Virtual surface bundle groups and Johnson’s trichotomy

J. A. Hillman

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, AUSTRALIA e-mail: jonh@ maths.usyd.edu.au

ABSTRACT

We give a new exposition of Johnson’s trichotomy for torsion free groups which are virtually extensions of one surface group by another, and show that all groups in his first two classes may be realized by aspherical closed smooth 4-manifolds.

AMS Subject Classification (1991): Primary 57N13. Secondary 20F34, 20J05. Key words and phrases: asperical. bundle. extension. 4-manifold. orbifold. surface.

Johnson has established a trichotomy for Surface bundle groups, in order to study in how many ways such a group could be an extension of one Surface group by another [Jo94]. (Following Johnson, we shall use the term Surface to mean closed surface with negative Euler characteristic. Thus a Surface group is a P D_{2}-group with trivial centre). He has since extended this trichotomy to torsion free groups commensurable with such Surface bundle groups, and asks whether every such group is the fundamental group of some aspherical closed 4-manifold [Jo96]. We shall give an alternative exposition of this trichotomy and of his uniqueness of structure results in §1, and show in §2 that all groups in his first two classes may be thus realized. We shall make some remarks on the cases with χ = 0 in §3.

§1. Johnson’s trichotomy

A virtual Surface bundle group (or vSb group) is a torsion free group which has a subgroup of finite index which is an extension of one Surface group by another. If π is a group then ζπ and \sqrt{π} shall denote its centre and Hirsch-Plotkin radical, respectively. If G is a subgroup of π then C_{π}(G) shall denote the centralizer of G in π.

Typeset by AMS-TEX

Lemma 1. Let G_{1} and G_{2} be groups with no nontrivial abelian normal subgroup. If H is a normal subgroup of G = G_{1} × G_{2} which contains no nontrivial direct product then either H ≤ G_{1} × {1} or H ≤ {1} × G_{2}. Proof. Let P_{i} be the projection of H onto G_{i}, for i = 1, 2. If (h, h′) ∈ H, g_{1} ∈ G_{1} and g_{2} ∈ G_{2} then ([h, g_{1}], 1) = [(h, h′), (g_{1}, 1)] and (1, [h′, g_{2}]) are in H. Hence [P_{1}, P_{1}] × [P_{2}, P_{2}] ≤ H. Therefore either P_{1} or P_{2} is abelian, and so is trivial, since P_{i} is normal in G_{i}, for i = 1, 2. □

Theorem 2. Let π be a group with a normal subgroup K such that K and π/K are P D_{2}-groups with trivial centres.

(i) If C_{π}(K) = 1 and K_{1} is a finitely generated normal subgroup of π then C_{π}(K_{1}) = 1 also. (ii) The index [π : KC_{π}(K)] is finite if and only if π is virtually a direct product of P D_{2}-groups. Proof. (i) Let z ∈ C_{π}(K_{1}). If K_{1} ≤ K then [K : K_{1}] < ∞ and ζK_{1} = 1. Let M = [K : K_{1}]!. Then f(k) = k^{−1}z^{M}kz^{−M} is in K_{1} for all k in K. Now f(kk_{1}) = k_{1}^{−1}f(k)k_{1} and also f(kk_{1}) = f(kk_{1}k^{−1}k) = f(k) (since K_{1} is a normal subgroup centralized by z), for all k in K and k_{1} in K_{1}. Hence f(k) is central in K_{1}, and so f(k) = 1 for all k in K. Thus z^{M} centralizes K. Since π is torsion free we must have z = 1. Otherwise p(K_{1}) is a nontrivial finitely generated normal subgroup of π/K, and so has trivial centralizer. Hence p(z) = 1. Now [K, K_{1}] ≤ K ∩ K_{1} and so K ∩ K_{1} ≠ 1, for otherwise K_{1} ≤ C_{π}(K). Since z centralizes the nontrivial normal subgroup K ∩ K_{1} in K we must again have z = 1.

