STABILITY OF LOWER CENTRAL SERIES OF COMPACT 3-MANIFOLD GROUPS

T.D. Cochran* and Kent E. Orr**

Abstract. The length of a group G is the least ordinal α such that G_{α} = G_{α+1} where G_{α} is the α^{th} term of the transfinite lower central series. We begin by establishing connections between lower central series length and the Parafree Conjecture, 4-dimensional topological surgery, and link concordance. We prove that the length of all surface groups and most Fuchsian groups is at most ω. We show that the length of the group a Seifert fibration over a base of non-positive even Euler characteristic is at most ω. Our major result is the existence of closed hyperbolic 3-manifolds with length at least 2ω. We observe that any closed orientable 3-manifold group has the same lower central series quotients as a hyperbolic one.

§1. Basic Definitions; History; Main Results; Motivation.

Recall that if G is a group and α an ordinal, the (transfinite) lower central series of G, G = G_{1} ⊃ G_{2} ⊃ ⋯ ⊃ G_{α} ⊃ …, is defined inductively by G_{1} = G, G_{α+1} = [G, G_{α}] and, if α is a limit ordinal, G_{α} = ⋂_{β<α} G_{α}. By cardinality considerations this series must stabilize and we shall call the least ordinal δ such that G_{δ} = G_{δ+1} the length of G. Herein the letter δ is reserved for this least ordinal and consequently G_{δ} is also the intersection of all terms of the (transfinite) lower central series of G. If G_{α} = {1} then G is said to be αnilpotent; if α is finite G is said to be nilpotent; if α = ω then G is said to be ω-nilpotent or residually nilpotent; if α is greater than ω then G is said to be transfinitely nilpotent. The groups G/G_{α} are called the lower central series quotients of G. At the end of this chapter we try to indicate some of the interrelationships between the lower central series of finitely presented groups and other important topics in topology and algebra such as: homology

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*partially supported by the National Science Foundation DMS-9100254, DMS-9400224, DNS-9205540

**partially supported by the National Science Foundation

cobordism, the Parafree Conjecture, link concordance, and the topological classification of non-simply-connected 4-manifolds. This paper is concerned with the questions:

a) What lengths are possible for compact 3-manifold groups? and, more generally

b) How are the quotients G_{α}/G_{α+1} restricted for compact 3-manifold groups? Herein we give a lot of new information about a).

The literature about the lower central series length seems sparse. In 1935 W. Magnus showed that free groups are ω-nilpotent and hence have length ω (length 2 if of rank 1) [Ma]. In 1949 A.I. Malcev showed that, for any ordinal α, there exists a group of length α [Mal]. In 1962 G. Baumslag showed that the fundamental groups of compact surfaces of even Euler characteristic (except the Klein bottle) are ω-nilpotent and hence of length at most ω [Ba1]. In 1966, P. Hall and B. Hartley showed that, for any countable ordinal α, there is a finitely-generated group of length α ([HH], see also [H1], [H2]). All of their examples were non-finitely presented. In 1991 J. Levine exhibited the first finitely-presented group with length greater than ω [L3]. His example has length ω +1*. The authors have recently expanded upon this to exhibit large families of finitely-presented groups with very large length (at least 2ω at present count) and other desirable properties [CO2]. On a different but related front, in 1969, C. Thomas described precisely those 3-manifolds whose groups are nilpotent [T]. P. Teichner’s recent paper is a good companion to ours [Te].

We now summarize our major results. Probably the most interesting result is the existence of closed, orientable 3-manifold groups, even hyperbolic, which have very large length. This is interesting given that any hyperbolic 3-manifold group has a subgroup of finite index which is ω-nilpotent!

Theorem 8.1, 8.2. There exist closed hyperbolic 3-manifolds whose fundamental groups have length at least 2ω (i.e., greater than ω + k for all integers k).

*This calculation is not in [L3] but was carried out by the authors. The interested reader may write to us for a copy of the calculations.

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We observe from combining previous work of D. Ruberman [R] and J. Stallings [St] that any 3-manifold has the same lower central series quotients as a hyperbolic one. Corollary 7.2. Suppose M is a closed, oriented 3-manifold with π_{1}(M) ≅ G. Then there exists a closed, oriented hyperbolic 3-manifold X with π_{1}(X) ≅ P and an epimorphism f_{∗}: P ⟶ G which induces an isomorphism P/P_{α} ≅ G/G_{α} for each ordinal α. In particular the length of G equals the length of P.

We do not know the precise lengths of these examples but they appear to be ω^{2}. The existence of these “pathologies” makes the following interesting. Theorem 3.3. Non-abelian fundamental groups of compact surfaces have length ω and are ω-nilpotent. Theorems 5.6 and 5.9. A Fuchsian or planar discontinuous group (see §5) whose underlying surface has even Euler characteristic or non-empty boundary, and is not S^{2}, has length at most ω. If the underlying surface is S^{2} then the group has length at most ω + 1. Theorem 6.10. Suppose G is the fundamental group of a compact 3-manifold which admits a Seifert fibering with base surface X. If X is neither S^{2} nor a connected sum of an odd number of projective planes, then length(G) ≤ ω. If X is S^{2} then length(G) ≤ ω+1.

On the other hand it is easy to show that almost all Fuchsian groups and Seifert fibered groups have length at least ω. For example, Theorem 5.10. If G is a Fuchsian group whose underlying surface X has π_{1}(X) nonabelian then the length of G is at least ω. Propositions 6.11 and 6.12. With G as in 6.10 above, if π_{1}(X) is not abelian then G has length at least ω. If rank H_{1}(G) ≥ 4 then G has length at least ω.

The reader interested only in the length calculations and not in connections to other areas of algebra and low-dimensional topology, should now skip to Chapter 2.

The relationship between the lower central series of finitely presented groups and homology cobordism

The basic interplay between the homology, in low dimensions, of a group or space and its lower central series was established by work of K.T. Chen, W.S. Massey, J. Milnor, J. Stallings and W. Dwyer [Ch1, Ch2, Mas, Mi, St, D]. In fact we shall employ the following theorem of Stallings so often in our present work that we state it for the readers convenience.

1.1 Stallings’ Theorem [St; p. 170]. Let N ◁ G be a normal subgroup contained in G_{2}. There is a natural exact sequence

H_{2}(G) ⟶^{π_{∗}} H_{2}(G/N) ⟶ \frac{N}{[G, N]} ⟶ 0.

If h : A ⟶ B is a homomorphism inducing an isomorphism on H_{1} and an epimorphism on H_{2} then, for any finite n, h induces an isomorphism A/A_{n} ≅ B/B_{n}; h induces an embedding A/A_{ω} ↪ B/B_{ω}. Moreover if h is surjective it induces an isomorphism A/A_{α} ≅ B/B_{α} for any ordinal α. (Consequently A and B have the same length).

There are similar results for the coefficient groups Z_{p} (which we shall use in §8) and ℚ but we shall not state these. As an easy consequence of 1.1, if 2 manifolds M^{i} i = 0, 1 are homology cobordant then their groups G^{i} are isomorphic modulo any term of the finite lower central series, i.e., G^{0}/G_{n}^{0} ≅ G^{1}/G_{n}^{1}.

Corollary 1.2. If M^{0} is homology cobordant to M^{1} and the length of π_{1}(M^{0}) = G^{0} is finite then the length of π_{1}(M^{0}) equals the length of π_{1}(M^{1}) = G^{1}. Therefore length(G^{0}) ≥ ω if and only if length(G^{1}) ≥ ω.

1.3 If one defines two groups G^{0}, G^{1} to be homology cobordant if there exist maps ϕ_{i} : G^{i} ⟶ H to some third group H, which induce isomorphisms on H_{1} and epimorphisms on H_{2} then the following question presents itself.

1.4 Question: Is length invariant under homology cobordism of groups?

An affirmative answer would prove the Parafree Conjecture (see below) for a very large class of groups. Unfortunately (or fortunately) we shall herein resolve 1.4 in the negative for general finitely-presented groups (see 9.1).

The relationship between the length of 3-manifold groups and the “Parafree Conjecture”

Recall that a group G is parafree of rank m if G/G_{n} ≅ F/F_{n}, n ∈ Z_{+}, where F is the free group of rank m, and G_{ω} = {e} [Ba2, Ba3] [Str]. The “Parafree Conjecture,” due to G. Baumslag, is: “A finitely generated parafree group G has H_{2}(G; Z) = 0” [Ki]. A parafree group of rank 1 is easily seen to be free using Stalling’s theorem. A “proof” of the parafree conjecture appeared in [Gu] but that proof has been found to have gaps. We remark that the nilpotent completion of F, \underleftarrow{lim}(F/F_{n}), is an infinitely generated parafree group of rank m whose second homology group is uncountably generated if m > 1 [B2]. The parafree conjecture has relations with link concordance invariants, homology cobordism, the Kervaire conjecture, Whitehead’s Asphericity conjecture and with lengths of 3-manifold groups. As regards the latter we have the following. A group G is called almost parafree if there is some free group F and homomorphism F ⟶ G which induces isomorphisms F/F_{n} ⟶ G/G_{n} for all positive integers n (this is called weakly parafree in [FT2] and almost parafree in [Ki]). Thus if G is almost parafree then G/G_{ω} is parafree.

Theorem 1.5. Assume the rank of the abelianization of all groups is greater than 1. The following are equivalent.

P: The Parafree Conjecture is true.

AP: Every finitely generated almost parafree group G has length ω (see [Ki]).

E: Every finite E-group has length ω. (Recall a finite E-group is the fundamental group of a finite 2-complex Y such that H_{1}(Y) is torsion-free and H_{2}(Y; Z) = O [C2]).

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T : If G is the fundamental group of a closed orientable 3-manifold and G is almost parafree, then G has length ω.

Remark. It is interesting that T , which is restricted to finitely presented 3-manifold groups, can imply P which has to do with all finitely generated groups.

Proof of 1.5. First note that AP ⇒ E since every E-group is almost parafree [C2; p. 643]. Suppose G is almost parafree. Then G/G_{ω} is finitely generated and parafree. It follows that P ⇒ AP since, by Stallings’ Theorem, if H_{2}(G/G_{ω}) = 0 then G_{ω} ≅ G_{ω+1}. Moreover T follows a fortiori from AP. The following will show E ⇒ P.

Lemma 1.6. Given any finitely generated almost parafree group G there exists a finite E-group E and an epimorphism E ⟶^{f} G which induces an isomorphism E/E_{ω} ⟶^{f} G/G_{ω}.

Proof of 1.6. Since G is almost parafree, H_{1}(G) ≅ Z^{m} for some m. Choose a generating set {x_{i}, y_{j} | 1 ≤ i ≤ m, 1 ≤ j ≤ k} for G such that {x_{i}} is a basis for G/G_{2}. Then, for each j there y_{j} = (∏_{i=1}^{m} x_{i}^{n_{ij}}) w_{j} for some n_{ij} where w_{j} lies in the commutator subgroup of the free group on {x_{i}, y_{j}}. Let E = ⟨x_{i}, y_{j} | y_{j} = (∏_{i=1}^{m} x_{i}^{n_{ij}}) w_{j} 1 ≤ j ≤ k, 1 ≤ i ≤ m⟩, and f : E ⟶ G be the obvious epimorphism. Clearly f induces an isomorphism on H_{1}. One sees by inspection that E is a finite E-group. Consider the maps F ⟶^{g} E ↠^{f} G where F is free on {x_{i}} and g(x_{i}) = x_{i}. Therefore f ∘ g induces an isomorphism abelianizations and hence induces epimorphisms F/F_{n} ⟶ G/G_{n} for each n. Since G is almost parafree G/G_{n} ≅ F/F_{n}. Since F/F_{n} is nilpotent, hence Hopfian, these maps F/F_{n} ⟶ G/G_{n} induced by f ∘ g are isomorphisms. Similarly g induces isomorphisms F/F_{n} ⟶ E/E_{n}. Therefore f induces isomorphisms E/E_{n} ⟶ G/G_{n}. Since f is surjective it follows that f induces an isomorphism E/E_{ω} ≅ G/G_{ω}. □

Applying 1.6 with G a finitely generated parafree group we see that H_{2}(G) ≅ H_{2}(E/E_{ω}) ≅ E_{ω}/E_{ω+1} (since H_{2}(E) = 0) so E ⇒ P.

It will suffice now to show that “not E” implies “not T ”. Suppose E is a finite E-group and γ ∈ E_{ω} − E_{ω+1}. Let C = E/ ⟨γ⟩, so C is finitely presented and E ↠^{π} C induces an isomorphism on abelianizations and indeed E/E_{ω} ≅ C/C_{ω}. In particular C is almost parafree. Moreover the class [γ] ∈ H_{2}(C) is not zero (form K(C, 1) by attaching a 2-cell along γ to the 2-complex associated to E then adding 3 and higher-dimensional cells as needed) since, indeed, its image in H_{2}(C/C_{ω}) ≅ H_{2}(E/E_{ω}) ≅ E_{ω}/E_{ω+1} is not zero (the fact that [γ] ∈ H_{2}(E/E_{ω}) maps to γ under the Stallings’ map is clear, for example, from the proof of Stallings’ theorem in [C3]).

