider the Coxeter group generated by 17 reflections in with graph given in Figure 5. It was discovered by Vinberg as being the maximal reflection subgroup of the group of units of the unimodular quadratic form of signature . The program CoxIter yields the following information.
Alternating sum of the components of the f-vector: 0
Euler characteristic: -87757/289236647411712000
Covolume: pi^7 * 87757/305359330843607040000
The Coxeter group with Coxeter graph given by Figure 6, also discovered by Vinberg, is the maximal reflection subgroup of the group of units of the unimodular quadratic form . Here, CoxIter provides the following data (see also , Table 2).
The Coxeter group with 19 generators and with Coxeter graph given by Figure 7 was discovered by Tumarkin . It is distinguished by the fact that it represents the single top-dimensional cofinite hyperbolic Coxeter group in with generators. Furthermore, is arithmetic and CoxIter provides the following further details (see also , Table 2).
In  Section 7, Vinberg constructed a particular quadratic form by considering the lattice , where is the even unimodular quadratic lattice of signature , and e is a long root of norm two. Recall that the automorphism group of the lattice yields the standard form , which is reflective with maximal reflection subgroup of index 2 (see Section 2.3 and Figure 1). By means of an algorithm developed earlier by Vinberg, he proved that the lattice L yields a reflective quadratic form as well, and this by construction of a finite volume Coxeter polyhedron with explicit description of the Coxeter graph. The associated arithmetic Coxeter group is given by Figure 8. In particular, by applying Guglielmetti’s program CoxIter to the Coxeter group , which is generated by 24 reflections, the Euler characteristic equals . Hence, the covolume can be computed as follows (see  Table 5).
In an earlier work, together with Kaplinskaja, Vinberg  used the algorithm mentioned above to prove that the unimodular quadratic forms are also reflective for and (while this is not the case for ). Furthermore, they provided the corresponding Coxeter graphs. By CoxIter, Guglielmetti computed the covolume of the Coxeter group , related to , which is generated by 37 reflections, and found that so that the covolume is (see  Table 4). Observe that the numerator of is identical to the one of , a direct consequence of the formula (4) and the fact that both Coxeter graphs have integer weights 2 and 3, only.
In the sequel of the two hyperbolic Coxeter groups and , there are, up until now, no such reflection groups in known with explicit presentation and covolume for . In , Ratcliffe and Tschantz considered arithmetic n-space forms given as quotients of by the principal congruence subgroups of level . The spaces are smooth manifolds for . The orbifolds , which are closely related to the quadratic form , are well understood for (see examples above). By means of Theorems 6, 8 and 25 in , Ratcliffe and Tschantz provide an explicit volume formula for for all and odd prime numbers p, by exploiting a result of Siegel related to the case .
3. Minimal Volume Cusped Hyperbolic Orbifolds
Let be a lattice with fundamental polyhedron such that the n-orbifold is non-compact. This implies that P is the convex hull of finitely many vertices, with at least one vertex q belonging to , whose stabiliser is a crystallographic group (see Section 2.2). Consider the volume spectrum:
of all cusped hyperbolic n-orbifolds together with its minimal element for each .
In , a universal lower volume bound for cusped hyperbolic n-manifolds has been established that also holds in the singular case (see  Section 2,  Section 2 and  Chapter 5). It provides a lower bound for in terms of (lattice) packing densities and orders of maximal point groups.
More precisely, denote by the Euclidean k-volume functional and by a Euclidean r-ball. Let be the maximal point group order of elements in a fixed -class of maximal, finite, absolutely irreducible subgroups of , and let be the maximal lattice packing density in Euclidean k-space. In particular, one has that , which is equal to twice the order of the Conway group , and that and (see ). Notice that for , the densest lattice packings are known and intimately related to root lattices. Moreover, by a recent fundamental result of Viazovska , the root lattice yields the densest sphere packing in , leading to a proof of similar flavour, by Cohn, Kumar, Miller, Radchenko and Viazovska , showing that the Leech lattice is an optimal sphere packing in .
The value denotes the simplicial horoball density, that is, the density of horoballs based at the vertices of an ideal regular simplex in . By means of the volume of an ideal regular n-simplex with its representation as an infinite series given by Milnor, can be expressed as follows (see , Theorem 2.1).
(Kellerhals [12,28]). Let , and let be a hyperbolic n-orbifold with cusps. Then,
As an example, by using the data of the Leech lattice in (8), the volume of a 25-dimensional hyperbolic orbifold with cusps can be bounded from below as follows.
