ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Pages 40–46 (May 27, 2005) S 1079-6762(05)00145-9

THE SMALLEST HYPERBOLIC 6-MANIFOLDS

BRENT EVERITT, JOHN RATCLIFFE, AND STEVEN TSCHANTZ

(Communicated by Walter Neumann)

Abstract. By gluing together copies of an all right-angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic −1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.

1. Introduction

The last few decades have seen a surge of activity in the study of finite volume hyperbolic manifolds—that is, complete Riemannian n-manifolds of constant sectional curvature −1. Not surprisingly for geometrical objects, volume has been, and continues to be, the most important invariant for understanding their sociology. The possible volumes in a fixed dimension form a well-ordered subset of ℝ, indeed a discrete subset except in 3 dimensions (where the orientable manifolds at least have ordinal type ω^{ω}). Thus it is a natural problem with a long history to construct examples of manifolds with minimum volume in a given dimension.

In 2 dimensions the solution is classical, with the minimum volume in the compact orientable case achieved by a genus 2 surface, and in the noncompact orientable case by a once-punctured torus or thrice-punctured sphere (the identities of the manifolds are of course also known in the nonorientable case). In 3 dimensions the compact orientable case remains an open problem with the Matveev-Fomenko-Weeks manifold [16, 30] obtained via (5, −2)-Dehn surgery on the sister of the figure-eight knot complement conjecturally the smallest. Amongst the noncompact orientable 3-manifolds the figure-eight knot complement realizes the minimum volume [17], and the Gieseking manifold (obtained by identifying the sides of a regular hyperbolic tetrahedron as in [14, 20]) does so for the nonorientable ones [1]. One could also add “arithmetic” to our list of adjectives and so have eight optimization problems to play with (so that the Matveev-Fomenko-Weeks manifold is known to be the minimum volume orientable, arithmetic compact 3-manifold; see [5]).

When n ≥ 4 the picture is murkier, although in even dimensions we have recourse to the Gauss-Bonnet Theorem, so that in particular the minimum volume a 2m-dimensional hyperbolic manifold could possibly have, is when the Euler characteristic χ satisfies |χ| = 1. The first examples of noncompact 4-manifolds with

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Reverts to public domain 28 years from publication

Received by the editors October 31, 2004.

1991 Mathematics Subject Classification. Primary 57M50.

The first author is grateful to the Mathematics Department, Vanderbilt University for its hospitality during a stay when the results of this paper were obtained.

χ = 1 were constructed in [23] (see also [13]). The compact case remains a difficult unsolved problem, although if we restrict to arithmetic manifolds, then it is known [2, 8] that a minimum volume arithmetic compact orientable 4-manifold M has χ ≤ 16 and M is isometric to the orbit space of a torsion-free subgroup of the hyperbolic Coxeter group [5, 3, 3, 3]. The smallest compact hyperbolic 4manifold currently known to exist has χ = 8 and is constructed in [8]. Manifolds of very small volume have been constructed in 5 dimensions [13, 24], but the smallest volume 6-dimensional example hitherto known has χ = −16 [13].

In this paper we announce the discovery of a number of noncompact nonorientable hyperbolic 6-manifolds with Euler characteristic χ = −1. The method of construction is classical in that the manifolds are obtained by identifying the sides of a 6-dimensional hyperbolic Coxeter polytope.

2. Coxeter polytopes

Let C be a convex (not necessarily bounded) polytope of finite volume in a simply connected space X^{n} of constant curvature. Call C a Coxeter polytope if the dihedral angle subtended by two intersecting (n − 1)-dimensional sides is π/m for some integer m ≥ 2. When X^{n} = S^{n} or the Euclidean space E^{n}, such polyhedra have been completely classified [11], but in the hyperbolic space H^{n}, a complete classification remains a difficult problem (see for example [27] and the references there).

If Γ is the group generated by reflections in the (n − 1)-dimensional sides of C, then Γ is a discrete cofinite subgroup of the Lie group Isom X^{n}, and every discrete cofinite reflection group in Isom X^{n} arises in this way from some Coxeter polytope, which is uniquely defined up to isometry. The Coxeter symbol for C (or Γ) has nodes indexed by the (n − 1)-dimensional sides, and an edge labeled m joining the nodes corresponding to sides that intersect with angle π/m (label the edge joining the nodes of nonintersecting sides by ∞). In practice the labels 2 and 3 occur often, so that edges so labeled are respectively removed or left unlabeled.