Given a group C as above, we shall construct a 4-manifold Y whose boundary will be the 3-manifold counterexample to T . For specificity consider a presentation P of C obtained from a finite presentation for E associated to a 2-complex with H_{2} = 0, by adding the relation γ. Construct a 4-manifold Y such that π_{1}(Y) ≅ C by adding a one-handle to B^{4} for each generator of P and a 2-handle for each relator of P. The final 2handle, attached to the null-homologous loop γ should be attached with framing ±1. Then H_{2}(Y) ≅ Z generated by this last 2-handle and the self-intersection of this class is ±1. It follows that the map H_{2}(Y) ⟶ H_{2}(Y, ∂Y) is an isomorphism (note H_{2}(Y, ∂Y) is free abelian since Tor H_{2}(Y, ∂Y) ≅ Tor H^{2}(Y) ≅ Tor H_{1}(Y) ≅ 0). Since Y has no 3-handles, H_{1}(Y, ∂Y) ≅ 0 and hence H_{1}(∂Y) ⟶ H_{1}(Y) is an isomorphism. Let G = π_{1}(∂Y). Then the inclusion i : G ↠ C is an epimorphism which induces an isomorphism G/G_{2} ≅ C/C_{2}. Consider a map F ⟶^{g} G which induces an isomorphism on abelianizations and hence epimorphisms g : F/F_{n} ⟶ G/G_{n}, and epimorphisms i ∘ g : F/F_{n} ⟶ C/C_{n} for all n. Since C/C_{n} ≅ F/F_{n} and F/F_{n} is Hopfian, the maps i ∘ g are isomorphisms from which it follows that F/F_{n} ≅ G/G_{n} ≅ C/C_{n}. Therefore G is almost parafree. Since i is surjective, G/G_{ω} ≅ C/C_{ω} and hence i_{∗} : H_{2}(G/G_{ω}) ⟶ H_{2}(C/C_{ω}) is an isomorphism. Suppose that G had length ω so G_{ω} ≅ G_{ω+1} and (from 1.1) H_{2}(G) ⟶ H_{2}(G/G_{ω}) is onto. It would follow that the image of [γ] under the map H_{2}(C) ⟶ H_{2}(C/C_{ω}) pulls back to a class α ∈ H_{2}(G) i.e., i_{∗} ∘ π_{∗}(β) = π_{∗}([γ]). This is a contradiction since H_{2}(∂Y) ⟶ H_{2}(Y) is the zero map so H_{2}(G) ⟶^{i_{∗}} H_{2}(C) is also the zero map implying π_{∗}([γ]) = i_{∗} ∘ π_{∗}(β) = π_{∗} ∘ i_{∗}(β) = 0, but we have previously observed that π_{∗}([γ]) ≠ 0 in H_{2}(C/C_{ω}). Therefore G is a counterexample to T and we have shown “not E” implies “not T ”. □

For a slightly smaller class of groups we can be much more specific about which 3manifold groups are related to the Parafree Conjecture.

Theorem 1.7. For the class S of all groups G such that rank(H_{1}(G)) = m > 1 and G is normally generated by m elements, the following are equivalent:

P: The Parafree Conjecture is true for groups in S.

AP: Every almost parafree group G in S has length ω.

R: The fundamental group of the exterior of every ribbon link in S^{3} has length ω.

Equivalently the group of the 0-surgery along a ribbon link has length ω. E: Every finite E-group E in S has length ω.

Proof. First note that P ⇒ AP ⇒ E as in 1.5. It is also well known that, for any ribbon link group G, the map G ↠ G/G_{ω} factors through an epimorphism to a finite E-group E of rank m [Hi; p. 23–25]. It follows from Stallings’ theorem that G/G_{ω} ≅ E/E_{ω} and hence that E ⇒ R. For if E_{ω} ≅ E_{ω+1} then H_{2}(E/E_{ω}) ≅ 0, implying H_{2}(G/G_{ω}) ≅ 0 which implies G_{ω} ≅ G_{ω+1} as above. Conversely, it is shown by P. Bellis [Be; Thm. 3.4] that, for any finitely-generated E-group in S, there exists a ribbon link in S^{3} with group G and an epimorphism ϕ : G ↠ E which is an isomorphism on abelianization. Therefore G/G_{ω} ≅^{ϕ} E/E_{ω} (by Stallings’ theorem) so H_{2}(G/G_{ω}) ≅ H_{2}(E/E_{ω}) ≅ E_{ω}/E_{ω+1}. If the length of G is ω then H_{2}(G) ⟶ H_{2}(G/G_{ω}) is surjective. From the naturality of Stallings’ exact sequence with respect to ϕ : G ⟶ E one easily concludes that H_{2}(E/E_{ω}) ≅ 0 and hence that E has length ω. Therefore R ⇒ E.

It remains only to show E ⇒ P. Suppose G is a finitely-generated parafree group of rank m lying in S, generated by {x_{i} | 1 ≤ i ≤ ℓ}. Suppose G is normally generated by {y_{i} | 1 ≤ j ≤ m} where x_{i} = ∏_{k=1}^{n_{i}} η_{ik}y_{ik}^{ϵ_{ik}}η_{ik}^{−1}. Let E ≡ ⟨x_{i}, y_{j} | x_{i} = ∏_{k=1}^{n_{i}} η_{ik}y_{ik}^{ϵ_{ik}}η_{ik}^{−1}⟩ and define ϕ : E ⟶ G by ϕ(x_{i}) = x_{i}, ϕ(y_{j}) = y_{j}. Note E is normally generated by m elements. The proof is then completed as above in 1.5 and 1.6.

Since the longitudes of a ribbon link lie in G_{δ}, the group of the 0-surgery has the same lower central series quotients and length as that of the ribbon link [ ]. □

For any integer N there exists a finitely presented group G in S such that F/F_{n} ≅ G/G_{n} for all n ≤ N but length(G) > ω [CO2]. However we are not able to resolve the Parafree Conjecture at this time.

Note that since every ribbon link is concordant to a trivial link, every ribbon link group is homology cobordant (in the sense of 1.3) to a free group. Hence if length were an invariant of general homology cobordism then R would be true, implying the Parafree Conjecture for groups in S. In fact it would suffice that length be invariant of homology cobordism for almost parafree groups.

The relationship between the lower central series of finitely-presented groups and 4-dimensional topological surgery

The lower central series has arisen repeatedly in the quest to identify, in dimension 4, for which fundamental groups G the “basic machinery of manifold theory, the surgery and 5-dimensional s-cobordism theorems exists in the topological category [FT2; p. 1]” [F1, F2, F3, FQ chapter 6, FT1]. This machinery is known to exist when G is “good” (subexponential growth) [FT1] and “suspected to fail for non-abelian free groups and hence for a “random” group [FT2]”. The general set-up for surgery involves a compact 4-manifold N with a degree 1 normal map f : N ⟶ X, and the “surgery kernel” in H_{2}(N, Z[π_{1}(X)]). The lower central series arises in the following way. To perform surgery on 4-manifolds one attempts to realize certain 2-dimensional homology classes in the kernel by embedded 2-spheres. Reducing to a relative version, one has a certain circle γ which bounds a surface in a 4-manifold and one hopes to find an embedded 2-disk with boundary γ in this relative homology class. All the successful work in this area has involved approaching this problem via certain infinite constructs involving either disks (Whitney disks) or surfaces (gropes, half-gropes, capped gropes); depending on the treatment ([CF], [FT1], [FQ]). For precise definitions the reader is referred to [FT2] and [FQ]. However we can give the reader the basic idea. If [γ] ∈ G_{2} then γ bounds a mapped in surface. If [γ] ∈ G_{3} then γ bounds a mapped in surface S_{2} with standard symplectic basis {a_{i}, b_{i} | i = 1, …, g} such that a_{i} bounds a mapped in surface S_{3,i} for each i = 1, …, g. Then S_{1} ∪ ⋃_{i=1}^{g} S_{3,i} is called a half-grope of class 3 for γ. The loop γ bounds a map of a half-grope of class k if and only if [γ] ∈ G_{k} [FT2; Lemma 2.1]. In infinite constructions one often desires an ∞-grope. Note that if [γ] ∈ G_{ω}, then [γ] ∈ G_{k} for each k but this merely guarantees the existence of a grope of class k for each k and these might not be related to each other. In fact, γ bounds a map of an ∞-grope if and only if γ ∈ G_{δ} [FT2; Lemma 2.2]! Therefore it would be highly desirable if the length of G were at most ω, for then G_{ω} = G_{δ}. In §8 we exhibit the first examples of compact 3-manifolds whose groups have length greater than omega.

We shall now make a specific connection between the Parafree Conjecture and 4dimensional topological surgery. Although this connection seems provocative, it may not be fundamental. But first, in order to make the connection we must refer more precisely to the major result of [FT2]. There N is a compact topological 4-manifold, Y is a π_{1}-null, codimension 0 submanifold such that H_{1}(Y) ≅ H_{1}(∂Y). N^{+} is the complex obtained from N by attaching 3-cells to 2-spheres in N representing a free basis for the image of H_{2}(Y). Let C = π_{1}(Y).

Freedman-Teichner Theorem 1.1 [FT2; Thm. 1.1]: Given (N, Y) as above, if H_{2}(C) ⟶ H_{2}(C/C_{n}) is the zero map for all n then there exists a 4-manifold N′(∂N′ = ∂N^{+}) and a simple homotopy equivalence h : (N′, ∂N′) ⟶ (N^{+}, ∂N^{+}) i.e., a manifold structure on N^{+}.

This theorem is then applied to successfully perform surgery on a map f : N ⟶ X with vanishing surgery kernel carried by such a Y (see [FT2; Corollary 1.2]). The power of the theorem, as pointed out by Freedman and Teichner, is that whereas previous results had relied on finding a submanifold Y whose second homology is “spherical”, that is H_{2}(C) = 0, their theorem only requires sphericity modulo each finite term of the lower central series.

One way in which their hypothesis could be satisfied is if, in fact, H_{2}(C) ⟶ H_{2}(C/C_{ω}) is the zero map, a priori stronger than their condition that H_{2}(C) ⟶ H_{2}(C/C_{n}) be zero for each finite n. We shall show below that the first condition is in fact equivalent to their condition if and only if the Parafree Conjecture is true, and thus is stronger than their condition if and only if there exists an almost parafree hyperbolic 3-manifold group of length greater than ω.

Theorem 1.8. Given 4-manifolds Y ⊂ N as above (π_{1}(Y) ⟶ π_{1}(N) is zero, H_{1}(∂Y) ≅ H_{1}(Y)) such that (N, Y) satisfies the hypotheses of Freedman-Teichner’s Theorem 1.1 in that H_{2}(C) ⟶ H_{2}(C/C_{n}) is zero for all finite n, but H_{2}(C) ⟶ H_{2}(C/C_{ω}) is not zero, there is a counterexample to the Parafree Conjecture and there is an almost parafree closed hyperbolic 3-manifold group which has length greater than ω. Conversely if there exists a counterexample to the Parafree Conjecture, then there is a pair of 4-manifolds (N, Y) as above which satisfies the hypotheses of Freedman-Teichner’s Theorem 1.1 that H_{2}(C) ⟶ H_{2}(C/C_{n}) is zero for all n, while H_{2}(C) ⟶ H_{2}(C/C_{ω}) is not zero.

Proof of 1.8. The condition H_{1}(∂Y) ≅ H_{1}(Y)) is quite strong. It implies H_{2}(Y) and H_{1}(Y) are free abelian and that H_{2}(∂Y) ⟶ H_{2}(Y) is the zero map [FT2]. Thus there is a map F ⟶^{f} C which induces isomorphisms on abelianizations. Using Dwyers’ extension of Stallings’ theorem, the fact that H_{2}(C) ⟶ H_{2}(C/C_{n}) is zero for each n implies f induces isomorphisms F/F_{n} ≈ C/C_{n}. Hence C/C_{ω} is a finitely generated parafree group and H_{2}(C/C_{ω}) ≠ 0 since the map H_{2}(C) ⟶ H_{2}(C/C_{ω}) was assumed to be non-zero. Thus C/C_{ω} is a counterexample to the Parafree Conjecture. By 1.5 “not P” implies “not T ” as desired.

Conversely if the Parafree Conjecture fails then “not P” implies “not E” as in 1.5. Looking back at the proof of “not E” implies “not T ” in 1.5, we see that we constructed a compact 4-manifold Y with almost parafree fundamental group C which satisfies the hypotheses of 1.8. This Y may then be embedded in 4-manifolds N, such that π_{1}(Y) ⟶ π_{1}(N) is zero, in many inequivalent ways. □

We remark that, while no example of 1.8 is currently known, one can write down an example (N, Y) which fails the Freedman-Teichner criterion in that H_{2}(C) ⟶ H_{2}(C/C_{n}) is non-zero for some n.

The relationship between lower central series and link concordance

The relationship between the lower central series of G = π_{1}(S^{3} − L) for a link L of m circles in S^{3} and the concordance class of L is well established. In the 1950’s, Milnor, building on work of Chen, defined his \overline{μ}-invariants, a sequence of increasingly discriminating invariants which inductively detect whether or not the meridional map F ⟶ G induces an isomorphism F/F_{k} ≅ G/G_{k} for finite k. Milnor’s invariants are zero if and only if G is almost parafree or, equivalently, the longitudes of L lie in G_{ω}. Since the exterior of a concordance is a homology cobordism between the link exteriors, Stallings’ theorem showed that the quotients G/G_{k} as well as Milnor’s invariants are invariant under concordance. It is known that fusions of boundary links have vanishing \overline{μ}-invariants yet are not in general concordant to trivial links or even any boundary link [C1, CO1]. Therefore efforts have turned to finding invariants “beyond G_{ω}” [L1, L2, O1, O2, LD, L3, C2]. For example it is still unknown whether or not the longitudes of a null-concordant link must lie in G_{δ} as they do for the trivial link (see property R above). All of this work is concerned with certain “universal groups” to which all link groups map and which, therefore, can be used to define invariants for all links. The invariants lie, loosely speaking, in H_{2} and H_{3} of these groups. The groups are F̂, the algebraic closure of the free group F, as defined by J. Levine [L2] and P. Vogel [LD], and \overline{F}, the algebraic closure of F in its nilpotent completion \underleftarrow{lim}(F/F_{n}), due to Levine [L1]. It was shown by Levine that \overline{F} ≅ F̂/F̂_{ω} and that H_{2}(F̂; Z) = 0. Thus, using Stallings’ exact sequence, H_{2}(\overline{F}) = F̂_{ω}/F̂_{ω+1} so H_{2}(\overline{F}) = 0 if and only if the length of F̂ is ω. F̂ is not very well understood but Levine showed that it is a direct limit of higher-dimensional link groups (n.b. these are always almost parafree — see AP above) so the length of F̂ is related to the length of such groups and hence to the “conjectures” listed above. Moreover it follows from Levine’s work that \overline{F} is a direct limit of finitely generated parafree groups and hence the Parafree Conjecture for groups in S directly implies H_{2}(\overline{F}) = 0 and length(F̂) = ω. The “Strong Parafree Conjecture”, that a finitely generated parafree group has H_{2} = 0 and is of cohomological dimension at most 2, implies H_{3}(\overline{F}) = 0 and hence that invariants associated to \overline{F} by Levine and Orr always vanish.