(ii) Since K has trivial centre KC_{π}(K) ≅ K × C_{π}(K) and so the condition is necessary. Suppose that f : G_{1}×G_{2} → π is an isomorphism onto a subgroup of finite index, where G_{1} and G_{2} are P D_{2}-groups. Let L = K ∩ f(G_{1} × G_{2}). Then [K : L] < ∞ and so L is also a P D_{2}-group, and is normal in f(G_{1} ×G_{2}). We may assume that L ≤ f(G_{1}), by Lemma 1. Then f(G_{1})/L is finite and is isomorphic to a subgroup of f(G_{1} × G_{2})/K ≤ π/K, so L = f(G_{1}). Now f(G_{2}) normalizes K and centralizes L, and [K : L] < ∞. Hence f(G_{2}) has a subgroup of finite index which centralizes K, as in part (i). Hence [π : KC_{π}(K)] < ∞. □

It follows immediately that if π and K are as in the theorem whether (i) C_{π}(K) ≠ 1 and [π : KC_{π}(K)] = ∞, (ii) [π : KC_{π}(K)] < ∞ or (iii)

C_{π}(K) = 1 depends only on π and not on the subgroup K. In [Jo94] these cases are labeled as types I, II and III, respectively. (The action of π by conjugation on K determines a homorphism θ : π/K → Out(K). In terms of this action: if Imθ is infinite and Kerθ ≠ 1 then π is of type I, if Imθ is finite then π is of type II, and if θ is injective then π is of type III). It also follows easily from the argument of part (i) that if ρ is a subgroup of finite index in π and L = K ∩ ρ then C_{ρ}(L) = 1 if and only if C_{π}(K) = 1. Since ρ is virtually a product of Surface groups if and only if π is, commensurate vSb groups are of the same type [Jo96].

Theorem 3. Let π be a group with normal subgroups K and K_{1} such that K, K_{1}, π/K and π/K_{1} are P D_{2}-groups with trivial centres. Suppose that C_{π}(K) ≠ 1. If [π : KC_{π}(K)] = ∞ then K_{1} = K is unique. If [π : KC_{π}(K)] < ∞ then either K_{1} = K or K_{1}∩K = 1; in the latter case K and K_{1} are the only such normal subgroups which are P D_{2}-groups with torsion free quotients. Proof. Let p : π → π/K be the quotient epimorphism. Then p(C_{π}(K)) is a nontrivial normal subgroup of π/K, since K ∩ C_{π}(K) = ζK = 1. Suppose that K_{1} ∩ K ≠ 1. Now Σ = K_{1} ∩ (KC_{π}(K)) contains K_{1} ∩ K, and K_{1} ∩ K ∩ C_{π}(K) = K_{1} ∩ ζK = 1. Hence Σ ≰ C_{π}(K). Since Σ is normal in KC_{π}(K) ≅ K × C_{π}(K) we must have Σ ≤ K_{1}, by Lemma 1. Hence Σ ≤ K_{1} ∩K. Hence p(K_{1})∩p(C_{π}(K)) = 1, and so p(K_{1}) centralizes the nontrivial normal subgroup p(C_{π}(K)) in π/K. Therefore K_{1} ≤ K and so [K : K_{1}] < ∞. Since π/K_{1} is torsion free we must have K_{1} = K.

If K_{1} ∩ K = 1 then [K, K_{1}] = 1 so K_{1} ≤ C_{π}(K) and [π : KC_{π}(K)] ≤ [π/K : p(K_{1})] < ∞. Suppose K_{2} is a normal subgroup of π which is a P D_{2}- group with ζK_{2} = 1 and such that π/K_{2} is torsion free and K_{2} ∩ K = 1. Then H = K_{2} ∩ (KK_{1}) is normal in KK_{1} ≅ K × K_{1} and [K_{2} : H] < ∞, so H is a P D_{2}-group with ζH = 1 and H ∩ K = 1. The projection of H to K_{1} is nontrivial since H ∩ K = 1. Therefore H ≤ K_{1}, by Lemma 1, and so K_{1} ≤ K_{2}. Hence K_{1} = K_{2}. □