Let Λ be an (n+1)-dimensional Lorentzian lattice, that is, an (n+1)-dimensional free ℤ-module equipped with a ℤ-valued bilinear form of signature (n, 1). For each n, there is a unique such Λ, denoted I_{n,1}, that is odd and self-dual (see [25, Theorem V.6], or [18, 19]). By [4], the group O_{n,1}ℤ of automorphisms of I_{n,1} acts discretely, cofinitely by isometries on the hyperbolic space H^{n} obtained by projectivising the negative norm vectors in the Minkowski space-time I_{n,1} ⊗ℝ (to get a faithful action one normally passes to the centerless version PO_{n,1}ℤ).

Vinberg and Kaplinskaya showed [28, 29] that the subgroup Reflec_{n} of PO_{n,1}ℤ generated by reflections in positive norm vectors has finite index if and only if n ≤ 19, thus yielding a family of cofinite reflection groups and corresponding finite volume Coxeter polytopes in the hyperbolic spaces H^{n} for 2 ≤ n ≤ 19. Indeed, Conway and Sloane showed ([9, Chapter 28] or [10]) that for n ≤ 19 the quotient of PO_{n,1}ℤ by Reflec_{n} is a subgroup of the automorphism group of the Leech lattice. Borcherds [3] showed that the (non-self-dual) even sublattice of I_{21,1} also acts cofinitely, yielding the highest-dimensional example known of a Coxeter group acting cofinitely on hyperbolic space.

When 4 ≤ n ≤ 9 the group Γ = Reflec_{n} has Coxeter symbol,

with n + 1 nodes and C a noncompact, finite volume n-simplex Δ^{n} (when n > 9, the polytope C has a more complicated structure).

Let v be the vertex of Δ^{n} opposite the side F_{1} marked on the symbol, and let Γ_{v} be the stabilizer in Γ of this vertex. This stabilizer is also a reflection group with symbol as shown, and is finite for 4 ≤ n ≤ 8 (being the Weyl group of type A_{4}, D_{5}, E_{6}, E_{7} and E_{8} respectively) and infinite for n = 9 (when it is the affine Weyl group of type Ẽ_{8}). Let

P_{n} = ⋃_{γ∈Γ_{v}} γ(Δ^{n}),

a convex polytope obtained by gluing |Γ_{v}| copies of the simplex Δ^{n} together. Thus, P_{n} has finite volume precisely when 4 ≤ n ≤ 8, although it is noncompact, with a mixture of finite vertices in H^{n} and cusped ones on ∂H^{n}. In any case, P_{n} is an all right-angled Coxeter polytope: its sides meet with dihedral angle π/2 or are disjoint. This follows immediately from the observation that the sides of P_{n} arise from the Γ_{v}-images of the side of Δ^{n} opposite v, and this side intersects the other sides of Δ^{n} in dihedral angles π/2 or π/4. Vinberg has conjectured that n = 8 is the highest dimension in which finite volume all right-angled polytopes exist in hyperbolic space.

The volume of the polytope P_{n} is given by

vol(P_{n}) = |Γ_{v}|vol(Δ^{n}) = |Γ_{v}|[PO_{n,1}ℤ : Γ]covol(PO_{n,1}ℤ),

where covol(PO_{n,1}ℤ) is the volume of a fundamental region for the action of PO_{n,1}ℤ on H^{n} (and for 4 ≤ n ≤ 9 the index [PO_{n,1}ℤ : Γ] = 1). When n is even, we have by [26] and [22],

covol(PO_{n,1}ℤ) = \frac{(2^{\frac{n}{2}} ± 1)π^{\frac{n}{2}}}{n!} ∏_{k=1}^{\frac{n}{2}} |B_{2k}|,

with B_{2k} the 2k-th Bernoulli number and with the plus sign if n ≡ 0, 2 mod 8 and the minus sign otherwise.