The material is organized as follows.

§1 History, Main Results, Motivation

§2 Elementary Facts and Elementary Examples §3–§5 Lengths of Fuchsian Groups

§6 Lengths of Groups of Seifert Fibered 3-Manifolds

§7 Every 3-Manifold has the Same Lower Central Series Quotients as a Hyperbolic

3-Manifold §8 3-Manifolds Whose Groups have Length at least 2ω §9 Length and Homology Cobordism §10 Questions §2. Notation, Elementary Facts, Simple Examples of Length of 3-Manifold Groups.

In this paper we use the convention [a, b] = aba^{−1}b^{−1}.

The following elementary results will be used throughout. Proposition 2.1 (see [MKS; Thm. 5.1, 5.3]). Let a, b, c be elements of a group G. Let n be a positive integer and α an ordinal.

a. If a ∈ G_{α} then ab ≡ ba mod G_{α+1}

b. If [a, b] ∈ G_{α} and [a, c] ∈ G_{α} then [a, bc] ≡ [a, b][a, c] modulo G_{α+1} and so [a, b]^{n} ≡

[a^{n}, b] ≡ [a, b^{n}] modulo G_{α+1}.

Proposition 2.2. Let A, B be groups with lengths ℓ(A) and ℓ(B).

a. If f : A ⟶ B is an epimorphism then ℓ(A) ≥ min{ω, ℓ(B)}.

b. If r : A ⟶ B is a retraction (i.e., there is a homomorphism i : B ⟶ A such that

r ∘ i is the identity) then ℓ(A) ≥ ℓ(B) c. (A × B)_{α} ≅ A_{α} × B_{α} and ℓ(A × B) = max{ℓ(A), ℓ(B)}.

Remark. b) is false for a general epimorphism since any group is a quotient of a free group with length ω. Proof of 2.2. a) Suppose not. Then ℓ(A) = n < ℓ(B) so there is some b ∈ B_{n} − B_{n+1}. One checks that the restriction f : A_{n} ⟶ B_{n} is onto (this fails if n is not finite), so there is some a ∈ A_{n} such that f(a) = b. Since ℓ(A) = n, a ∈ A_{n+1} implying b ∈ B_{n+1}, a contradiction. For b), if r is a retraction then r : A_{α} ⟶ B_{α} is onto for any ordinal α so the argument for a) applies. Part c) is clear. □

The following ideas are important to our construction of elements in G_{δ} and G_{ω}. Definition 2.3. Let p be a positive integer. An element g ∈ G is called a generalized periodic element of period p, or a generalized p-element if, for any positive integer n, there is an integer k = k(n) such that g^{p^{k}} ∈ G_{n}. Equivalently, g is of order a power of p in each G/G_{n} for n finite. Example. The group K of the Klein bottle has a presentation ⟨u, t | [u, t] = t^{−2}⟩ and it follows that [u, [u, …, [u, t] …]] = t^{(−2)^{n}} and hence that t is a generalized 2-element. The dihedral group D^{n} of order 2n has a presentation ⟨u, t | u^{2} = utu^{−1}t = t^{n} = 1⟩ and t^{2^{k}} ∈ D_{k+1}^{n}. Thus t is a generalized 2-element. Proposition 2.4. If a and b are generalized periodic elements of G of relatively prime orders p, q then [a, b] ∈ G_{ω}. Proof of 2.4. We proceed by induction. Suppose [a, b] ∈ G_{n}. It suffices to show [a, b]^{p^{k}} ∈ G_{n+1} and [a, b]^{q^{t}} ∈ G_{n+1} for some k, t. Choose k, t such that a^{p^{k}} ∈ G_{n} and b^{q^{t}} ∈ G_{n}. By 2.1a, [a, b]^{p^{k}} ≡ [a^{p^{k}}, b] modulo G_{n+1} and so [a, b]^{p^{k}} ∈ G_{n+1}. Similarly for [a, b]^{q^{t}}. □

Proposition 2.5. Let a and b be periodic elements (torsion) of the group G of relatively prime orders p and q. Then [a, b] ∈ G_{δ}. (It suffices that a^{p^{k}} commutes with b and b^{q^{t}} commutes with a). Proof of 2.5. Suppose a^{p^{k}} commutes with b and b^{q^{t}} commutes with a. By induction suppose [a, b] ∈ G_{α}. Then merely apply the argument of the proof of 2.4. □

A few words are in order about the relationship between nilpotence, transfinite nilpotence and length. If G_{α} = {e} then clearly the length of G is at most α. Thus a nilpotent group is of finite length equal to its nilpotency class and an ω-nilpotent group (G_{ω} = {e}) has length at most ω and, in fact, equal to ω unless it is nilpotent. We shall often use residually nilpotent (for any g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N is nilpotent) instead of ω-nilpotent since they are equivalent. But the reader must be careful not to conclude anything in the other direction, for there are many groups which are of length 1 (perfect groups) or length 2 (see below), for example, which are not transfinitely nilpotent.

Proposition 2.6. If the abelianization of G ≠ e is cyclic then G has length 2. Proof. The surjection h : G ⟶ G/G_{2} satisfies the hypothesis of 1.1 so G/G_{n} ≅ (G/G_{2})/(G/G_{2})_{n}. Since G/G_{2} is abelian, the latter quotient is G/G_{2} if n > 1. Therefore G_{n} = G_{2}. □

Proposition 2.7. Let p be a prime. The groups Z ∗ Z, Z ∗ Z_{p^{k}}, Z_{p^{k}} ∗ Z_{p^{m}} are of length ω and residually nilpotent. Proof of 2.7. The group Z_{p^{k}} ∗ Z_{p^{m}} is not nilpotent since a nilpotent group which is generated by a finite number of elements of finite order must be itself finite which these groups are clearly not. It follows that Z ∗ Z and Z ∗ Z_{p^{k}} fail to be nilpotent. It is a theorem of A.I. Malcev that these groups are residually nilpotent ([Mal], see also [Lic]). Alternatively, a free product of residually finite p-groups is residually a finite p-group [MKS; p. 417]. Since finite p-groups are nilpotent, these groups are residually nilpotent. Hence in each case G_{ω} = {e} and so length(G) ≤ ω. If the length of G were n < ω then G_{n} = G_{ω} = {e} which is a contradiction. □

The reader is warned that Z_{2} ∗ Z_{3} and Z ∗ Z_{6} are not even transfinitely nilpotent since commutators of elements of order 2 with elements of order 3 lie in G_{δ} by 2.5. By 2.6, ℓ(Z_{2} ∗ Z_{3}) = 2 while ℓ(Z ∗ Z_{6}) ≥ ω since it maps onto Z ∗ Z_{2}. Proposition 2.8. The dihedral group D of order 2^{k+1} is (k + 1)-nilpotent and has length k + 1. The “infinite dihedral group” ⟨u, t | u^{2} = utu^{−1}t = 1⟩ has length ω and is residually nilpotent. Proof of 2.8. If D = ⟨u, t | u^{2} = utu^{−1}t = t^{2^{k}} = 1⟩ then there is an split exact sequence 1 ⟶ Z_{2^{k}} ⟶ D ⟶ Z_{2} ⟶ 1. It can be shown directly (by induction) that D_{m+1} ≅ 2^{m}Z_{2^{k}} so D_{k+1} ≅ {e} and D_{k} ≠ {e}. The infinite dihedral group maps onto any finite dihedral group so has length at least ω by 2.2a. But the infinite dihedral group is Z_{2} ∗ Z_{2} hence is residually nilpotent by 2.7. Therefore it has length ω. □

Now we list examples of 3-manifold groups of length 1, 2, n and ω to give the reader a feeling for the situation.

(2.10) A group has length 1 if and only if it is perfect so the group of any homology

3-sphere has length 1. (2.11) If G is a non-trivial abelian group then G_{2} = {e} so G has length 2. Therefore

3-manifolds with abelian group such as S^{1}×S^{1}×S^{1} have length 2. There are many more examples with length 2 however. According to 2.6, if H_{1}(M) is cyclic then π_{1}(M) has length 2. Examples are knot complements and their 0-framed surgeries

(2.12) To find closed 3-manifolds with any prescribed finite length it is convenient to look among those with finite, nilpotent fundamental groups [T]. For example the generalized quaternion group Q^{n} of order 4n, presented by ⟨u, t | u^{2} = t^{n} utu^{−1} = t^{−1}⟩ is the group of the Euler class n circle bundle over RP (2) with orientable total space. Killing u^{2} gives the dihedral group of order 2n. When n = 2^{k} the latter is (k + 1)-nilpotent and has length k + 1 (see 2.8). Since Q^{n} is a Z_{2} central extension of D^{n}, it is easy to see that Q^{2^{k}} is (k + 2)-nilpotent and has length (k + 2). Therefore, for any positive integer k there is a closed, orientable 3-manifold with finite fundamental group of length k.

(2.13) The circle bundle over a torus with Euler class 1 has the Heisenberg group, F/F_{3}, as its group and hence has length 3. The Euler class n bundles also have nilpotent groups of length 3 (2 if n = 0).

(2.14) The length of a non-abelian free group is ω by [Mal], so if M = #_{i=1}^{k}S^{1}×S^{2} for k ≥ 2 then the length of π_{1}(M) is ω. Moreover if M = M_{1}#M_{2} were H^{1}(M_{i}; Z_{p}) ≠ 0 for i = 1, 2 then π_{1}(M) maps onto Z_{p} ∗ Z_{p} and hence has length at least ω by 2.2a and 2.7. This idea can be formalized as follows.

Suppose M is a closed, connected, 3-manifold and M = #M_{i} its prime decomposition. Let M̂ be obtained by eliminating all M_{i} which are homology spheres. Then there is a degree one map M ⟶ M̂ which induces an isomorphism on H_{1} so, by Stallings’ theorem we have: Proposition 2.15. π_{1}(M) and π_{1}(M̂) have the same length and isomorphic lower central series quotients.

Similarly, let M̂_{p} be obtained by eliminating all M_{i} which are Z_{p}-homology spheres. We say M is Z_{p}-homology prime if M̂_{p} is prime. Proposition 2.16. If M is not Z_{p}-homology prime then π_{1}(M) maps onto Z_{p} ∗ Z_{p} so has length at least ω.

In §6 we observe that almost all Seifert fibered 3-manifold groups have length at least ω. Together with 2.16, we begin to see that it seems clear that “most” 3-manifold groups will have length at least ω, but we cannot formalize this.

§3. Surface Groups.

Let X be a compact, connected surface and G its fundamental group. In this chapter we see that, if χ(X) < 0, then G has length ω and is residually nilpotent. We also give partial results about when G is residually a finite p-group and prove a key proposition about finite extensions of such groups. These facts will be crucial in our analysis of the lengths of Fuchsian groups and the groups of Seifert-fibered 3-manifolds. Proposition 3.1. If H ◁ G where H is residually a finite p-group (p prime), and G/H is a finite p-group then G is residually a finite p-group. Proof. It suffices to find, for an arbitrary non-trivial h ∈ H, a normal subgroup N of G such that h ∉ N and G/N is a p-group. Since H is residually a finite p-group, there is M ◁ H of index p^{n} which does not contain h. If {g_{i} | i = 1, …, p^{m}} are coset representatives for H in G then let N = ⋂_{i=1}^{p^{m}} M_{i} where M_{i} = g_{i}Mg_{i}^{−1}. Then N is normal in G and H. Since h ∉ N, it remains to show that G/N has order p^{k}. Since the order of G/H is a power of p it suffices to show that the order of H/N is a power of p. Consider the projections π_{i} : H/N ⟶ H/M_{i}. Noting that M_{i} is of index p^{n} for all i, if [x] ∈ H/N then x^{p^{n}} ∈ M_{i} so x^{p^{n}} ∈ N. Therefore H/N is a p-group. □

We can now combine 3.1 with results of G. Baumslag to show the following. Proposition 3.2. All surface groups are residually finite 2-groups and hence residually nilpotent. If χ(X) is even and X is not the Klein bottle, then G is residually a finite p-group for any odd prime p. Proof. Baumslag has shown that closed surface groups of even Euler characteristic (except the Klein bottle) are residually free [Ba1]. Since free groups are residually finite p-groups for any p [MKS; p. 417], the above groups are residually finite p-groups for all primes p. Any closed non-orientable surface group has an index 2-subgroup which is the group of an orientable surface so, applying 3.1, such groups are residually finite 2-groups. Since finite p-groups are nilpotent, all surface groups are residually nilpotent. □

Theorem 3.3. Let X be a compact, connected surface and G its fundamental group. If χ(X) < 0 then the length of G is ω. If X is the Klein bottle then the length of G is ω. Otherwise G is abelian and its length is 2 (or 1 if G is trivial). Proof. In the cases where G is not abelian it is easy to see that G maps onto Z_{2} ∗ Z_{2}, the infinite dihedral group, and hence has length at least ω by 2.2a and 2.7. By 3.2, G has length at most ω. □

3.4 Remarks. If G is a group containing a non-trivial generalized q-element t, then G is not residually a finite p-group if (p, q) = 1; for if ϕ : G → P is any epimorphism and P is a finite p-group then ϕ factors through G/G_{n} for some finite n so ϕ(t) is of order q^{k} and hence is trivial in P. It follows a fortiori that such a group is not residually free. Therefore the group of the Klein bottle is not residually a finite p group for odd p since the presentation ⟨u, t | [u, t] = t^{−2}⟩ reveals that t is a non-trivial generalized 2-element. The groups of the other surfaces of even Euler characteristic contain no generalized p-elements for any prime p since they are residually free. The group of RP (2)#RP (2)#RP (2) is not residually free. We do not know if the odd Euler characteristic surface groups are residually finite p-groups for odd p and this ignorance obstructs the extension of our results to all Seifert fibered spaces. We suspect that these surfaces of odd genus greater than 1 do not have generalized p-elements for any p.

§4. Free Products of Finite Cyclic Groups.

