COMPACT HYPERBOLIC 4-MANIFOLDS OF SMALL VOLUME

Marston Conder^{1}  Colin Maclachlan
Department of Mathematics  Department of Mathematical Sciences
University of Auckland  University of Aberdeen
Private Bag 92019 Auckland  Aberdeen AB24 3UE
New Zealand  Scotland, U.K.

conder@math.auckland.ac.nz      c.maclachlan@maths.abdn.ac.uk

Abstract

We prove the existence of a compact non-orientable hyperbolic 4manifold of volume 32π^{2}/3 and a compact orientable hyperbolic 4manifold of volume 64π^{2}/3, obtainable from torsion-free subgroups of small index in the Coxeter group [5, 3, 3, 3]. At the time of writing these are the smallest volumes of any known compact hyperbolic 4-manifolds.

Mathematics Subject Classification 2000: 57M50, 20F55(primary), 51M20, 20B40(secondary)

1Research supported by grants from the N.Z. Marsden Fund (grant no. UOA 124) and the N.Z. Centres of Research Excellence Fund (grant no. UOA 201)

1 Introduction

The smallest volume (area) of a compact orientable hyperbolic 2-manifold is 4π and is achieved by any closed hyperbolic surface of genus 2 and so Euler characteristic −2. For non-compact orientable 2-manifolds, the smallest volume is 2π, achieved by a once-punctured torus which has Euler characteristic −1. For hyperbolic 3-manifolds, the work of Thurston and Jørgensen (see [18, 2]) has shown the existence of a smallest volume for compact orientable hyperbolic 3-manifolds which can be achieved by a finite number of manifolds. The prime candidate is the Weeks-Matveev-Fomenko manifold

[21, 14], whose volume can be given by the closed formula \frac{12(23^{3/2}) ζ_{k}(2)}{(4π^{2})^{2}} where the Dedekind zeta function ζ_{k} is over the field k of degree 3 over the rationals with one complex place and discriminant −23. This volume is approximately 0.942707. In the case of non-compact orientable hyperbolic

3-manifolds, the minimal volume is known to be \frac{12(3^{3/2}) ζ_{k}(2)}{4π^{2}} (≈ 2.029883) where k = ℚ(\sqrt{−3}) and is achieved by the figure 8 knot complement and its sister manifold [5]. These two manifolds and the Weeks manifold are known to be arithmetic and if one restricts to arithmetic 3-manifolds, then the minimum volume is known to be achieved by the Weeks manifold [6].

In the case of dimension 4, as with all even dimensions, the volume of a hyperbolic manifold is a constant multiple of its Euler characteristic. In dimension 4, this is given by Vol(M) = 4π^{2}χ(M)/3 (see [17, 11]). Furthermore the Euler characteristic of a compact orientable hyperbolic 4-manifold is always even. It is known that there exist non-compact orientable hyperbolic 4-manifolds of minimal Euler characteristic 1 [15, 9]. A well-studied example of a compact orientable hyperbolic 4-manifold is the Davis manifold [8, 10, 16], which has Euler characteristic 26.

In this paper, we establish the existence of a compact non-orientable hyperbolic 4-manifold of Euler characteristic 8, and an orientable double cover of this manifold, of Euler characteristic 16. These examples give the smallest known volumes so far in the compact case. Furthermore, these two 4manifolds and the Davis manifold are all arithmetic, and have the same arithmetic structure, and hence are commensurable in that they then have a common finite cover (see comments in §2). We will not make use of this fact here, but remark that, following the identification of the compact arithmetic hyperbolic 4-orbifold of smallest volume in [1] as the quotient IH^{4}/Γ_{1}, where Γ_{1} is defined in §2, it follows, as is shown in [1], that the as-yet-unknown compact arithmetic orientable hyperbolic 4-manifold of smallest volume has the form IH^{4}/Γ_{0} where Γ_{0} is a torsion-free subgroup of Γ_{1} of finite index. This is precisely how our manifolds are obtained.