Corollary. If α and β are automorphisms of π such that α(K) ∩ K = 1 then π ≅ K × K and β(K) = K or α(K). In particular, Aut(K × K) ≅ Aut(K)^{2} ˜×(Z/2Z). □

Johnson shows also that each finitely generated normal subgroup K such that π/K is a P D_{2}-group with trivial centre has only finitely many distinct images ϕ(K) under automorphisms of π. (The key point is that a P D_{2}-group with trivial centre has only finitely many normal subgroups with a given number of generators). He conjectures that in all cases there are only finitely many such normal subgroups which are themselves P D_{2}-groups with trivial centre. The next theorem is a partial result in this direction.

Theorem 4. Let π be a group with normal subgroups K and K_{1} such that K, K_{1} and π/K are P D_{2}-groups, π/K_{1} is torsion free and χ(π/K) < 0. Then either K_{1} = K or K_{1} ∩ K = 1 and π ≅ K × K_{1} or χ(K_{1}) < χ(π/K). Proof. Let p : π → π/K be the quotient epimorphism. If K_{1} ≤ K then K_{1} = K, as in Theorem 3. Otherwise p(K_{1}) has finite index in π/K and so χ(K_{1}) ≤ χ(p(K_{1})) ≤ χ(π/K). We may assume that χ(K_{1}) ≥ χ(π/K). Hence χ(K_{1}) = χ(π/K) and so p|_{K_{1}} is an epimorphism. Therefore K_{1} and π/K have the same orientation type, by the nondegeneracy of Poincaré duality and the Wu relation, and so K_{1} ≅ π/K. Since P D_{2}-groups are hopfian p|_{K_{1}} is an isomorphism. Hence [K, K_{1}] ≤ K ∩ K_{1} = 1 and so π = K.K_{1} ≅ K × K_{1}. □

The next two corollaries follow by elementary arithmetic.

Corollary A. If χ(K) = 0 or χ(K) = −1 and π/K_{1} is a P D_{2}-group then either K_{1} = K or π ≅ K × K_{1}. □

Corollary B. If K and π/K are P D_{2}-groups, χ(π/K) < 0, and χ(K)^{2} ≤ χ(π) then either K is the unique such subgroup or π ≅ K × K. □

Since there is a bijective correspondance between extensions of one Surface group by another and Surface bundles over Surfaces (see Theorem IV.1 of [Hi]), this has the following topological consequence.

Corollary C. Let M be the total space of a bundle ξ with base B and fibre F aspherical closed surfaces such that χ(B) < χ(F). If M′ is the total space of another surface bundle ξ′ with fibre F and π_{1}(M′) ≅ π_{1}(M) then M′ is diffeomorphic to M via a fibre-preserving diffeomorphism. □

§2. Realization

As remarked above every extension of one Surface group by another may be realized by some Surface bundle. We shall show that every vSb group of type I or II is the fundamental group of some aspherical closed smooth 4-manifold.

Theorem 5. Let π be a torsion free group with normal subgroups K < G < π such that K and G/K are P D_{2}-groups and [π : G] < ∞. Then π is the fundamental group of an aspherical closed smooth 4-manifold which is the total space of a bundle over a 2-dimensional orbifold. Proof. Let p : π → π/K be the quotient homomorphism. Since π is torsion free the preimage in π of any finite subgroup of π/K is a P D_{2}-group. As the finite subgroups of π/K have order at most [π : G], we may assume that π/K has no nontrivial finite normal subgroup, and so is the orbifold fundamental group of some 2-dimensional orbifold B. Let F be the aspherical closed surface with π_{1}(F) ≅ K. If π/K is torsion free then B is a closed aspherical surface, and the result follows from Theorem IV. 1 of [Hi]. In general, we may assume that B is the union of a punctured surface B_{o} with finitely many suborbifolds U_{i} which are quotients of D^{2} by finite subgroups of O(2) (i.e., either cone discs or segments of a disc with one or two reflector sides and possibly a corner point at the junction of two reflector sides).