Any free product A of cyclic groups has length at most ω. In this section we prove this in the case that each factor is finite cyclic. In the case that the order of every factor is a power of a single prime p, it follows from early work of Malcev that A is residually nilpotent so A_{ω} ≅ {e} [Mal]. But in all other cases A will fail to be transfinitely nilpotent, so showing A_{ω} ≅ A_{ω+1} is more difficult. Definition. If G is a group, T (G) is the subgroup generated by the set of elements of the form [a, b] where a and b are of relatively prime (finite) orders in G. Each such element is called a T -element.

Note that T (G) is a characteristic subgroup of G and, by 2.5, is contained in G_{δ}, the intersection of the terms of the lower central series. Therefore we have T (G) ⊂ G_{δ} ⊂ G_{ω} in general. Definition. G has Property T if G_{ω} = T (G).

Hence a group with Property T is of length at most ω. A residually nilpotent group has Property T (trivially). A torsion-free group cannot have Property T unless it is residually nilpotent. Thus, for example, the fundamental group of the exterior of a non-trivial knot or non-trivial ribbon link does not have Property T.

Theorem 4.1. Suppose A ≠ {e} is a free product of finite cyclic groups. Then A has Property T. A has length ω unless the orders of the free factors are pairwise coprime in which case A has length 2. Moreover if A = ⟨c_{1}, …, c_{m} | c_{i}^{n_{i}} = 1⟩ n_{i} ≥ 2, and {p_{1}, …, p_{ℓ}} is the set of distinct primes in the prime decomposition of n_{1}n_{2} … n_{m} and n_{i} = p_{1}^{r_{i1}} … p_{j}^{r_{ij}} … p_{ℓ}^{r_{iℓ}}, 1 ≤ i ≤ m, 1 ≤ j ≤ ℓ, then A_{ω} is normally generated by

K = {[c_{i}^{k_{ij}}, c_{t}^{k_{tv}}] | j ≠ v, k_{ij} = n_{i}/p_{j}^{r_{ij}}, k_{tv} = n_{t}/p_{v}^{r_{tv}}} and A/A_{ω} ≅^{f} ×_{j=1}^{ℓ}B_{p_{j}} where

B_{p_{j}} = Z_{p_{j}^{r_{1j}}} ∗ Z_{p_{j}^{r_{2j}}} ∗ ⋯ ∗ Z_{p_{j}^{r_{mj}}}, and Z_{p_{j}^{r_{ij}}} is generated by f(c_{i}^{k_{ij}}), k_{ij} = n_{i}/p_{j}^{r_{ij}}.

Proof. It is easily seen that we may assume that A is a finite free product since any element of A lies in such a finite free product which is a retract of A (see 2.2b). Therefore suppose A ≅ ⟨c_{1}, …, c_{m} | c_{i}^{n_{i}} = 1⟩ where n_{i} ≥ 2, and n_{i} = ∏_{j=1}^{ℓ} p_{j}^{r_{ij}} for {p_{1}, …, p_{ℓ}} distinct primes. Note that for each i there is an obvious epimorphism ϕ_{i} : Z_{p_{1}^{r_{i1}}} ∗ ⋯ ∗ Z_{p_{ℓ}^{r_{iℓ}}} ⟶ Z_{n_{i}} (given by abelianization) and that this map is an isomorphism on H_{1}. Therefore, letting Ã = ∗_{i=1}^{m} ∗_{j=1}^{ℓ} Z_{p_{j}^{r_{ij}}} there is an obvious epimorphism ϕ : Ã ⟶ A which induces an isomorphism on H_{1} and an epimorphism on H_{2} (since H_{2}(A) = 0) and hence induces an isomorphism modulo any term of the lower-central-series. If Ã is generated by α_{ij}, 1 ≤ i ≤ m, i ≤ j ≤ ℓ then ϕ(α_{ij}) = c_{i}^{k_{ij}} where k_{ij} = n_{i}/p_{j}^{r_{ij}}. We can express Ã ≅ ∗_{j=1}^{ℓ}B_{j} where B_{j} ≅ ∗_{i=1}^{m}Z_{p_{j}^{r_{ij}}} generated by {α_{ij}}. Then the kernel of the epimorphism π : Ã ⟶ ×_{j=1}^{ℓ}B_{j} = B is normally generated by the set of T -elements {[α_{ij}, α_{tv}] | j ≠ v}. Examining ϕ_{i} above, one sees that the kernel of ϕ is normally generated by {[α_{ij}, α_{iv}] | j ≠ v} so that π factors through ϕ inducing an epimorphism f : A ↠ ×_{j=1}^{ℓ}B_{j}. Since the kernel of π lies in Ã_{δ}, π induces an isomorphism π : Ã/Ã_{δ} ⟶ B/B_{δ}. Hence f induces an isomorphism f : A/A_{δ} ⟶ B/B_{δ}. But a free product of residually finite p-groups is a residually finite p-group hence B_{j} is residually nilpotent [MKS p. 417]. It follows that B is residually nilpotent. Thus f : A/A_{δ} ⟶ B is an isomorphism, A_{ω} ≅ A_{δ}, and f : A/A_{ω} ⟶ ×_{j=1}^{ℓ}B_{j} is an isomorphism. The kernel of f is normally generated by {[ϕ(α_{ij}), ϕ(α_{tv})] | j ≠ v} as desired. Since these are T -elements, A_{ω} ⊂ T (A) and hence A_{ω} = T (A). Hence A has Property T.

Since A has Property T its length is at most ω. If the orders of free factors are pairwise relatively prime then map from A to its abelianization is an isomorphism on H_{1} and an epimorphism on H_{2}. By Stallings’ theorem, then A has the same length as an abelian group, that is, 2. Otherwise there is an epimorphism from A to Z_{p} ∗ Z_{p} for some prime p so ℓ(A) ≥ ℓ(Z_{p} ∗ Z_{p}) = ω by 2.2a. Thus the length of A is ω. □

§5. Fuchsian and Planar Discontinuous Groups.

In this section we discuss the length of planar discontinuous groups in preparation for studying the groups of Seifert fibered 3-manifold groups.

In this paper, for simplicity, a Fuchsian group will be defined as a group which admits a presentation of one of the following two forms:

5.1 G = ⟨a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{m}, d_{1}, …, d_{k} | c_{i}^{n_{i}} = 1 ∏_{j=1}^{m} c_{j} ∏_{ℓ=1}^{k} d_{ℓ} = ∏_{i=1}^{g}[a_{i}, b_{i}]⟩

where n_{i} ≥ 2.

5.2  G = ⟨a_{1}, …, a_{g}, c_{1}, …, c_{m}, d_{1}, …, d_{k} | c_{i}^{n_{i}} = 1 ∏_{j=1}^{m} c_{j} ∏_{ℓ=1}^{k} d_{ℓ} = ∏_{i=1}^{g} a_{i}^{2}⟩

where n_{i} ≥ 2 and g ≥ 1.

This is a serious abuse of language although not without precedent in the literature (e.g. [JS]). If one sets μ(G) = 2 − 2g − k − ∑_{j=1}^{m} (1 − \frac{1}{n_{i}} ) in 5.1 (in case of 5.2 replace 2 − 2g by 2 − g). Then it is known that if μ(G) < 0, G can be realized as a group of orientation-preserving motions on the hyperbolic plane, i.e., a true Fuchsian group. If μ(G) = 0 then G can be realized as a group of motions on the Euclidean plane, i.e., a crystallographic group. If μ(G) > 0 then G can be realized as a group of motions on the 2-sphere [ZVC; p. 122, p. 155].

The quotient of G by {c_{i}} is the group of a compact surface X of genus g with k boundary components which is orientable in 5.1 and non-orientable in 5.2. This is called the surface underlying G and adjectives such as closed, orientable, non-empty boundary, genus and Euler characteristic will be applied to G but refer to its underlying surface. It will also be convenient to introduce the signature of G, (ϵ, g, n_{1}, …, n_{m}) where ϵ = ±1 according as G is orientable or not. Note that free products of cyclic groups (k ≠ 0) and surface groups (m = 0) are examples of Fuchsian groups so our work of sections 3 and 4 will not be in vain.

As is standard in treatments of Fuchsian groups, it will be convenient to define a specific “capped surface” Y of which G is the fundamental group. Given a presentation of form 5.1 or 5.2, let X denote the underlying surface (with boundary) and let X_{0} be X minus the disjoint union of m open 2-disks. Now form Y from X_{0} by adjoining m 2-disks to these new m components of ∂X_{0} via maps of degree n_{i}. Then π_{1}(Y) ≅ G and Y shall be called the capped-surface of G (actually corresponding to the particular presentation). Moreover, borrowing from the language of orbifolds, we shall use cone point of G to refer to one of the c_{i} and the order of the cone point shall be the corresponding n_{i}.

Theorem 5.3. Let G be a Fuchsian group whose cone points are each of order a power of a fixed prime p.

i) if p = 2, G is a residually finite 2-group, hence residually nilpotent, Hopfian and of length at most ω.

ii) if p ≠ 2 and we assume the underlying surface of G has either even Euler characteristic or non-empty boundary, then G is residually nilpotent, Hopfian, of length at most ω and, if G is not the group of the Klein bottle, is residually a finite p-group.

5.4 Remarks. Such Fuchsian groups almost always have length at least ω and hence, in the cases covered by 5.3, almost always have length precisely ω. For, if g > 1, G maps onto a surface group of negative Euler characteristic (see 3.3). If k ≥ 1 then G maps onto some Z_{p} ∗ Z_{p} unless ϵ = ±1, g = 0, k = 1 and the n_{i} are pairwise coprime; or ϵ = −1, k = 1, g = 1 and m = 0. In the latter cases G has length 2. This leaves only the cases k = 0, g ≤ 1 untreated. If g = 1, k = 0 then G maps onto H ∗ ⟨c_{i} | c_{i}^{n_{i}} = 1, ∏_{i=1}^{m} c_{i} = 1⟩ where H = Z × Z if ϵ = +1 and H = Z_{2} if ϵ = −1. Therefore, if ϵ = +1 G will map onto Z ∗ Z_{p} (and hence have length at least ω) unless the group ⟨c_{i} | c_{i}^{n_{i}} = 1, Πc_{i} = 1⟩ is perfect. If ϵ = −1 the same holds unless this group has H^{1}( ; Z_{2}) = 0. We shall not pursue the scattered remaining cases.

Proof of 5.3. If G has non-empty boundary then G is isomorphic to a free product of copies of Z and Z_{p^{n_{i}}}. Since a free product of residually finite p-groups is again a residually finite p-group, we are done in this case [MKS; p. 417].

We now assume that we are in the closed case. We may also assume that G is not the group of the Klein bottle since 3.2 handles that case. The proof is accomplished by induction using 3.1 and the following Lemma which, in the orientable, positive genus case can be found in [EEK; Theorem 3.1]. Lemma 5.5 (compare [H; p. 119], [EEK; Theorem 3.1]). Let G be a closed Fuchsian group with signature (ϵ, g, p^{n_{1}}, …, p^{n_{m}}), for some prime p, which we suppose is not (−1, 2), i.e., G is not the Klein bottle group. Then there is a finite filtration, G_{k} ◁ G_{k−1} ◁ … ◁ G_{1} = G where G_{i}/G_{i+1} is a finite abelian p-group and G_{k} is the group of a closed surface which, in case p is odd, has signature (ϵ, g′) which is not (−1, 2), and where g′ ≡ g mod 2. (It follows that G has a normal subgroup G′ such that G/G′ is a finite p-group and G′ is a surface group as above).

First let us see that 5.5 implies 5.3 for the closed case. If p = 2, 3.1, 3.2 and 5.5 imply 5.3 directly, since all surface groups are residually finite 2-groups.

If p is odd and the Euler characteristic of the underlying surface of G is even then g is even as is g_{k}. By 3.2, G_{k} is a residually finite p-group and so, using 3.1 we see that G is a residually finite p-group. Therefore the proof of 5.5 will complete the proof of 5.3. Proof of 5.5. We loosely follow [H; p. 119–120]. Given a capped surface Y as defined above we define the complexity of Y, c(Y), to be the maximum of the orders of the cone points. We begin with the capped surface of G with complexity (p^{n_{1}}, …, p^{n_{m}}). If m = 0 then G is not the Klein bottle and so G = G_{k} satisfies the conclusion of 5.5. Otherwise, given such a Y, we shall find a regular covering space Ỹ such that Ỹ is a capped surface whose cone points are each of order a power of p, c(Ỹ) < c(Y), G/π_{1}(Ỹ) is a finite abelian p-group and, if p is odd, has signature not equal to (−1, 2) and g̃ ≡ g mod 2. This then defines G_{2} = π_{1}(Ỹ) and is the inductive step.

Clearly any p^{r}-fold regular cover of Y will be a capped surface Ỹ whose underlying punctured surface X̃_{0} is a p^{r}-fold cover of X_{0}. Hence if p is odd, Ỹ will be orientable if and only if Y is, and g̃ ≡ g mod 2. Finally, we claim that G_{2} = π_{1}(Ỹ) is not the group of the Klein bottle. For if so then μ(Y) = 2 − g − ∑_{i=1}^{m}(1 − 1/p^{n_{i}}) would have to be zero since it is well known that this “orbifold Euler characteristic” multiplies under covers [ZVC]. But it is easy to check that this happens only for m = 0, g = 2 or p = 2, m = 2 which are excluded since p is odd and G is not the group of the Klein bottle.

Moreover a “cone point” or “cap” of Y of order p^{n_{i}} will lift to q_{i} disjoint caps of order p^{n_{i}}/q_{i} where q_{i} divides p^{r}. Therefore it suffices to find an epimorphism ϕ from G onto a finite abelian p-group such that, for each i where p^{n_{i}} = c(Y), ϕ(c_{i}) ≠ 0 since the cover corresponding to such a ϕ will reduce complexity.