PSfrag replacements Δ_{1} Δ_{2} Δ_{3} Δ_{4} Δ_{5}

Figure 1: Five compact Coxeter simplices in hyperbolic 4-space

2 Preliminaries

In hyperbolic 4-space IH^{4}, there are five compact Coxeter simplices — that is, simplices whose faces are geodesic and whose dihedral angles between faces of codimension 1 are submultiples of π (see [13, 19]). If Γ is the group generated by reflections in the 3-dimensional faces of such a simplex, then Γ is a discrete subgroup of Isom IH^{4}, and the simplex is a fundamental domain for Γ so that its images under Γ tessellate IH^{4} (see [19]). The Coxeter symbols Δ_{i} representing these simplices are given in Figure 1, and the corresponding reflection groups, in the same numbering, are denoted by Γ_{i} (i = 1, 2…. , 5).

The group Γ_{1} is also conveniently denoted as [5, 3, 3, 3]. Each of these reflection groups Γ has a torsion-free subgroup Γ_{0} of finite index, so that one can define a rational Euler characteristic for Γ by χ(Γ) = \frac{χ(Γ_{0})}{[Γ : Γ_{0}]} where χ(Γ_{0})

is the Euler characteristic of the manifold IH^{4}/Γ_{0}. This is well-defined, and furthermore, χ(Γ) can be computed directly from the simplex by the formula

χ(Γ) = ∑_{τ} \frac{(−1)^{dim τ}}{∘(Stab τ)}

where the sum is over all the cells τ of the simplex (see [4] for example). The resulting Euler characteristics in the five compact cases are:

χ(Γ_{1}) = \frac{1}{14400},  χ(Γ_{2}) = \frac{17}{28800},  χ(Γ_{3}) = \frac{26}{14400},
χ(Γ_{4}) = \frac{17}{14400},  and  χ(Γ_{5}) = \frac{11}{5760}.

Any torsion-free subgroup inside one of these reflection groups will give rise to a compact hyperbolic manifold, so that, for minimal volume, we endeavour to find torsion-free subgroups of as small an index as possible. The manifold will be orientable if and only if the torsion-free subgroup is contained in the index 2 subgroup generated by products of pairs of the generating reflections (or equivalently, contains no element expressible as a word of odd length in the generating reflections). For a systematic approach to Coxeter groups in dimensions 4 and higher, see [9].

Note that if H is a finite subgroup of the reflection group Γ, then for any torsion-free subgroup Γ_{0}, the index [Γ : Γ_{0}] must be divisible by the order of H (since H ∩ Γ_{0} has to be trivial). Thus for each of the five groups concerned, any resulting compact orientable manifold IH^{4}/Γ_{0} will have Euler characteristic a multiple of 2, 34, 26, 34 or 22 respectively. For details on Γ_{1}, see §3. The Davis manifold arises from a normal torsion-free subgroup of Γ_{3} of index precisely 14400 — indeed the unique such subgroup — so has Euler characteristic 26 (see [8, 10, 16]). We also note that the groups Γ_{1}, …, Γ_{5} are arithmetic, the first four have the same arithmetic structure and are hence commensurable [20, 12]. Thus any subgroups of finite index in these four groups will also be pairwise commensurable.

3 Torsion elements of the group Γ_{1} = [5, 3, 3, 3]

In order to find torsion-free subgroups of finite index in the Coxeter group Γ_{1}, we first determine representatives of conjugacy classes of torsion elements of prime order in Γ_{1}. Let a, b, c, d, e represent the reflections in the faces labelled A, B, C, D, E in the symbol Δ_{1} in Figure 2.

Figure 2: Coxeter symbol for the group Γ_{1} = [5, 3, 3, 3]

Note that if a subgroup contains a torsion element x, of order n, say, and p is any prime divisor of n, then x^{n/p} is a torsion element of order p lying in the same subgroup, and hence we may restrict our attention to prime orders.