Let H_{i} be the preimage in π of π_{1}^{orb}(U_{i}). Then [H_{i} : K] < ∞ and H_{i} embeds as a discrete uniform subgroup of Isom(𝕏^{2}), where 𝕏^{2} = ℝ^{2} if χ(K) = 0 and 𝕏^{2} = ℍ^{2} otherwise. The obvious action of H_{i} on 𝕏^{2} ×D^{2} determines a bundle with general fibre F over the orbifold U_{i}. The action of π/K on K determines an action of π_{1}(B_{o}) on K and hence an F -bundle over B_{o}.

These pieces may be assembled to give a bundle with general fibre F over the orbifold B, whose total space is an aspherical closed smooth 4-manifold with fundamental group π. □

Corollary. Virtual Surface bundle groups of type I are realizable. Proof. Let π be torsion free, and suppose that π has a subgroup σ of finite index with a normal subgroup ν such that ν and σ/ν are P D_{2}-groups with ζν = ζ(σ/ν) = 1. Let G = ∩hσh^{−1} and K = ν ∩ G. Then [π : G] < ∞, G is normal in π, and K and G/K are P D_{2}-groups. If G is of type I then K is characteristic in G, by Theorem 3, and so is normal in π. As π/K is virtually a P D_{2}-group, Theorem 5 applies. □

This settles the realization question for groups of type I (and also for many others - see the corollaries to Theorem 4). It would also handle type III groups if it could be shown that if such a group G has a normal subgroup K such that both K and G/K are Surface groups then K is characteristic in G.

Groups of type II may not have such a characteristic surface subgroup (although this is nearly true). Nevertheless, all such groups may be realized geometrically, as we shall show next. A closed ℍ^{2}×ℍ^{2}-manifold M is reducible if it has a finite cover isometric to a product of closed surfaces.

Theorem 6. A group π is the fundamental group of a reducible ℍ^{2} × ℍ^{2}- manifold if and only if it is torsion free, \sqrt{π} = 1 and π has a subgroup of finite index which is isomorphic to a product of P D_{2}-groups. Proof. The conditions are clearly necessary. Suppose that they hold. Then π is a P D_{4}-group and has a subgroup of finite index which is a direct product αβ ≅ α × β, where α and β are P D_{2}-groups. Let N be the intersection of the conjugates of α.β in π. Then N is normal in π, so \sqrt{N} = 1 also, and [π : N] < ∞. Let K = α ∩ N and L = β ∩ N. Then K and L are P D_{2}-groups with trivial centre, and KL ≅ K × L is normal in N and has finite index in π. Moreover N/K and N/L are isomorphic to subgroups of finite index in β and α, respectively, and so are also P D_{2}-groups. Since any automorphism of N must either fix these subgroups or interchange them, by Theorem 3, KL is normal in π and [π : N_{π}(K)] ≤ 2.

Let ν = N_{π}(K). Then L ≤ C_{π}(K) ≤ ν and ν = N_{π}(L) also. After enlarging K and L, if necessary, we may assume that L = C_{π}(K) and K = C_{π}(L). Hence ν/K and ν/L have no nontrivial finite normal subgroup. (For if K_{1} is normal in ν and contains K as a subgroup of finite index then K_{1} ∩ L is finite, hence trivial, and so K_{1} ≤ C_{π}(L)). The action of ν/L by conjugation on K has finite image in Out(K), and so ν/L embeds as a discrete cocompact subgroup of Isom(ℍ^{2}), by the Nielsen conjecture [Ke83]. Together with a similar embedding for ν/K we obtain a homomorphism from ν to a discrete cocompact subgroup of Isom(ℍ^{2} × ℍ^{2}).