If there are k caps of maximal order p^{r} where k ≥ 2, say corresponding to c_{1}, …, c_{k}, then define ϕ : G ⟶ ×_{i=1}^{k−1}Z_{p^{r}} by ϕ(c_{1}) = (1, 0, …), ϕ(c_{2}) = (0, 1, …), ϕ(c_{k−1}) = (0, …, 1), ϕ(c_{k}) = (−1, −1, …, −1), ϕ(c_{i}) = 0 if i > k, ϕ(a_{i}) = ϕ(b_{i}) = 0. In both orientable and non-orientable cases this ϕ suffices.

Now suppose G is orientable and there is only one cap of maximal order p^{n_{1}}. If g > 0 we can do a preliminary p-fold cover of Y defined by ϕ(a_{1}) = 1, ϕ(a_{i}) = ϕ(c_{j}) = ϕ(b_{k}) = 0 if i ≠ 1. The resulting Ỹ will have p caps of maximal order and c(Ỹ) = c(Y) and we can then apply the argument above. If g = 0 and m ≤ 2 then G is cyclic of order a power of p and we can take Ỹ to be the universal cover of Y. If g = 0 and m ≥ 3 then map ϕ : G ↠ Z_{p^{n_{2}}} by ϕ(c_{1}) = 1, ϕ(c_{2}) = −1, ϕ(c_{i}) = 0 if i > 2. This completes all cases when X is orientable.

Now suppose X is non-orientable with one cap of maximal order p^{n_{1}}. If p is odd define ϕ : G ↠ Z_{p^{n_{1}}} by ϕ(c_{1}) = 1, ϕ(c_{i}) = 0 if i > 1, ϕ(2a_{1}) = 1 and ϕ(a_{i}) = 0 if i > 1. In case p is 2, we can do a preliminary 2-fold cover ϕ : G ⟶ Z_{2} by ϕ(a_{1}) = 1, ϕ (all others)= 0 which results in Ỹ with 2 caps of maximum order without increasing complexity and reduces to an earlier case.

The last remark of 5.5 may be shown as follows. The group G_{k} may not be normal in G_{k−2} but it is normal in G_{k−1}. Redefine G_{k} to be the intersection of its conjugacy classes in G_{k−2}. This new G_{k} is normal in G_{k−1} and G_{k−2}, and, by the proof of 3.1 is of index p^{n} in G_{k−2} for some n. Now ignore G_{k−1} and consider G_{k} ◁ G_{k−2} ◁ G_{k−3} and proceed as above to shrink G_{k} to get it normal in G_{k−3}. Eventually we shrink G_{k} to G′ which is normal in G_{1} = G and of index a power of p. Finally note that we have replaced the filtration by one where G_{k} is normal in G_{1} but the general remarks above still hold to restrict the topological type of the surface of which G_{k} is the fundamental group. □

Theorem 5.6. Let G be a Fuchsian group whose underlying surface is not S^{2} and either has even Euler characteristic or has non-empty boundary (excludes precisely a connected sum of an odd number of projective planes). Then G has Property T, hence length at most ω. Proof of 5.6. We assume the notation of 5.1, 5.2 and our earlier discussion of capped surfaces. Let A = ⟨c_{1}, …, c_{m} | c_{i}^{n_{i}} = 1⟩ as in 4.1. Let N be the subgroup of G normally generated by {c_{i}}. Let S be π_{1}(X) where X is the surface underlying G. □

Lemma 5.7. There is a monomorphism ∗_{s∈S}A ⟶^{ϕ} G whose image is N. Proof of 5.7. Since X ≠ S^{2}, X ≠ RP^{2}, its universal cover X̃ is a contractible surface with deck translations S and contractible fundamental domain P. We shall assume S has a presentation obtained from 5.1 or 5.2 by setting c_{i} = 0 so that we are implicitly choosing lifts of the generators of S to generators of G. S acts on X̃, tiling X̃ by copies of P. Delete from X̃ the orbit of m disjoint open 2-disks in P so that the result is the regular S-cover of X_{0}. Then in each translate of P, adjoin to the i^{th} boundary circle a set of 2-disks {Ẽ_{i}^{s} | s ∈ S} by maps of degree n_{i}. The resulting space Ỹ has on it a free, properly discontinuous action of S so p : Ỹ ⟶ Ỹ/S ≡ Y is a regular cover. Clearly N = p_{∗}(π_{1}(Ỹ)). Moreover π_{1}(Ỹ) ≅ ∗_{s∈S}A in a natural way. Under this identification, letting ϕ = p_{∗}, we complete the proof of 5.7. □

Consider the exact sequence 1 ⟶ A ⟶^{ϕ} G ⟶^{π} S ⟶ 1 where A ≡ ∗_{s∈S}A. Recall from 4.1 that A_{ω} is normally generated by the set K = {[c_{i}^{k_{ij}}, c_{t}^{k_{tv}}] | j ≠ v, k_{ij} = n_{i}/p_{j}^{r_{ij}}, k_{tv} = n_{t}/p_{v}^{r_{tv}}} where n_{i} = ∏_{j=1}^{ℓ} p_{j}^{r_{ij}}, 1 ≤ i ≤ m. Let K_{s} be the corresponding set in the s^{th}-copy of A in A and let K^{∗} be the normal closure of ∪K_{s} in A. Clearly A/K^{∗} ≅^{f} ∗_{s∈S}(B_{p_{1}} × … × B_{p_{ℓ}}) ≡ B where B_{p_{j}} is as in 4.1. Now we shall force all copies of B_{p_{j}} to commute with copies of B_{p_{v}}, j ≠ v, if we kill all elements of the form [b_{p_{j}}^{s_{1}}, b_{p_{v}}^{s_{2}}] where j ≠ v, s_{1} ≠ s_{2} and b_{p_{j}}^{s_{1}} denotes a generator of one of the free factors of the s_{1} copy of B_{p_{j}} in B. Recall from 4.1 that these factors are generated by {f((c_{i}^{k_{ij}})^{s_{1}}) | 1 ≤ i ≤ m, k_{ij} = n_{i}/p_{j}^{r_{ij}}}. Therefore let L = {[(c_{i}^{k_{ij}})^{s_{1}}, (c_{t}^{k_{tv}})^{s_{2}}] | j ≠ v, s_{1} ≠ s_{2}} and L^{∗} be the normal closure of

L in A. Hence A/L^{∗}K^{∗} ≅^{f} B/f(L^{∗}) ≅ ×_{j=1}^{ℓ}M_{j} where M_{j} ≡ ∗_{s∈S}B_{p_{j}}. Also observe that

ϕ(∪K_{s} ∪ L) ⊆ G is precisely the set of elements of the form [s_{1}c_{i}^{k_{ij}}s_{1}^{−1}, s_{2}c_{t}^{k_{tv}}s_{2}^{−1}] where j ≠ v, k_{ij} = n_{i}/p_{j}^{r_{ij}}, k_{tv} = n_{t}/p_{v}^{r_{tv}}. Let \overline{N} = ϕ(L^{∗}K^{∗}). Since L^{∗}K^{∗} is normal in A and since ∪_{s}K_{s} ∪ L is “S-invariant”, \overline{N} is normal in G.

Since the orders of c_{i}^{k_{ij}} and c_{t}^{k_{tv}} are p_{j}^{r_{ij}} and p_{v}^{r_{tv}} respectively, and since j ≠ v, \overline{N} ⊂ T (G). Let \overline{G} = G/\overline{N}. It suffices to show \overline{G}_{ω} ≅ \{e\} since this will imply that G_{ω} ≅ T (G) and hence that G has Property T.

Note that we have an exact sequence 1 ⟶ ⨁_{i=1}^{ℓ} M_{i} ⟶^{ϕ} \overline{G} ⟶^{π} S ⟶ 1. Suppose g ∈ \overline{G}_{ω}. Since S is residually nilpotent by 3.2, g = ϕ(x_{1}, …, x_{ℓ}). It clearly suffices to show each x_{i} is zero. By symmetry it suffices to consider x_{1}. Consider the projection Ψ_{1} : ×_{i=1}^{ℓ}M_{i} ⟶ M_{1}, and the commutative diagram below:

\begin{matrix}1 ⟶ ×_{i=1}^{ℓ} & M_{i} ⟶^{ϕ} \overline{G} ⟶^{π} S ⟶ 1 \\ Ψ_{1} & ↓  ↓Ψ  ∥ \\ 1 ⟶ & M_{1} ⟶ \overline{G}_{1} ⟶ S ⟶ 1. \end{matrix}

One verifies that ϕ(kernel Ψ_{1}) is normal in \overline{G} since M_{1} is an “S-invariant” subgroup. It will then suffice to show (\overline{G}_{1})_{ω} = \{e\} since then Ψ(g) = 0 implies Ψ_{1}(x_{1}, …, x_{ℓ}) = x_{1} = 0. For this purpose it is necessary to identify \overline{G}_{1}.

We claim that \overline{G}_{1} ≅ \overline{G}/ ⟨S⟩ where S = \{c_{i}^{p_{1}^{r_{i1}}} | i = 1, …, m\}. The kernel of Ψ_{1} is normally generated in ×_{i=1}^{ℓ}M_{j} by f((c_{i}^{k_{ij}})^{s}) where s ∈ S and j ≠ 1. Therefore the kernel of Ψ in \overline{G} is normally generated by \{c_{i}^{k_{ij}} | 1 ≤ i ≤ m, j ≠ 1\}. Since k_{ij} = n_{i}/p_{j}^{r_{ij}} and j ≠ 1, p_{1}^{r_{i1}} divides the exponent of c_{i}^{k_{ij}}. Thus kernel Ψ_{1} ⊂ ⟨S⟩. On the other hand, for fixed i, gcd\{k_{ij} | j ≠ 1\} = p_{1}^{r_{i1}} so S ⊂ kernel Ψ_{1}. We also can see that G/ ⟨S⟩ ≅ \overline{G}/ ⟨S⟩ so we get a presentation for \overline{G}_{1} by adding S to the relations of 5.1 or 5.2. We then see immediately that \overline{G}_{1} is a Fuchsian group with underlying surface group S with cone points each of order a power of p_{1}. By 5.3, \overline{G}_{1} is residually nilpotent. This completes the proof of 5.6. □

We can handle the genus zero case but unfortunately it seems to require special techniques and our conclusion is weaker. We are only able to show that the length of such a group is at most ω + 1. Yet we do not know an example of such a group with length ω + 1. Theorem 5.8. Let G be a Fuchsian group with genus zero. Then G_{ω} ⊆ T (G)N where N ◁ G and [N, G] ⊂ T (G). Corollary 5.9. The length of a Fuchsian group of genus zero is at most ω + 1. Proof of 5.9. G_{ω+1} ⊂ [T (G)N, G] ⊂ T (G) ⊂ G_{δ}. Thus G_{ω+1} = G_{δ}. Proof of 5.8. G has a presentation ⟨c_{i} 1 ≤ i ≤ m | c_{i}^{n_{i}} = 1, ∏_{i=1}^{m} c_{i} = 1⟩.

Our first step will be to prove that we can assume each n_{i} is a prime power (the primes will vary). Suppose n_{1} = p_{1}^{r_{1}} … p_{ℓ}^{r_{ℓ}} where the p_{j} are distinct primes. Consider the Fuchsian group G̃ = ⟨b_{1}, …, b_{ℓ}, c_{1}, …, c_{m} | b_{j}^{p_{j}^{r_{j}}} = 1, c_{i}^{n_{i}} = 1, ∏_{j=1}^{ℓ} b_{j} ∏_{i=2}^{m} c_{i} = 1⟩. We shall define ϕ : G̃ ⟶ G and show that ϕ : G̃/T (G̃) ⟶ G/T (G) is an isomorphism. Then it is easy to verify that G satisfies the conclusion of 5.8 if and only if G̃ does. Since G̃ has at most (m − 1) cone points of non-prime-power order, we can continue this reduction process until we can assume all orders are prime powers. It remains to define ϕ. Since gcd{n_{1}/p_{j}^{r_{j}} | 1 ≤ j ≤ ℓ} = 1, we may choose integers x_{j} such that ∑_{j=1}^{ℓ} x_{j}(n_{1}/p_{j}^{r_{j}}) = 1. Let y_{j} = x_{j}(n_{1}/p_{j}^{r_{j}}) and note that (y_{j}, p_{j}) = 1 and that y_{j} is a multiple of p_{i}^{r_{i}} if i ≠ j. Define ϕ(b_{j}) = c_{1}^{y_{j}} and ϕ(c_{i}) = c_{i} for i ≥ 2. An inverse Ψ to the induced map ϕ : G̃/T (G̃) ⟶ G/T (G) is constructed by setting Ψ(c_{1}) = b_{1} … b_{ℓ} and Ψ(c_{i}) = c_{i} if i > 1. To see that Ψ is well-defined, note that [b_{i}, b_{j}] ∈ T (G̃) if i ≠ j so Ψ(c_{1}^{n_{1}}) = (b_{1} … b_{ℓ})^{n_{1}} ≡ b_{1}^{n_{1}} … b_{ℓ}^{n_{1}} ≡ 0 in G̃/T (G̃). Furthermore Ψ ∘ ϕ(b_{j}) ≡ b_{1}^{y_{j}} … b_{ℓ}^{y_{j}} ≡ b_{j}^{y_{j}} in G̃/T (G̃). Since y_{j} ≡ 1 modulo p_{j}^{r_{j}}, it follows, with a little checking, that Ψ is ϕ^{−1}. This completes the first step of the proof.