Alternatively (when n is even), we have recourse to the Gauss-Bonnet Theorem, so that vol(P_{n}) = κ_{n}|Γ_{v}|χ(Γ), where χ(Γ) is the Euler characteristic of the Coxeter group Γ and κ_{n} = 2^{n}(n!)^{−1}(−π)^{n/2}(n/2)!. The Euler characteristic of Coxeter groups can be easily computed from their symbol (see [6, 7] or [13, Theorem 9]). Indeed, when n = 6, χ(Γ) = −1/L where L = 2^{10} 3^{4} 5 and so vol(P_{6}) = 8π^{3}|E_{6}|/15L = π^{3}/15.

The Coxeter symbol for P_{n} has a nice description in terms of finite reflection groups. If v′ is the vertex of Δ^{n} opposite the side F_{2}, let Γ_{e} be the pointwise stabilizer of {v, v′}: the elements thus stabilize the edge e of Δ^{n} joining v and v′.

Now consider the Cayley graph C_{v} for Γ_{v} with respect to the generating reflections in the sides of the symbol for Γ_{v}. Thus, C_{v} has vertices in one-to-one correspondence with the elements of Γ_{v} and for each generating reflection s_{α}, an undirected edge labeled s_{α} connecting vertices γ_{1} and γ_{2} if and only if γ_{2} = γ_{1}s_{α} in Γ_{v}. In particular, C_{v} has s_{2} labeled edges corresponding to the reflection in F_{2}. Removing these s_{2}-edges decomposes C_{v} into components, each of which is a copy of the Cayley graph C_{e} for Γ_{e}, with respect to the generating reflections.

Take as the nodes of the symbol for P_{n} these connected components. If two components have an s_{2}-labeled edge running between any two of their vertices in C_{v}, then leave the corresponding nodes unconnected; otherwise, connect them by an edge labeled ∞. The resulting symbol (respectively the polytope P_{n}) thus has |Γ_{v}|/|Γ_{e}| nodes (resp. sides). The number of sides of P_{n} for n = 4, 5, 6, 7, 8 is 10, 16, 27, 56 and 240 respectively.

3. Constructing the manifolds

We now restrict our attention to the case n = 6. We work in the hyperboloid model of hyperbolic 6-space

H^{6} = {x ∈ ℝ^{7} : x_{1}^{2} + x_{2}^{2} + ⋯ + x_{6}^{2} − x_{7}^{2} = −1 and x_{7} > 0}

and represent the isometries of H^{6} by Lorentzian 7 × 7 matrices that preserve H^{6}. The right-angled polytope P_{6} has 27 sides each congruent to P_{5}. We position P_{6} in H^{6} so that 6 of its sides are bounded by the 6 coordinate hyperplanes x_{i} = 0 for i = 1, …, 6 and these 6 sides intersect at the center e_{7} of H^{6}. Let K_{6} be the group of 64 diagonal Lorentzian 7 × 7 matrices diag(±1, …, ±1, 1). The set Q_{6} = K_{6}P_{6}, which is the union of 64 copies of P_{6}, is a right-angled convex polytope with 252 sides. We construct hyperbolic 6-manifolds, with χ = −8, by gluing together the sides of Q_{6} by a proper side-pairing with side-pairing maps of the form rk with k in K_{6} and r a reflection in a side S of Q_{6}. The side-pairing map rk pairs the side S′ = kS to S (see §11.1 and §11.2 of [21] for a discussion of proper side-pairings). We call such a side-pairing of Q_{6} simple. We searched for simple side-pairings of Q_{6} that yield a hyperbolic 6-manifold M with a freely acting ℤ/8 symmetry group that permutes the 64 copies of P_{6} making up M in such a way that the resulting quotient manifold is obtained by gluing together 8 copies of P_{6}. Such a quotient manifold has χ = −8/8 = −1. This is easier said than done, since the search space of all possible side-pairings of Q_{6} is very large. We succeeded in finding desired side-pairings of Q_{6} by employing a strategy that greatly reduces the search space. The strategy is to extend a side-pairing in dimension 5 with the desired properties to a side-pairing in dimension 6 with the desired properties.

Let Q_{5} = {x ∈ Q_{6} : x_{1} = 0}. Then Q_{5} is a right-angled convex 5-dimensional polytope with 72 sides. Note that Q_{5} is the union K_{5}P_{5} of 32 copies of P_{5} where P_{5} = {x ∈ P_{6} : x_{1} = 0} and K_{5} is the group of 32 diagonal Lorentzian 7 × 7 matrices diag(1, ±1, …, ±1, 1). A simple side-pairing of Q_{6} that yields a hyperbolic 6-manifold M restricts to a simple side-pairing of Q_{5} that yields a hyperbolic 5-manifold which is a totally geodesic hypersurface of M. All the orientable hyperbolic 5-manifolds that are obtained by gluing together the sides of Q_{5} by a simple side-pairing are classified in [24].