Every element of finite order in Γ_{1} has a fixed point in IH^{4} and so is conjugate in Γ_{1} to an element stabilising a vertex. The vertex stabilisers are the five 4-generator subgroups obtained by deleting one generator, namely ⟨a, b, c, d⟩, ⟨a, b, c, e⟩, ⟨a, b, d, e⟩, ⟨a, c, d, e⟩ and ⟨b, c, d, e⟩. These five subgroups are all finite, of orders 14400, 240, 60, 48 and 120 respectively, and their structure is well known (and also can be found with the help of the Magma system [3]). The conjugacy classes of the elements of prime order in each of these finite subgroups is given in the table below (where in some cases, for information, we have included more than one representative).

(i) Subgroup ⟨a, b, c, d⟩, the group of the 120-cell, of order 14400:

Order of element Class size Class representative(s)
2  1  (abcd)^{15}
2  60  a, b, c, d
2  60  (abc)^{5}

\begin{matrix}2 & 450 & ac, ad, bd \\ 3 & 40 & (abcd)^{10} \\ 3 & 400 & bc, cd \\ 5 & 24 & (abcd)^{6} \\ 5 & 24 & (abcd)^{12} \\ 5 & 144 & ab \\ 5 & 144 & (ab)^{2} \\ 5 & 288 & (abcd)^{10}(abc)^{−2} \end{matrix}

(ii) Subgroup ⟨a, b, c, e⟩ ≅ A_{5} × C_{2} × C_{2}, of order 240:

Order of element Class size Class representative(s)
2  1  e
2  1  (abc)^{5}
2  1  (abc)^{5}e
2  15  a, b, c
2  15  ac
2  15  ae, be, ce
2  15  ace
3  20  bc
5  12  ab
5  12  (ab)^{2}

(iii) Subgroup ⟨a, b, d, e⟩ ≅ D_{5} × D_{3}, of order 60:

Order of element Class size Class representative(s)
2  5  a, b
2  3  d, e
2  15  ad, ae, bd, be
3  2  de
5  2  ab
5  2  (ab)^{2}

(iv) Subgroup ⟨a, c, d, e⟩ ≅ S_{4} × C_{2}, of order 48:

Order of element Class size Class representative(s)
2  1  a
2  3  ce
2  3  ace
2  6  ac, ad, ae
2  6  c, d, e
3  8  cd, de

(v) Subgroup ⟨b, c, d, e⟩ ≅ S_{5}, of order 120:

Order of element Class size Class representative(s)
2  10  b, c, d, e
2  15  bd, be, ce
3  20  bc, cd, de
5  24  bcde

From these observations the following theorem follows easily:

Theorem 1 A subgroup of the Coxeter group Γ_{1} is torsion-free if and only if it contains no element that is conjugate to one of the elements in the list L = [ a, ab, ac, bc, ace, (abc)^{5}, (abc)^{5}e, (abcd)^{15}, (abcd)^{10}, (abcd)^{6}, bcde, (abcd)^{10}(abc)^{−2}].

Note that the subgroup ⟨a, b, c, d⟩ is isomorphic to the Coxeter group [5, 3, 3], also known as the automorphism group of the 120-cell. Any torsion-free subgroup of Γ_{1} must intersect this subgroup trivially, and therefore its index in Γ_{1} has to be divisible by 14400. The torsion-free subgroup Σ we find in section 5 has index 8 × 14400.

4 Subgroups of small index in Γ_{1}

Using the LowIndexSubgroups command in Magma [3], it is very easy to find conjugacy classes of small index in the group Γ_{1}. In fact there are 24 classes of subgroups of index up to 240: one of index 1, one of index 2, two of index 85, two of index 120, two of index 136, two of index 156, four of index 170, and ten of index 240.

Of particular interest to us are the two conjugacy classes of index 120.