We now assume each n_{i} = p_{i}^{r_{i}} for some prime from {p_{1}, …, p_{ℓ}}. Re-order and re-index the c’s so that C_{j} = {c_{j1}, …, c_{jm_{j}}} is the set of those c’s whose orders are powers of the prime p_{j}. Note that commutators of elements from distinct C_{j} are T -elements. Let N be the normal closure of the set S = {δ_{1}, …, δ_{ℓ}} where δ_{j} = ∏_{k=1}^{m_{j}} c_{jk}, the product of the elements of C_{j}. We claim that [N, G] ⊆ T (G). For if c_{k} ∉ C_{j} then [δ_{j}, c_{k}] ⊂ T (G) since the order of c_{k} is relatively prime to p_{j}; and if c_{k} ∈ C_{j} then, using the relation ∏_{i=1}^{m} c_{i} = 1, we see that δ_{j} = ∏_{i≠j} δ_{i} modulo T (G), implying [δ_{j}, c_{k}] ≡ [∏_{i≠j} δ_{i}, c_{k}] ≡ 0 in G/T (G) since δ_{i} is a product of c’s which do not lie in C_{j}. It remains only to show G/T (G)N is residually nilpotent. Merely note that G/N is a free product of Fuchsian groups F_{1} ∗ F_{2} ∗ ⋯ ∗ F_{ℓ}, each of genus zero, such that F_{j} has cone points of order various powers of p_{j}. Therefore the map G/N ⟶ G/T (G)N factors through ×_{j=1}^{ℓ}F_{j}. By 5.3 and 2.2c the latter group is residually nilpotent, so the map G ⟶ G/T (G)N factors through G/G_{ω}. It follows immediately that G_{ω} ⊂ T (G)N as desired. □ Theorem 5.10. Let G be a non-trivial Fuchsian group whose underlying surface is X. If π_{1}(X) is not abelian then length(G) ≥ ω. If the first betti number of G is at least 3 then length(G) ≥ ω. More precisely, if X ≠ S^{2}, ≠ RP^{2} then either G G ≅ Z, G ≅ Z × Z or length(G) ≥ ω. Therefore if X is neither S^{2} nor a connected sum of an odd number of projective planes then either the length of G is precisely ω or G ≅ Z or G ≅ Z × Z or G ≅ ∗Z_{n_{i}} where (n_{i}, n_{j}) = 1, in which cases the length of G is 2.

Proof of 5.10. G maps onto π_{1}(X) so the first claim of 5.10 follows from 2.2a and 3.3. If β^{1}(G) ≥ 3 then π_{1}(X) cannot be abelian so the second claim of 5.10 follows. Now suppose X ≠ S^{2} and X ≠ RP^{2} and π_{1}(X) is abelian. Then X is either an annulus, a Moebius band, a 2-disk or S^{1} × S^{1}. If X is an annulus or a Moebius band then G maps onto some Z ∗ Z_{p} unless m = 0 in which case G ≅ Z. If X = D^{2} then G maps onto some Z_{p} ∗ Z_{p} unless G = ∗Z_{n_{i}} where (n_{i}, n_{j}) = 1. If X = S^{1} × S^{1} then either G = Z × Z or G maps onto the group ⟨a, b | [a, b]^{p} = 0⟩ for some prime p, which can be shown to have infinite length. For the last claim of 5.10, apply the above and 5.6. □

§6. Seifert Fibered 3-Manifold Groups.

In this section we show that the fundamental group of a compact Seifert fibered 3manifold with a base surface of even Euler characteristic which is not S^{2} (or with nonempty boundary) has length at most ω. If the base surface is S^{2} we can only show the length is at most ω + 1. In the case that base surface is closed and of odd Euler characteristic we can say nothing only because of our ignorance concerning those surface groups! We would expect that all Seifert fibered 3-manifold groups have length at most ω. Conversely, at the end of this section we observe that the length of most Seifert fibered groups is at least ω, in particular if the first betti number is at least 4 then this is true.

Theorem 6.1. Let G be a central extension 1 ⟶ N ⟶^{i} G ⟶^{π} P ⟶ 1 of a group P where P has Property T (see §5). Then G_{ω} = (N ∩ G_{ω})G_{δ}, and G_{ω+1} = G_{δ}. More generally, upon omitting the word “central” above, it is sufficient that T (P) is the normal closure of {[π(α_{i}), π(β_{i})]} where [N, β_{i}] = [α_{i}, N] = {e}.

Proof. We identify N with i(N). Suppose g ∈ G_{ω}, so π(g) ∈ P_{ω} = T (P). It follows that π(g) = ∏_{i=1}^{k} η_{i}[a_{i}, b_{i}]η_{i}^{−1} where a_{i}, b_{i} are of relatively prime finite orders, say p_{i} and q_{i} in P. Choose elements ξ_{i}, α_{i}, β_{i} of G in the inverse image of η_{i}, a_{i}, b_{i} respectively. Let h = ∏_{i=1}^{k} ξ_{i}[α_{i}, β_{i}]ξ_{i}^{−1} so g = nh for some n ∈ N. It will suffice to show that, for any ordinal γ, [α_{i}, β_{i}] ∈ G_{γ}. Since π(α_{i}) is of order p_{i}, α_{i}^{p_{i}} lies in the center of G and similarly for β_{i}^{q_{i}}. Then apply 2.5 and we are done. □

Theorem 6.2. Suppose G is the fundamental group of a compact 3-manifold which has an orientable Seifert fibration (the circle fibers can be continuously oriented) over a base surface X which satisfies one of the following:

i) ∂X ≠ ∅ ii) X is closed and orientable iii) X is closed, non-orientable with even genus

In cases i) and iii) G has length at most ω. In case ii) G has length at most ω + 1, and, if X ≠ S^{2}, G_{ω}/G_{ω+1} is a (possibly trivial) cyclic group generated by a power of the regular fiber. Proof of 6.2. Let N be the subgroup generated by a regular fiber t. It is well known that G has a presentation of one of the following forms [H]:

6.3  G = ⟨t, a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{m}, d_{1}, …, d_{k} | c_{i}^{n_{i}} = t^{s_{i}}, [c_{i}, t]

= [a_{i}, t] = [b_{i}, t] = [d_{i}, t] = 1, t^{b} ∏_{j=1}^{m} c_{j} ∏_{ℓ=1}^{k} d_{ℓ} = ∏_{i=1}^{g}[a_{i}, b_{i}]⟩

6.4  G = ⟨t, a_{1}, …, a_{g}, c_{1}, …, c_{m}, d_{1}, …, d_{k} | c_{i}^{n_{i}} = t^{s_{i}}, [c_{i}, t]

= [a_{i}, t] = [d_{i}, t] = 1, t^{b} ∏_{j=1}^{m} c_{j} ∏_{ℓ=1}^{k} d_{ℓ} = ∏_{i=1}^{g} a_{i}^{2}⟩

where n_{i} ≥ 2 and (n_{i}, s_{i}) = 1.

Thus there is an exact sequence 1 ⟶ N ⟶ G ⟶^{π} P ⟶ 1 where P is a Fuchsian group with underlying surface X, satisfying the hypotheses of 5.6 except in the case that X is S^{2}. The latter case will be handled separately. Therefore P has Property T and 6.1 applies to G to show that G_{ω} = ⟨t^{N}⟩ G_{ω+1} for some integer N. In cases i) and iii) it is easy to see from 6.3 or 6.4 that t represents an element of infinite order in the abelianization so t^{N} ∈ G_{ω} implies N = 0 and consequently G_{ω} ≅ G_{ω+1}.

Now assume X = S^{2}. Then we have the exact sequence 1 ⟶ N ⟶ G ⟶^{π} P ⟶ 1 where 5.8 applies to P to say that P_{ω} ⊂ T (P)M where M ◁ P and [M, P] ⊂ T (P).

It suffices to show that, if g ∈ G_{ω}, then the element [c_{1}, g] lies in G_{δ} (since G_{ω+1} is normally generated by {[c_{i}^{±1}, g], [t_{i}^{±1}, g] | g ∈ G_{ω}}). We proceed by induction. Recall from 5.8 that M ◁ P is the normal closure of {δ_{1}, …, δ_{k}} where δ_{j} = ∏_{j=1}^{m} c^{n_{i}x_{ij}/p_{j}^{r_{ij}}} where ∑_{j=1} n_{i}x_{ij}/p_{j}^{r_{ij}} = 1 and n_{i} = ∏_{j=1}^{ℓ} p_{j}^{r_{ij}}. Therefore, proceeding as in the non-zero genus case, we can choose n ∈ N, m̃ ∈ G such that g = nhm̃ where h ∈ G_{δ} and π(m̃) ∈ M. Furthermore we can choose m̃ to be in the normal closure in G of {δ_{1}, …, δ_{k}} viewed as elements of G. Therefore it will suffice to show [c_{1}, δ_{j}] ∈ G_{δ} for fixed j. Assume [c_{1}, δ_{j}] ∈ G_{β} for all j. The proof will be complete if we show [c_{1}, δ_{j}] ∈ G_{β+1} for any fixed j.

Lemma 6.5. [c_{1}^{q}, δ_{j}] ∈ G_{δ} whenever q is an integer dividing n_{1} such that (n_{1}/q, p_{j}) = 1.

Proof of 6.5. Consider [c_{1}^{q}, c_{i}^{n_{i}x_{ij}/p_{j}^{r_{ij}}} ]. Since c_{1}^{q} is of order n_{1}/q in G modulo its center, and similarly c_{i}^{n_{i}x_{ij}/p_{i}^{r_{ij}}} is of order p_{j}^{r_{ij}}, the commutator lies in G_{δ} by 2.5. It follows that [c_{1}^{q}, δ_{j}] lies in G_{δ} since δ_{j} is a product of such powers of c_{i}’s (use 2.16). □

It follows from 6.5 (by taking q = p_{j}^{r_{1j}}) that [c_{1}, δ_{j}]^{p_{j}^{r_{1j}}} ∈ G_{β+1}. To conclude that [c_{1}, δ_{j}] ∈ G_{β+1} it will now suffice to show [c_{1}, δ_{j}]^{q} ∈ G_{β+1} where q = n_{1}/p_{j}^{r_{1j}} since (q, p_{j}^{r_{1j}}) = 1.

Lemma 6.6. ∏_{j=1}^{ℓ} δ_{j} ∈ G_{δ}N.

Proof of 6.6. Recall that ∏_{j=1}^{ℓ} δ_{j} = ∏_{j=1}^{ℓ} ∏_{i=1}^{m} c_{i}^{n_{i}x_{ij}/p_{j}^{r_{ij}}}. The factors of δ_{j} commute in G/G_{δ} with the factors of δ_{k} if j ≠ k, since the commutator of two such factors is of the form [a, b] where a^{p_{j}^{n}} ∈ N, b^{p_{k}^{m}} ∈ N, so [a, b] ∈ G_{δ} by 2.5. Therefore ∏_{j=1}^{ℓ} δ_{j} ≡ ∏_{i=1}^{m} ∏_{j=1}^{ℓ} c_{i}^{n_{i}x_{ij}/p_{j}^{r_{ij}}} in G/G_{δ}. Since ∑_{j=1}^{ℓ} n_{i}x_{ij}/p_{j}^{r_{ij}} = 1, ∏_{j=1}^{ℓ} δ_{j} ≡ c_{1} … c_{m} ≡ t^{b} in G/G_{δ}, completing the proof. □

Using 6.6, we see that [c_{1}, δ_{j}]^{q} ≡ [c_{1}, δ_{1} … δ̂_{j} … δ_{ℓ}]^{q} modulo G_{β+1}. Since we assumed [c_{1}, δ_{k}] ∈ G_{β} for all k, the latter term is congruent to (∏_{k≠j}[c_{1}, δ_{k}])^{q} and hence ∏_{k≠j}[c_{1}^{q}, δ_{k}] modulo G_{β+1} (using 2.1). Since (n_{1}/q, p_{k}) = (p_{j}^{r_{1j}}, p_{k}) = 1, we can apply 6.5 to conclude that [c_{1}^{q}, δ_{k}] ∈ G_{δ} and hence that [c_{1}, δ_{j}]^{q} ∈ G_{β+1}, thus finishing the proof that G_{ω+1} = G_{δ} in the genus zero, closed, case of Theorem 6.2. This completes the proof of 6.2. □

In general the fundamental group of a compact Seifert fibered 3-manifold has one of the following forms.

⟨t, a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{m}, d_{1}, …, d_{k} | c_{i}^{n_{i}} = t^{s_{i}}, [c_{i}, t] = 1, (6.7)

= a_{i}ta_{i}^{−1} = t^{ϵ_{i}}, b_{i}tb_{i}^{−1} = t^{n_{i}}, d_{i}td_{i}^{−1} = t^{θ_{i}}, t^{e}Πc_{j}Πd_{ℓ} = Π[a_{i}, b_{i}]⟩

⟨t, a_{1}, …, a_{g}, c_{1}, …, c_{m}, d_{1}, …, d_{k} | c_{i}^{n_{i}} = t^{s_{i}}, [c_{i}, t] = 1, (6.8)

= a_{i}ta_{i}^{−1} = t^{ϵ_{i}}, d_{i}td_{i}^{−1} = t^{θ_{i}}, t^{e}Πc_{j}Πd_{ℓ} = Πa_{i}^{2}⟩

Theorem 6.9. Suppose G is the fundamental group of a compact 3-manifold which admits a Seifert fibration whose base surface is not S^{2} or a connected sum of an odd number of RP (2)’s. Then no non-zero power of the class represented by the regular fiber lies in G_{ω}. (The result holds except when the base is S^{2} or RP (2) but we do not need this stronger fact.) Proof of 6.9. The proof breaks into several cases but in each case we shall construct a homomorphism ϕ : G ⟶ H where H is residually nilpotent (H_{ω} = {e}) and ϕ(t) is of infinite order. Clearly this is sufficient. Throughout, with regard to the presentations 6.7, 6.8, let x = ∏_{i=1}^{m} n_{i}, let β_{i} = s_{i} (∏_{j≠i} n_{j}), and let ϵ = ex + ∑_{i=1}^{m} β_{i}.

First we consider the cases in which there is an orientable Seifert fibering (i.e., all ϵ_{i}, η_{i}, θ_{i} are +1). If the base surface has non-empty boundary or is non-orientable then the regular fiber t can easily be seen to be of infinite order in the abelianization (as was noted in the proof of 6.2). Therefore we need only consider the case that the base surface is orientable with genus at least one and G has a presentation as in 6.3. Let H = ⟨a, b, t | [a, t] = [b, t] = e, t^{ϵ} = [a, b]⟩ and let ϕ(a_{1}) = a, ϕ(b_{1}) = b, ϕ(a_{i}) = ϕ(b_{i}) = 0 for i > 1, ϕ(t) = t^{x}, ϕ(c_{i}) = t^{β_{i}}. Note that H is one of the “Heisenberg groups” of 2.14 and is nilpotent. Since H is the fundamental group of an orientable circle bundle over the torus, it is torsion-free.