We started with the hyperbolic 5-manifold N, numbered 27 in [24], obtained by gluing together the sides of Q_{5} by the simple side-pairing with side-pairing code 2B7JB47JG81. The manifold N has a freely acting ℤ/8 symmetry group that permutes the 32 copies of P_{5} making up N in such a way that the resulting quotient Table 1. Side-pairing codes and homology groups of the seven examples.

N SP  H_{1}  H_{2}  H_{3}  H_{4}  H_{5}
0248 0248 0248 0248 0248

    1 GW8dNEEdN4ZJO1k2l1PIY 0401 1910 4821 1500 0000
    2 HX9dNFEcM5aKU6f3f6UKa 0401 1810 8710 5500 0000
    3 HX9dNFEcM5YIO1l3l1OIY 0401 2900 7810 4400 1000
    4 HX9dNFEcM5YIO6l3l6OIY 0401 2800 7910 4400 1000
    5 HX9dNFEcM5YIOxl3lyOIY 0211 2800 4821 1400 1000
    6 HX9dNFEcM5YIOyl3lxOIY 0211 2800 4930 1400 1000
    7 HX9dNFEcM5aKUxf3fyUKa 0301 1900 5630 2500 0000

manifold is obtained by gluing together 4 copies of P_{5}. A generator of the ℤ/8 symmetry group of N is represented by the Lorentzian 6 × 6 matrix

( \begin{matrix}1 & 0 & 0 & 1 & 0 & −1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ −1 & 0 & −1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & −1 & 0 & 1 \\ −1 & 0 & −1 & −1 & 0 & 2 \end{matrix} ) .

The strategy is to search for simple side-pairings of Q_{6} that yield a hyperbolic 6-manifold with a freely acting ℤ/8 symmetry group with generator represented by the following Lorentzian 7 × 7 matrix that extends the above Lorentzian 6 × 6 matrix:

( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & −1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & −1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 & −1 & 0 & 1 \\ 0 & −1 & 0 & −1 & −1 & 0 & 2 \end{matrix} ) .

For such a side-pairing the resulting quotient manifold can be obtained by gluing together 8 copies of P_{6} by a proper side-pairing. By a computer search we found 14 proper side-pairings of 8 copies of P_{6} in this way, and hence we found 14 hyperbolic 6-manifolds with χ = −1. Each of these 14 manifolds is noncompact with volume 8vol(P_{6}) = 8π^{3}/15 and five cusps. These 14 hyperbolic 6-manifolds represent at least 7 different isometry types, since they represent 7 different homology types. Table 1 lists side-pairing codes for 7 simple side-pairings of Q_{6} whose ℤ/8 quotient manifold has homology groups isomorphic to ℤ^{a} ⊕ (ℤ/2)^{b} ⊕ (ℤ/4)^{c} ⊕ (ℤ/8)^{d} for nonnegative integers a, b, c, d encoded by abcd in the table. In particular, all 7 manifolds in Table 1 have a finite first homology group.

All of our examples, with χ = −1, can be realized as the orbit space H^{6}/Γ of a torsion-free subgroup Γ of PO_{6,1}ℤ of minimum index. These manifolds are the first examples of hyperbolic 6-manifolds having the smallest possible volume. All these manifolds are nonorientable. In the near future, we hope to construct orientable examples of noncompact hyperbolic 6-manifolds having χ = −1.

References

[1] C. Adams, The noncompact hyperbolic 3-manifold of minimum volume, Proc. AMS 100 (1987), 601–606. MR0894423 (88m:57018)

[2] M. Belolipetsky, On volumes of arithmetic quotients of SO(1, n), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear).