If [π : ν] = 2 let t be an element of π − ν, and let j : ν/K → Isom(ℍ^{2}) be an embedding onto a discrete cocompact subgroup S. Then tKt^{−1} ∩ K = 1 and t induces an isomorphism f : ν/G → ν/(tKt^{−1}). Together j and j ∘ f^{−1} determine an embedding J : ν → Isom(ℍ^{2} × ℍ^{2}) onto a discrete cocompact subgroup of finite index in S × S. Now t^{2} ∈ ν and J(t^{2}) = (s, s), where s = j(t^{2}K). We may extend J to an embedding of π in Isom(ℍ^{2} × ℍ^{2}) by defining J(t) to be the isometry sending (x, y) to (y, s.x). Thus (in either case) π acts isometrically and properly discontinuously on ℍ^{2} ×ℍ^{2}. Since π is torsion free the action is free, and so π = π_{1}(M), where M = π\(ℍ^{2}×ℍ^{2}). □

Corollary. Let M be a ℍ^{2} × ℍ^{2}-manifold. Then M is reducible if and only if it has a 2-fold covering space which is homotopy equivalent to the total space of a bundle over a hyperbolic 2-orbifold and with general fibre a hyperbolic surface. Proof. The isometry group Isom(ℍ^{2} × ℍ^{2}) contains P SL(2, ℝ) × P SL(2, ℝ) as a normal subgroup of index 8. An irreducible lattice in P SL(2, ℝ) × P SL(2, ℝ) cannot have any nontrivial normal subgroups of infinite index, by Theorem IX.6.14 of [Ma]. Hence an ℍ^{2} × ℍ^{2}-manifold which is finitely covered by the total space of a surface bundle is reducible. The converse was proven in the theorem. □

In general, we cannot assume that M is itself fibred over a 2-orbifold. Let G be a P D_{2}-group with ζG = 1 and let x be a nontrivial element of G. A uniform free action of G on ℍ^{2} determines a uniform free action of π = ⟨G × G, t | t(g_{1}, g_{2})t^{−1} = (xg_{2}x^{−1}, g_{1}) ∀(g_{1}, g_{2}) ∈ G × G, t^{2} = (x, x)⟩ on ℍ^{2} × ℍ^{2}, by (g_{1}, g_{2}).(h_{1}, h_{2}) = (g_{1}.h_{1}, g_{2}.h_{2}) and t.(h_{1}, h_{2}) = (x.h_{2}, h_{1}), for all (g_{1}, g_{2}) ∈ G × G and (h_{1}, h_{2}) ∈ ℍ^{2} × ℍ^{2}. The group π has no normal subgroup which is a P D_{2}-group. (See Lemma 1).

§3. The cases with χ = 0

If π is an extension of Z^{2} by a normal Surface subgroup K (with ζK = 1) then C_{π}(K) = \sqrt{π}, and [π : KC_{π}(K)] < ∞ if and only if π is virtually K ×Z^{2}, so Johnson’s trichotomy extends to such groups. The three types may be characterized by (I) \sqrt{π} ≅ Z, (II) \sqrt{π} ≅ Z^{2}, and (III) \sqrt{π} = 1. As these properties are shared by commensurate torsion free groups the trichotomy extends further to torsion free groups which are virtually such extensions. There is at present no uniqueness result corresponding to Theorem 3 above for such subgroups K < π, and (excepting for groups of type II) it is not known whether every such group is realized by some aspherical closed 4-manifold. (In fact, it also appears to be unknown in how many ways a 3-dimensional mapping torus may fibre over S^{1}).

On the other hand, if π is virtually an extension of a P D_{2}-group by a P D_{2}-group K with ζK ≠ 1 then either \sqrt{π} ≅ Z^{2} (and so is a canonical normal P D_{2}-subgroup) or π is virtually poly-Z. The Johnson trichotomy is inappropriate here as Out(Z^{2}) ≅ GL(2, ℤ) is virtually free, and so the action θ is never injective. However all such groups may be realized by aspherical closed 4-manifolds. If π is virtually poly-Z then it is the fundamental group of an infrasolvmanifold. (See Chapter VI of [Hi]). Otherwise π is the fundamental group of an aspherical Seifert fibred 4-manifold, with general fibre a torus or Klein bottle, by Theorem 5.

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