The questions about the explicit value and the realisation of the minimal volume as have only partial answers so far. By the above theorem, one deduces the bound for , which is a key ingredient in answering the question for . In fact, the classical results for the dimensions , due to Siegel, and , due to Meyerhoff, were extended by Hild-Kellerhals  for and by Hild [13,14] for , with the consequence that, for these dimensions, the unique covolume minimising groups are given by certain hyperbolic Coxeter groups (up to index two in dimension ). For a survey, see . It turns out that all these groups are arithmetic and related to a tessellation of by a 1-cusped Coxeter simplex.
3.1. The Arithmetic Case
In view of the situation just described and when looking to dimensions , it makes sense to study the (proper) subset of all volumes of orientable cusped hyperbolic n-orbifolds with arithmetic fundamental groups and to ask corresponding questions about the minimal element in , denoted by .
For an arbitrary dimension n, there is the standard arithmetic group of automorphisms that leave the form invariant. This group provides a first candidate for small volume. As already mentioned, for , the group is reflective and can be written as the semi-direct product of its cofinite maximal reflection subgroup and the symmetry group of its (fundamental) Coxeter polyhedron . For the covolume of , one has the following result for n even.
with a plus sign if mod 8 and the minus sign if mod 8.
3.1.1. Even Dimensions
In a much more general context, Belolipetsky and Emery (see [2,3,4,5] and ) successfully exploited a relevant structural result of Prasad and determined the explicit value of for the cases of orientable cusped arithmetic orbifolds of even dimensions and of odd dimensions , respectively (notice that non-compactness is not a constraint in their works). In particular, for even dimensions, there is the following result in terms of the Euler characteristic.
(Belolipetsky [2,3]). For each dimension there is a unique orientable cusped arithmetic hyperbolic n-orbifold of minimal volume. It has Euler characteristic:
where if , and if .
While the evaluation of Belolipetsky’s theorem for even dimensions coincides with the previously mentioned (and more explicit) results of Kellerhals and Hild (see  and ), it yields the following Table 4 for even dimensions .
Let us compare the results in Belolipetsky’s theorem with the values and obtained in Section 2.6. Some of the Coxeter group examples presented in Section 2.3 have Coxeter graphs admitting a non-trivial symmetry group of order k, say, which corresponds to the symmetry group of the same order of the associated Coxeter polyhedron. By extending the Coxeter group by , we pass to a group of -times the covolume of the original group. Furthermore, since reflections are orientation reversing isometries, we need to pass to the index two orientation preserving subgroup. By taking into account the uniqueness property in Belolipetsky’s result, we can deduce the following explicit volume minimality result.
Let n be even with . Then, the unique orientable cusped arithmetic hyperbolic n-orbifold is given by the action on of the index two orientation preserving subgroup of the group given by Table 5.
3.1.2. Odd Dimensions
For odd dimensions , the results of Hild provide a complete picture about minimal volume cusped hyperbolic n-orbifolds, arithmetically defined or not, including proofs for uniqueness, a presentation of the fundamental group, and the value of . The orbifolds are closely related to Coxeter simplices, which do not exist for dimensions (see Example 1 and Table 1). Combinatorially very close are pyramids over a product of two simplices of positive dimensions, which have been studied and classified in the Coxeter case by Tumarkin (see Section 2.3). These groups are generated by reflections, they are all non-compact and exist in for all with . By Vinberg’s arithmeticity criterion (see Section 2.4), one verifies easily their arithmeticity when . According to the corresponding commensurability classification performed in , one has five Coxeter pyramid groups in falling into two commensurability classes, three Coxeter pyramid groups in forming one commensurability class, and finally the single Coxeter pyramid group that is closely related to the automorphism group of the even unimodular group (see Example 2). Among the five arithmetic Coxeter pyramid groups , which fall into two commensurability classes, the group given by the graph in Figure 9 has smallest covolume, and among the three commensurable Coxeter pyramid groups in , the group given by Figure 11 has smallest covolume (see  and ).
In order to identify explicitly—if possible—the minimal volume orientable cusped arithmetic hyperbolic n-orbifolds for odd, we provide details of the corresponding result of Belolipetsky and Emery (see Section 3.1.1).
(Belolipetsky, Emery [4,5]). For each dimension , there is a unique orientable arithmetic cusped hyperbolic n-orbifold of minimal volume. Its volume is given by the following formula.