[3] R. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133–153. MR0913200 (89b:20018)

[4] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962), no. 3, 485–535. MR0147566 (26:5081)

[5] T. Chinburg, E. Friedman, K. N. Jones, and A. W. Reid, The arithmetic hyperbolic 3-manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30 (2001), no. 4, 1–40. MR1882023 (2003a:57027)

[6] I. M. Chiswell, The Euler characteristic of graph products and Coxeter groups, in Discrete Groups and Geometry, W. J. Harvey and Colin Maclachlan (Editors), London Math. Soc. Lect. Notes, vol. 173, 1992, pp. 36–46. MR1196914 (94a:05090)

[7] I. M. Chiswell, Euler characteristics of groups, Math. Z. 147 (1976), 1–11. MR0396785 (53:645)

[8] M. D. E. Conder and C. Maclachlan, Small volume compact hyperbolic 4-manifolds, to appear, Proc. Amer. Math. Soc.

[9] J. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Second Edition, Springer, 1993. MR1194619 (93h:11069)

[10] J. Conway and N. J. A. Sloane, Leech roots and Vinberg groups, Proc. Roy. Soc. London Ser. A 384 (1982), no. 1787, 233–258. MR0684311 (84b:10048)

[11] H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. 35 (1934), no. 2, 588–621. MR1503182

[12] M. W. Davis, A hyperbolic 4-manifold, Proc. Amer. Math. Soc. 93 (1985), 325–328. MR0770546 (86h:57016)

[13] B. Everitt, Coxeter groups and hyperbolic manifolds, Math. Ann. 330 (2004), no. 1, 127–150. MR2091682

[14] B. Everitt, 3-Manifolds from Platonic solids, Top. App. 138 (2004), 253–263. 2035484 (2004m:57031)

[15] B. Everitt, J. Ratcliffe and S. Tschantz, Arithmetic hyperbolic 6-manifolds of smallest volume, (in preparation).

[16] V. Matveev and A. Fomenko, Constant energy surfaces of Hamilton systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds, Russian Math. Surveys 43 (1988), 3–24. MR0937017 (90a:58052)

[17] C. Cao and G. R. Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Invent. Math. 146 (2001), no. 3, 451–478. MR1869847 (2002i:57016)

[18] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer, 1973. MR0506372 (58:22129)

[19] A. Neumaier and J. J. Seidel, Discrete Hyperbolic Geometry, Combinatorica 3 (1983), 219– 237. MR0726460 (85h:51025)

[20] I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (1998), no. 2, 497–515. MR1642716 (99i:52029)

[21] J. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer 1994. MR1299730 (95j:57011)

[22] J. Ratcliffe and S. Tschantz, Volumes of integral congruence hyperbolic manifolds, J. Reine Angew. Math. 488 (1997), 55–78. MR1465367 (99b:11076)

[23] J. Ratcliffe and S. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000), no. 1, 101–125. MR1758804 (2001b:57048)

[24] J. Ratcliffe and S. Tschantz, Integral congruence two hyperbolic 5-manifolds, Geom. Dedicata 107 (2004), 187–209. MR2110762

[25] J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer, 1973. MR0344216 (49:8956)

[26] C. L. Siegel, Über die analytische Theorie der quadratischen Formen II, Ann. Math. 37 (1936), 230–263. MR1503276 [27] E. B. Vinberg, Hyperbolic reflection groups, Uspekhi Mat. Nauk 40 (1985), no. 1, 29–66;

English transl., Russian Math. Surveys 40 (1985), no. 1, 31–75. MR0783604 (86m:53059)

[28] E. B. Vinberg and I. M. Kaplinskaya, On the groups O_{18,1}(ℤ) and O_{19,1}(ℤ), Dokl. Akad. Nauk. SSSR 238 (1978), no. 6, 1273–1275; English transl., Soviet Math. Dokl. 19 (1978), no. 1, 194–197. MR0476640 (57:16199)

[29] E. B. Vinberg, On the groups of units of certain quadratic forms, Math. Sb. 87 (1972), no. 1, 18–36; English transl., Math. USSR, Sb. 16 (1972), no. 1, 17–35. MR0295193 (45:4261)

[30] J. Weeks, Hyperbolic structures on 3-manifolds, Ph.D. thesis, Princeton University, 1985.

Department of Mathematics, University of York, York YO10 5DD, England

E-mail address: bje1@york.ac.uk

Department of Mathematics, Vanderbilt University, Nashville, TN 37240

E-mail address: ratclifj@math.vanderbilt.edu

Department of Mathematics, Vanderbilt University, Nashville, TN 37240

E-mail address: tschantz@math.vanderbilt.edu