Now consider the case of a non-orientable Seifert fibration over a closed non-orientable base surface of genus at least two (since RP (2) is excluded). Here we have a presentation as in 6.8 with k = 0. Without loss of generality we may assume ϵ_{i} = −1 for 1 ≤ i ≤ r < g and ϵ_{i} = ϵ_{g} for r < i ≤ g. Let H = ⟨a, b, t | ata^{−1} = t^{−1}, btb^{−1} = t^{±1}, t^{ϵ} = a^{2}b^{2}⟩ where we use the + sign if the number of ϵ_{i} equal to −1 is odd. Let ϕ(t) = t^{x}, ϕ(c_{i}) = t^{β_{i}} and ϕ(a_{1}) = a. If the number of ϵ_{i} equal to −1 is odd then let ϕ(a_{2}) = a, ϕ(a_{3}) = a^{−1}, ϕ(a_{4}) = a, ϕ(a_{5}) = a^{−1}, et cetera, ending with ϕ(a_{r}) = a^{−1}, and let ϕ(a_{g}) = b and ϕ(a_{i}) = 0 for r < i < g. Otherwise let ϕ(a_{2}) = b, ϕ(a_{3}) = b^{−1}, ϕ(a_{4}) = b, …, et cetera and ϕ(a_{i}) = 0 if ϵ_{i} = +1. We shall argue that H is residually a finite 2-group, hence residually nilpotent. Let H̃ be the kernel of the epimorphism Ψ : H ⟶ Z_{2} × Z_{2} obtained by Ψ(a) = (1, 0), Ψ(b) = (0, 1) and Ψ(t) = 0. Since there is an exact sequence 1 ⟶ ⟨t⟩ ⟶ H ⟶^{π} K ⟶ 1 where K = ⟨a, b | a^{2}b^{2} = 1⟩ is the group of the Klein Bottle, there is an exact sequence 1 ⟶ ⟨t⟩ ⟶ H̃ ⟶ Z × Z ⟶ 1 since the kernel of induced map K ⟶ Z_{2} × Z_{2} is the group of a torus. Moreover t is central in H̃ since H̃ is generated by a^{2}, b^{2}, [a, b], ba^{2}b^{−1} and ab^{2}a^{−1}. Therefore H̃ is one of the Heisenberg groups as discussed above. By 3.1 it will suffice to show that the Heisenberg group ⟨a, b, t | [a, t] = [b, t] = 1, t^{e} = [a, b]⟩ is residually a finite 2-group. For this it suffices, for any integer n, to find a quotient which is a finite 2-group in which t^{n} survives. But ⟨a, b, t | [a, t] = [b, t] = a^{2^{n}} = b^{2^{n}} − t^{2^{n}} = 1, t^{e} = [a, b]⟩ is such a group.

Next consider the case of a non-orientable Seifert fibration with a closed orientable base surface (of genus at least 1). Here we have a presentation as in 6.7 with k = 0 and g ≥ 1. We can assume ϵ_{1} = −1. Then let H = ⟨a, b, t | ata^{−1} = t^{−1}, btb^{−1} = t^{η_{1}}, t^{ϵ} = [a, b]⟩ and let ϕ(c_{i}) = t^{β_{i}}, ϕ(t) = t^{x}, ϕ(a_{1}) = a, ϕ(b_{1}) = b and otherwise ϕ(a_{i}) = 0 if ϵ_{i} = +1 or ϕ(a_{i}) = a if ϵ_{i} = −1 and ϕ(b_{i}) = 0 if η_{i} = +1 or ϕ(b_{i}) = a if η_{i} = −1. It is easy to see that t is of infinite order in H by calculating, for example, that t is of infinite order in the abelianization of the kernel of the epimorphism H ⟶ Z_{2} given by sending a and b to 1. We see that H is residually nilpotent as follows. The commutator subgroup H_{2} is normally generated by [a, b], [a, t] and [b, t] and hence normally generated by t^{r}, where r = 1 if e is odd, and r = 2 if e is even. Thus H_{2} is the subgroup generated by t^{r}. It follows by induction that H_{n} is the infinite cyclic subgroup (r2^{n−1})Z ⊆ Z generated by t^{r2^{n−1}}. Hence H_{ω} = {e}.

Now consider the case of a non-orientable Seifert fibration with a non-orientable base surface with non-empty boundary. Let H = ⟨t, a_{1}, …, a_{g}, d_{1}, …, d_{k−1} | a_{i}ta_{i}^{−1} = t^{ϵ_{i}}, d_{j}td_{j}^{−1} = t^{θ_{j}}, [a_{i}, d_{j}] = [a_{i}, a_{j}] = [d_{i}, d_{j}] = 1⟩, and let ϕ(c_{i}) = t^{β_{i}}, ϕ(t) = t^{x}, ϕ(a_{i}) = a_{i}, ϕ(d_{j}) = d_{j} if j ≠ k and ϕ(d_{k}) = (t^{ϵ} ∏_{j≠k} d_{j})^{−1} ∏ a_{i}^{2}. Then H is clearly a semi-direct product of a free abelian group with Z (generated by t) and H_{n} = 2^{n−1}Z ⊆ Z by [L4; 3.1] or by direct calculation. Thus H is residually nilpotent.

Finally we have the case of a non-orientable Seifert fibration with an orientable base surface with non-empty boundary. Then the procedure just above works (solve for d_{k} and let all a_{i}, b_{i}, d_{i} commute) to yield a map to a semi-direct product as above. Note that g = 0, k = 1 is not possible for a non-orientable Seifert fibration.

This completes the proof of 6.9. □

Theorem 6.10. The fundamental group G of a compact Seifert fibered 3-manifold whose base surface is neither S^{2} nor a connected sum of an odd number of projective planes has length at most ω. If the base surface is S^{2}, G has length at most ω + 1. Proof of 6.10. G is the extension of a Fuchsian group P by the cyclic subgroup N generated by the regular fiber t. From the proof of 5.6 we see that T (P) is the normal closure of a set of elements [π(α_{i}), π(β_{i})] where α_{i} and β_{i} are conjugates of a power of some c_{j}. But any conjugate of c_{j} commutes with t since c_{j} commutes with t. Hence, by 6.1, G_{ω} = (N ∩ G_{ω})G_{δ}. By 6.9 N ∩ G_{ω} = {e} so G_{ω} = G_{δ} and G has length at most ω.

If the base surface is S^{2} then the Seifert fibration must have orientable fibers and hence is covered by 6.2. □

On the other hand, almost all Seifert fibered 3-manifold groups have length at least ω. Proposition 6.11. If π_{1}(X) is not abelian then G, the group of a Seifert fibered 3manifold, has length at least ω. Proof. There is an epimorphism G ⟶ π_{1}(X) so by 2.2a and 3.3, length(G) ≥ ω. Proposition 6.12. If M is Seifert fibered and rank (H_{1}(M)) ≥ 4 then length (π_{1}(M)) ≥ ω. Proof. From 6.3 and 6.4 we see that rank (H_{1}(M)) ≤ 2g + k + 1 if X is orientable and g + k + 1 if X is non-orientable. By 6.11 we may assume X is RP (2), a Moebius band, an annulus, a torus, D^{2} or S^{2}. But in these cases rank (H_{1}(M)) ≤ 3. □ Theorem 6.13. If G is the fundamental group of a compact Seifert fibered 3-manifold with base surface X such that π_{1}(X) is not abelian and X is not a connected sum of an odd number of projective planes, then G has length ω.

§7. 3-Manifold Groups Have the Same Lower Central Series as Hyperbolic 3-Manifold Groups

In this section we will show that, for any closed, oriented 3-manifold there is a hyperbolic 3-manifold (in particular a prime 3-manifold with torsion-free fundamental group) whose groups have the same length and the same lower central series quotients. This is very similar in spirit to earlier work of Robert Myers and Chuck Livingston on homology cobordisms between 3-manifolds [My2] [Li1]. In our situation, however, an homology cobordism to a hyperbolic 3-manifold is not sufficient since such only preserves the isomorphism type of G/G_{α} for finite values of α, and therefore does not preserve length (see 9.1). The stronger result needed here is implied by a theorem of D. Ruberman [R; Theorem 2.6 and Corollary 2.7]. Recent work of Akio Kawauchi on “almost identical imitations” is also closely related [K1, K2]. Ruberman’s result proves even more than we need so we include our own very similar (but shorter) proof.

Theorem 7.1 (D. Ruberman [R; Theorem 2.6]). Let M be a closed, connected, oriented 3manifold. There exists a closed, connected, oriented hyperbolic 3-manifold X and a degree one map f: X ⟶ M which induces isomorphisms on all integral homology groups.

Corollary 7.2. Suppose M is a closed, connected, oriented 3-manifold with π_{1}(M) ≅ G. Then there is a closed, connected, oriented hyperbolic 3-manifold X with π_{1}(X) ≅ P and an epimorphism f_{∗}: P ⟶ G which induces an isomorphism f_{α}: P/P_{α} ⟶ G/G_{α} for each ordinal α. The length of P is equal to the length of G.

Proof of 7.2. Using 7.1, the degree one map induces an epimorphism f_{∗}: P ⟶ G which satisfies the hypotheses of Stalling’s theorem (1.1). It follows that each f_{α} is an isomorphism. It then follows easily from this that the lengths are equal. □

Proof of 7.1. Let T be a “standard trivial” embedding of the wedge W of k circles into S^{3}. A tame embedding g : W ⟶ S^{3} is called characteristic if there is an orientation-preserving homeomorphism ϕ : S^{3} ⟶ S^{3} such that ϕ∘g = T and ϕ(S^{3}−image g) = S^{3}−image T. In other words there is a degree one map rel boundary from E(g) (the exterior of image(g)) to E(T). It is well known that g is characteristic if (and only if) g defines a boundary tangle, one such that the k component “knots” g(S^{1}) bound pairwise disjoint (except at wedge point) embedded, oriented surfaces in S^{3} [CS, Mil]. The theorem will follow from the existence of characteristic, simple, Haken tangles. By this we mean a characteristic embedding g whose exterior N = E(g) is irreducible, with incompressible boundary, such that every properly embedded incompressible torus or annulus is boundary parallel.

Lemma 7.3. For any k ≥ 1 there exists a characteristic tangle g whose exterior N is a simple, Haken 3-manifold (with incompressible boundary).

Assuming 7.3, choose a Heegard splitting of genus k with k ≥ 2 for M. Identifying E(T) with the handlebody of genus k and M with E(T) ∪_{ψ} E(T) for some homeomorphism ψ of ∂E(T), there is a degree one map f : E(g) ∪_{ψ} E(g) ⟶ E(T) ∪_{ψ} E(T) ≡ M which is an isomorphism on homology. Moreover X = E(g) ∪_{ψ} E(g) is a simple Haken manifold [My1; Lemma 3.3] which is hyperbolic by W. Thurston [Mo].

Proof of 7.3. Fix k ≥ 2. Let h : W ⟶ S^{3} be a 2k-string simple Haken tangle (see [My2; Prop. 4.1 and proof of 5.2]). Let F′ denote S^{1} × S^{1} with an open disk deleted. Choose an identification ϕ of (♮_{i=1}^{k}F′) × [0, 1] with a regular neighborhood of image(h). Let F be the complement in F′ of a small open collar of ∂F′, and let F_{i} be a copy of F lying in the i^{th} copy of F in ♮_{i=1}^{k}F′. Let N be the manifold obtained from E(h) by identifying ϕ(F_{i} × {0}) with ϕ(F_{i} × {1}) for i = 1, …, k. Note that N = E(g) where g : ⋁_{i=1}^{k} S^{1} ⟶ S^{3} and g restricted to the i^{th} circle is h restricted to ϕ(∂F_{i} × {1/2}). Since {ϕ(∂F_{1}), ϕ(∂F_{2}), …, ϕ(∂F_{k})} is a boundary link, g is a boundary tangle and hence characteristic. It remains only to show that N is simple, irreducible and has incompressible boundary. This is a standard glueing result [My3; Lemma 2.1]. □

This completes the proof of Theorem 7.1.

§8. 3-Manifold Groups With Length Greater than ω.

Let K be the fundamental group of the Klein bottle. We shall show that, for any odd integer q, C = K ∗ Z_{q} has length at least 2ω. This demonstrates that there are closed, orientable 3-manifolds whose groups have length at least 2ω. For there is a twisted interval bundle over the Klein bottle whose total space is an orientable 3-manifold Y with boundary a torus. Letting X be the “double” of Y along its boundary observe, that Y is a retract of X and hence π_{1}(Y) ≅ K is a retract of π_{1}(X). Therefore C is a retract of π_{1}(X) ∗ Z_{q} and so the length of π_{1}(X) ∗ Z_{q} is at least 2ω by 2.2 and 8.1. But this group is the fundamental group of the closed, orientable 3-manifold X#L(q, 1). Theorem 8.1. Let K = ⟨u, t | utu^{−1} = t^{−1}⟩ be the Klein bottle group and q be an odd positive integer. Then C = K ∗ Z_{q} has length at least 2ω. If K′ has K as retract then K′ ∗ Z_{q} has length at least 2ω. Corollary 8.2. There exist closed, hyperbolic 3-manifold groups of length at least 2ω. Proof of 8.2. The argument in the first paragraph above shows that the group of X#L(q, 1) has length at least 2ω. Then apply Corollary 7.2. Proof of 8.1. We can assume q is prime since if q = mn then K ∗ Z_{m} is a retract of K ∗ Z_{mn}. The proof is accomplished in the following stages. We consider the presentation C = ⟨u, s, t | utu^{−1} = t^{−1}, s^{q} = 1⟩. Lemma 8.3. C_{ω} is the normal subgroup generated by [s, t].

The following enables us to find a presentation for C/C_{ω+k} for any k ∈ Z_{+}. Lemma 8.4. If A ◁ C is the subgroup normally generated by {a_{α} | α ∈ J} and B ◁ C is the subgroup generated by {b_{β} | β ∈ J′} then [A, B] is the subgroup normally generated by {[a_{α}, b_{β}] | α ∈ J, β ∈ J′}. Lemma 8.5. [s, t] is of order q in H_{2}(C/C_{ω}).