If r is even:
In , Emery described in more detail the fundamental group of the orientable arithmetic cusped orbifold of minimal volume as follows. For (mod 8), the group PSO(II is conjugate to in , while for (mod 8) the group PSO(I is conjugate to a subgroup of index 3 of in . For (mod 4), the group is commensurable to the group PO of integral automorphisms of the Lorentzian form of signature given by:
By a result of Mcleod , the group PO is reflective for . As an extension of their work for PO to the group PO, Ratcliffe and Tschantz determined the covolumes of the groups PO and, for each (mod 4), they computed furthermore the commensurability ratio of and PO showing that (see , (35)).
In view of these results and the knowledge of hyperbolic Coxeter group candidates in for and , we provide the following new characterisation of for and mention briefly the known result of Emery  in the case .
The case . Since (mod 4), the group of minimal covolume is commensurable to the group PO, whose cofinite maximal reflection subgroup was described by Mcleod. More precisely, the Coxeter graph of , given by Figure 10, has 15 nodes and shows a two-fold symmetry. Denote by the Coxeter polyhedron of .
In particular, by the result (, Table 1), one gets the value
where is Dirichlet’s L-function according to (, (12)), as well as (see , Section 7). This implies that and that .
Now, consider the group which has smallest covolume among all Coxeter pyramid groups in and let be its Coxeter pyramid. Based on an observation of Tschantz  when comparing the corresponding Coxeter graphs, there is a close combinatorial relation between the polyhedron P associated to Mcleod’s Coxeter group and the Coxeter pyramid . In fact, pass to the double of the polyhedron P by reflecting it in the bounding hyperplane depicted by in the graph of Figure 8. Then, the polyhedron is bounded by 16 hyperplanes. Reflect recursively the pyramid in its facets while staying inside the polyhedron . The image pyramids match along their facets or line up with the facets of . It takes exactly 4030 copies of the pyramid to fill . As a consequence and by (12), one obtains:
Putting everything together, one deduces that . By passing to the index two subgroup of orientation preserving isometries in the Coxeter pyramid group , one finally obtains the following result.
The orientable arithmetic cusped hyperbolic orbifold of minimal volume is the quotient of by the rotation subgroup of . The value is given by (13).
The case . Since 13 satisfies (mod 8), the fundamental group of the orientable arithmetic cusped orbifold of minimal volume is commensurable to the special unimodular group PSO(I, with ratio of their covolumes equal to 3 (see , Proposition 5). By the result , Theorem 6, of Ratcliffe and Tschantz mentioned in Remark 1, the covolume of PSO(I, being of index two in PO(I, can be expressed as follows.
Since the group PO(I is known to be reflective and equal to the Coxeter pyramid group , with the graph given by Figure 11, we can deduce the following result.
The orientable arithmetic cusped hyperbolic 13-orbifold of minimal volume is the quotient of by the rotation subgroup of . Its volume is given by (14).
The case . Let us finish by mentioning the result , Theorem 2, of Emery. It states that for (mod 8), the minimal volume orientable arithmetic cusped hyperbolic n-orbifold is the quotient space . For , the group PO(II) is reflective and is the semi-direct product of the reflection group with the symmetry group of , where is Tumarkin’s Coxeter pyramid with graph given in Figure 1 and described in Example 2. By exploiting the theorem above, one gets the following volume identification (see , Corollary 3, Corollary 4).
As mentioned by Emery in , Section 3, the space has minimal volume among all orientable arithmetic hyperbolic n-orbifold , compact or not, for . This means that for all , .
When looking at realisations of orbifolds with volumes equal to the minimal values for , there remains a need to study the case and to look for a candidate in the commensurability class of PO that is conjugate to the fundamental group of the minimal volume hyperbolic orbifold of dimension 15.
It is an interesting but difficult question whether, and to what extent, non-arithmetic considerations can perturb the picture described in Section 3 in such a way that for some .
This work arose during a research stay at the ICTP Trieste. The author would like to thank Fernando Rodriguez Villegas for the hospitality. Very stimulating are the works of Vincent Emery, and thanksworthy help when considering odd dimensions came from Steven Tschantz. We were participants at the AIM Square program "Hyperbolic geometry beyond dimension three", and we thank the American Institute of Mathematics AIM for their support. The author is grateful to Rafael Guglielmetti for having put at her disposal some tables and graphics of his paper  and to Simon Drewitz for some programming support. The author was partially supported by Schweizerischer Nationalfonds 200021–172583.
Conflicts of Interest
The author declares no conflict of interest.
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