One of these contains the subgroup H_{1} generated by a, b, c, dcbabacebabcd, dedcbabacbabcdcbabcababcded and edcbabacbabdcbaebcdabcababcde. Now in the permutation representation of Γ_{1} on (right) cosets of this subgroup of index 120 by (right) multiplication, the elements of the list L in Theorem 1 have cycle structures 1^{32}2^{44}, 1^{10}5^{22}, 1^{8}2^{56}, 1^{6}3^{38}, 1^{8}2^{56}, 1^{32}2^{44}, 1^{32}2^{44}, 1^{120}, 3^{40}, 5^{24}, 5^{24} and 5^{24} respectively. (Note: here the notation r^{c_{r}} indicates c_{r} cycles of length r.) As an element g ∈ Γ_{1} fixes the coset H_{1}x in this permutation representation if and only if xgx^{−1} ∈ H_{1}, it follows that H_{1} contains conjugates of each of the torsion elements a, ab, ac, bc, ace, (abc)^{5}, (abc)^{5}e and (abcd)^{15}, but contains no conjugates of any of the torsion elements (abcd)^{10}, (abcd)^{6}, bcde or (abcd)^{10}(abc)^{−2}.

The other class of subgroups of index 120 contains the subgroup H_{2} generated by ac, bd, ababcdedcababa, adcedcbabcdeba and ababacbaebcdaba, and in the permutation representation on cosets of this subgroup the elements of the list L have cycle structures 2^{60}, 5^{24}, 1^{8}2^{56}, 3^{40}, 1^{8}2^{56}, 2^{60}, 2^{60}, 1^{120}, 1^{6}3^{38}, 1^{10}5^{22}, 5^{24} and 5^{24} respectively.

Now consider the subgroup H_{1} ∩ H_{2}. This has index 120^{2} = 14400 in Γ_{1} and in the permutation representation on its cosets the elements of the list L have cycle structures 2^{7200}, 5^{2880}, 1^{64}2^{7168}, 3^{4800}, 1^{64}2^{7168}, 2^{7200}, 2^{7200}, 1^{14400}, 3^{4800}, 5^{2880}, 5^{2880} and 5^{2880} respectively. Hence this intersection contains conjugates of each of the torsion elements ac, ace and (abcd)^{15}, but not of any of the other elements of the list L.

In order to eliminate conjugates of the three elements ac, ace and (abcd)^{15}, we need to dig deeper into the subgroup lattice of Γ_{1}. Another good candidate subgroup is a subgroup H_{3} of index 2 in H_{2} (and index 240 in Γ_{1}), generated by ac, bd, ababcdedcababa and ababcbabadcbabcde. In the permutation representation of Γ_{1} on the 240 cosets of this subgroup, the elements of the list L in Theorem 1 have cycle structures 2^{120}, 5^{48}, 1^{16}2^{112}, 3^{80}, 2^{120}, 2^{120}, 2^{120}, 1^{240}, 1^{12}3^{76}, 1^{20}5^{44}, 5^{48} and 5^{48} respectively.

The subgroup H_{1} ∩ H_{3} has index 28800 in Γ_{1}, and in the permutation representation on its cosets the elements of the list L have cycle structures 2^{14400}, 5^{5760}, 1^{128}2^{14336}, 3^{9600}, 2^{14400}, 2^{14400}, 2^{14400}, 1^{28800}, 3^{9600}, 5^{5760}, 5^{5760} and 5^{5760} respectively. Hence this intersection still contains conjugates of each of ac and (abcd)^{15}, but none of any of the other elements of the list L.

Similar observations can be made for other subgroups of index 240, although the outcome is best for H_{1} and H_{3}.

5 Construction of the 4-manifolds

A further application of the LowIndexSubgroups command in Magma [3] to the subgroup H_{3} described in section 4 produces a subgroup H_{4} of index 4 in H_{3} (and index 960 in Γ_{1}), generated by ababcbabadcbabcde and abacbdcbaedecbabcaba. In the permutation representation of Γ_{1} on the 960 cosets of this subgroup, the elements of the list L in Theorem 1 have cycle structures 2^{480}, 5^{192}, 2^{480}, 3^{320}, 2^{480}, 2^{480}, 2^{480}, 2^{480}, 1^{48}3^{304}, 1^{80}5^{176}, 5^{192} and 5^{192} respectively.

The intersection Σ = H_{1} ∩ H_{4} is a subgroup of index 115200 in Γ_{1}, and in the permutation representation on its cosets the elements of the list L have cycle structures 2^{57600}, 5^{23040}, 2^{57600}, 3^{38400}, 2^{57600}, 2^{57600}, 2^{57600}, 2^{57600}, 3^{38400}, 3^{9600}, 5^{23040}, 5^{23040} and 5^{23040} respectively. In particular, none of the elements of the list L fix any points, and hence by Theorem 1, this subgroup is torsion-free.

Thus we have found a torsion-free subgroup Σ of index 8 × 14400 in Γ_{1}, and therefore of Euler characteristic 8, giving a compact hyperbolic 4-manifold of volume 4π^{2}/3 × 8 = 32π^{2}/3.

This manifold is non-orientable. In fact the subgroup Σ = H_{1} ∩ H_{4} is generated by the elements cbdcbabcdedcbabcda (of length 18), babacbabacbabcdcbabc dedcbabcdab (length 31), ababacdedcbabacbabcdcbabcababcded (length 33), and bedcbabacbabdcbabacbadcbaedcbabcdabcababcdbabcbaba (length 50). Its unique subgroup of index 2 containing only words of even length is a torsion-free subgroup Σ^{o} of index 16 × 14400 in Γ_{1}, giving a compact hyperbolic 4-manifold of Euler characteristic 16 and volume 64π^{2}/3.

Our findings can be summed up in the following theorem:

Theorem 2 In the Coxeter group Γ_{1} = [5, 3, 3, 3], the intersection Σ of the index 120 subgroup generated by the elements a, b, c, dcbabacebabcd, dedcbabacbabcdcbabcababcded and edcbabacbabdcbaebcdabcababcde with the index 960 subgroup generated by ababcbabadcbabcde and abacbdcbaedecbabcaba is torsion-free and of index 115200 in Γ_{1}, and the quotient space IH^{4}/Σ is a compact but non-orientable hyperbolic 4-manifold of volume 32π^{2}/3 and Euler characteristic 8. Furthermore, Σ has a subgroup Σ^{o} of index 2 such that the quotient space IH^{4}/Σ^{o} is a compact orientable hyperbolic 4-manifold of volume 64π^{2}/3 and Euler characteristic 16.

6 Closing remarks

There may still be torsion-free subgroups in Γ_{1} of smaller index than the subgroups we have described above, however we have not been able to find any using the computational methods available and a number of new approaches to this question. There is certainly no torsion-free subgroup of smaller index that is obtainable as an intersection of two subgroups of index up to 480 in Γ_{1}, or as an intersection of two subgroups of index up to 4 in any of these, but of course there may be other subgroups, possibly maximal in Γ_{1}, or otherwise lying inside some maximal subgroup of index larger than 480.

The abelianisations of the subgroups H_{1}, H_{2}, H_{3} and H_{4} are ZZ_{2} ⊕ ZZ_{2} ⊕ ZZ_{2}, ZZ_{2} ⊕ ZZ_{4}, ZZ_{2} ⊕ ZZ and ZZ respectively, while those of the intersections H_{1} ∩ H_{2}, H_{1} ∩ H_{3} and Σ = H_{1} ∩ H_{4} are ZZ_{2} ⊕ ZZ_{2} ⊕ ZZ_{2} ⊕ ZZ_{4}, ZZ_{2} ⊕ ZZ_{2} ⊕ ZZ ⊕ ZZ and ZZ_{2} ⊕ ZZ_{2} ⊕ ZZ ⊕ ZZ respectively. Finally, the abelianisation of Σ^{o} is ZZ^{2} ⊕ ZZ_{4} ⊕ ZZ_{2}^{2}. These were found with the help of the Rewrite and AQInvariants commands in Magma [3]. From this we obtain the integral homology of M = IH^{4}/Σ^{o} using Poincaré duality and the Euler characteristic. Thus H_{1}(M) = ZZ^{2} ⊕ZZ_{4} ⊕ZZ_{2}^{2}, H_{2}(M) = ZZ^{18} ⊕ZZ_{4} ⊕ZZ_{2}^{2}, and H_{3}(M) = ZZ^{2}.

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