Lemma 8.6. H_{2}(C) ⟶ H_{2}(C/C_{ω+k}) is not surjective.

Lemma 8.6 implies Theorem 8.1 since by Stallings’ theorem the cokernel of π_{∗} : H_{2}(C) ⟶ H_{2}(C/C_{ω+k}) is C_{ω+k}/C_{ω+k+1}. It follows that the length of C is greater than ω + k for all k ∈ Z_{+}, and hence at least 2ω. Lemma 8.5 is not strictly necessary but is much easier to prove than 8.6 and is enough to show that the length of C is greater than ω, since H_{2}(C) ≅ H_{2}(K) ⊕ H_{2}(Z_{q}) ≅ 0 and so H_{2}(C/C_{ω}) ≅ C_{ω}/C_{ω+1} by Stallings’ theorem.

Proof of 8.3. Since [u, t^{k}] = t^{(−2)^{k}}, [u, [u, …, [u, t]] …] = t^{(−2)^{n}} so t^{2^{n}} ∈ C_{n+1} (in fact it lies in K_{n+1}). Thus t is a generalized 2-element because it has order a power of 2 in each C/C_{j}, j ∈ Z_{+}. By 2.4, [s, t] ∈ C_{ω}. Now it suffices to show that B = C/ ⟨[s, t]⟩ is residually nilpotent. The group B has presentation ⟨u, t, s | utu^{−1} = t^{−1} , s^{q}, [s, t]⟩. Note that the subgroup J generated by t is normal and infinite cyclic. Moreover there is a semi-direct product decomposition B = J ⋊(Z∗Z_{q}) where Z∗Z_{q} = ⟨u, s | s^{q}⟩. J is a Z[Z∗Z_{q}] module where u_{∗}(t^{k}) = t^{−k} and s_{∗}(t^{k}) = t^{k}. Levine has shown that, since Z ∗ Z_{q} is residually nilpotent (see 2.7), B_{ω} ≅ ⋂_{j=1}^{∞} I^{j}J where I is the augmentation ideal of Z[Z ∗ Z_{q}] [L4]. Suppose w = w(u, s) is an element of Z ∗ Z_{q}. Since s acts trivially on J, the action w_{∗} is equal to u_{∗}^{m} where m is the exponent sum of u in w. Thus IJ = I′J where I′ is the augmentation ideal of J as a Z[u, u^{−1}] module. Similarly I^{j}J = (I′)^{j}J and the latter is easily seen to be 2^{j}J. Therefore B_{ω} = ⋂_{j=1}^{∞} 2^{j}J = {e} as desired. □

Proof of 8.4. Since [A, B] is normally generated by elements of the form [x, y] where x ∈ A, y ∈ B. We know x = Πη_{i}a_{i}^{±1}η_{i}^{−1}. Using the identities [ab, c] = a[b, c]a^{−1}[a, c] and [a, bc] = [a, b]b[a, c]b^{−1} we see that [x, y] is a product of conjugates of elements of the form [b^{±1}, ηa^{±1}η^{−1}] and their inverses, where b is a generator of B and a is a normal generator of A. Now use the identity [b, ηaη^{−1}] = γ[b, a]γ^{−1}η[[η^{−1}, b], a]η^{−1} where γ = bηb^{−1} and then expand the triple commutator by using [ab, c] = a[b, c]a^{−1}[a, c] and the fact that [η^{−1}, b] is a product of generators of B (or their inverses). This shows that [A, B] is normally generated by elements of the form [a_{α}^{±1}, b_{β}^{±1}]. Since [x^{±1}, y^{−1}] is conjugate to [x^{∓1}, y] and [x^{−1}, y] = yx^{−1}y^{−1}[x, y]^{−1}yxy^{−1}, the result follows. □

Proof of 8.5. Let B = C/C_{ω} as in 8.3 above. Let D be the normal subgroup of B generated by {s, t} so there is a split short exact sequence

1 ⟶ D ⟶^{f} B ↠^{π} Z ⟶ 1

where Z is generated by u. A presentation for D is ⟨s_{i}, t, i ∈ Z | s_{i}^{q} = 1 [s_{i}, t^{ϵ_{i}}] = 1 ϵ_{i} = (−1)^{i}⟩ where f(s_{i}) = u^{i}su^{−i} and f(t) = t. Thus D ≅ Z × (∗_{i=−∞}^{∞}Z_{q}) generated by t and {s_{i}} and H_{2}(D) ≅ ×_{i=−∞}^{∞}Z_{q} generated by {[s_{i}, t^{ϵ_{i}}] | i ∈ Z} which we denote {e_{i} | i ∈ Z}. There is then an exact sequence:

H_{2}(D) ⟶^{u_{∗}−id} H_{2}(D) ⟶^{f_{∗}} H_{2}(B) ⟶^{∂_{∗}} H_{1}(D) ⟶^{u_{∗}−id_{∗}} H_{1}(D) ⟶ .

Since u_{∗}(t) = t^{−1} and u_{∗}(s_{i}) = s_{i+1}, u_{∗}([s_{i}, t^{ϵ_{i}}]) = [s_{i+1}, t^{ϵ_{i+1}}]. Thus u_{∗}(e_{i}) = e_{i+1} and so H_{2}(D)/ ⟨u_{∗} − id⟩ is Z_{q} generated by [s, t] = e_{0}. It follows from the exact sequence that H_{2}(B) ≅ Z_{q} generated by [s, t]. □

Lemma 8.6. H_{2}(C) ⟶ H_{2}(C/C_{ω+k}) is not surjective (so the length of C is at least 2ω). Proof of 8.6. Let D′ be the normal subgroup of B = C/C_{ω+k} generated by {s, t}. From 8.3 and 8.4 we know that C_{ω+k} is the normal closure of {[δ_{1}, [δ_{2}, …, [δ_{k}, [s, t]] …]] | δ_{i} ∈ {u, s, t}}. There is an exact sequence 1 ⟶ D′ ⟶ B ⟶ Z ⟶ 1 and D′ has a presentation ⟨s_{i}, t | s_{i}^{q} = 1, [δ_{1}^{i}, [δ_{2}^{i}, …, [δ_{k}^{i}, [s_{i}, t_{i}]] …]]⟩ where the latter relations are rewritten using ut_{i}ū = t_{i+1}^{−1} and us_{i}u^{−1} = s_{i+1} (so here s_{i} = u^{i}su^{−i}, t_{i} = u^{i}tu^{−i} = t^{(−1)i}). We now analyze H_{2}(D′, Z_{q}). Let F = the free group on {s_{i}, t | i ∈ Z}, and R denote the relation subgroup of D′ so D′ = F/R. It will be more convenient to work with the mod q lower central series [St]. This differs from the ordinary lower central series in that, for non-limit ordinals, G_{α+1,q} ≡ G#G_{α,q} where, in general, for any subgroup U of G, G#U is the subgroup generated by all elements of the form [g, u]v^{p} for g ∈ G u, v ∈ U. Since R ⊆ F_{2,q} there is a natural homomorphism ϕ : R/R#F ⟶ F_{2,q}/F_{3,q}. Note that if x ∈ F_{m}, then [u, x] = (uxū)x̄ lies in F_{m} also. Therefore, the relation [δ_{1}^{i}, … [s_{i}, t_{i}]] lies in F_{3,q} unless each δ_{i} is u. And the relation [u, [u, … [s_{i}, t_{i}]] equals ∏_{i=1}^{k}(u − id)_{∗}([s_{i}, t_{i}]) in the abelian group F_{2,q}/F_{3,q}. It follows that the image of ϕ is generated by {s_{i}^{q}} and {(u − id)_{∗}^{k}([s_{i}, t_{i}]) | i ∈ Z}. We have the long exact sequence with coefficients in Z_{q}:

H_{2}(D′) ⟶^{u_{∗}−id} H_{2}(D′) ⟶^{i_{∗}} H_{2}(B) ⟶ H_{1}(D′) ⟶ H_{1}(D′) ⟶ .

By [St; Thm. 2.1], H_{2}(D′) ≅ R/R#F, so the cokernel of (u_{∗} − id) maps, via ϕ, onto the image of ϕ modulo the image of ϕ ∘ (u_{∗} − id). The image of ϕ ∘ (u_{∗} − id) is the subgroup generated by {s_{i+1}^{q}−s_{i}^{q}, (u−id)_{∗}^{k+1}([s_{i}, t_{i}]) | i ∈ Z}. Since t_{i} = t_{i}^{(−1)^{i}} and [s_{i}, t^{−1}] = −[s_{i}, t] in F_{2,q}/F_{3,q}, all the t_{i}’s in the above may be replaced by t’s. Thus it suffices to show that the subgroup generated by {(u − id)_{∗}^{k}([s_{i}, t])} modulo the subgroup generated by {(u − id)_{∗}^{k+1}([s_{i}, t])} is non-trivial for each fixed k ∈ Z_{+}. It is well known that F_{2,q}/F_{3,q} is a Z_{q}-vector space with basis S = {s_{i}^{q}, t^{q}, [s_{i}, t], [s_{i}, s_{j}] | i > j}. It is an easy exercise that the following is also a basis: (here let α_{i} = [s_{i}, t]) S′ = {s_{i}^{q}, t^{q}, [s_{i}, s_{j}], α_{0}, α_{1} − α_{0}, …, (u − id)_{∗}^{k}α_{0}, (u − id)_{∗}^{k+1}α_{i}}. Since image of ϕ contains (u − id)_{∗}^{k}α_{0}, for example, it is non-trivial even upon quotient by {(u−id)_{∗}^{k+1}α_{i}}. In fact, F_{2,q}/F_{3,q} modulo {(u−id)_{∗}^{k+1}α_{i}, s_{i+1}^{q} −s_{i}^{q}} has basis S′′ = {s_{0}^{q}, t^{q}, α_{0}, α_{1} − α_{0}, …, (u − id)_{∗}^{k}α_{0}} ≅ Z_{q}^{k+3}. The elements (u − id)_{∗}^{k}α_{0} and s_{0}^{q} are basis elements. It follows that the image of H_{2}(D′) ⟶^{i_{∗}} H_{2}(B) maps onto Z_{q} × Z_{q} with [u, [u, … [u, [s, t]]] mapping to (1, 0) and s^{q} mapping to (0, 1). Therefore H_{2}(B; Z_{q}) is of Z_{q} rank at least 2. Since H_{2}(C; Z_{q}) ≅ Z_{q}, the map π_{∗} : H_{2}(C) ⟶ H_{2}(B) is not surjective. Since π : C ⟶ B is an isomorphism on H_{1}, by the naturality of the Universal Coefficient Theorem, it follows that H_{2}(C) ⟶ H_{2}(B) is not surjective on integral homology. □

This completes the proof of 8.6 and of Theorem 8.1.

There are other 3-manifolds with very long fundamental groups. Let X_{p} be the oriented 3-manifold obtained by (p, p, 0) Dehn surgery on the Borromean Rings as shown in Figure 8.7

figure 8.7

Theorem 8.8. If (q, p) = 1 then X_{p}#L(q, ℓ) (or connected sum of X_{p} with any manifold with π_{1} equal to Z_{q}) has a fundamental group whose length is at least 2ω.

We shall not prove this theorem since its proof is similar to but much harder than that of 8.1. Letting s generate the Z_{q} as before, first one shows that C = A ∗ Z_{q} = π_{1}(X_{p}) ∗ π_{1}(L(q, 1)) has a presentation

⟨x, y, z, s | [x, y] = 1, x^{p} = [y, z^{−1}], y^{p} = [x, z], z^{q} = 1⟩ .

Then one shows by induction that x and y are generalized p-elements, specifically that x^{p^{k}} and y^{p^{k}} lie in A_{k+1} where A = π_{1}(X_{p}). It follows that [x, s] and [y, s] lie in G_{ω}. One then shows that G/ ⟨[x, s], [y, s]⟩ is ω-nilpotent by observing that the latter group is a semidirect product Z × Z ⋊ (Z ∗ Z_{q}) where Z × Z = ⟨x, y⟩ and calculating that the action is nilpotent. The final (messy) step is to show that the cokernel of H_{2}(G; Z_{q}) ⟶ H_{2}(G/G_{ω+k}; Z_{q}) maps onto Z_{q} × Z_{q} by examining the exact sequences related to the subgroup D (of infinite cyclic index) of G/G_{ω+k} which is normally generated by {x, y, s}. The interested reader may write to the authors for a copy of these calculations.

Finally, replacing X_{p} of 8.8 by (p, 0) surgery on a Whitehead link also works although a complete proof has been written down only for p = 4.

§9. Length of the lower central series and Homology Cobordism.

We shall say that two finitely-presented groups G and G′ are homology cobordant if there exists a third finitely-presented group H and homomorphisms f : G ⟶ H, f′ : G′ ⟶ H which induce isomorphisms on H_{1} and epimorphisms on H_{2}. Then if two compact manifolds are homology cobordant in the usual sense, their fundamental groups will be homology cobordant. If we also insist that the images of f and f′ normally generate H, two groups are homology cobordant if and only if they have isomorphic algebraic closures, in the sense of J. Levine ([L2], see especially Propositions 5 and 6). As mentioned in §1, Stallings’ theorem implies that the isomorphism type of G/G_{n}, n finite, is an invariant of homology cobordism. It follows that the length of G is an invariant of homology cobordism if the length of G is finite. If it were true that length were an invariant of homology cobordism, then all slice link exteriors would have groups of length ω since any slice link exterior is homology cobordant to a trivial link exterior. It would follow that the Parafree Conjecture is true for a large class of groups (see 1.5). However we shall observe below that length is not an invariant of homology cobordism. In fact an example was already found in [L3; §6] although it was not explicitly pointed out. Since the known examples will fail to be “weakly parafree” (F/F_{n} ≅ G/G_{n} for all finite n (see §1)), this leaves open the possibility that length might be invariant under homology cobordism of weakly parafree groups for the simple reason that all such groups may have length at most ω.

Proposition 9.1 (compare [L3; Section 6]). There exist finitely presented groups B and C and a homomorphism f : B ⟶ C whose image normally generates C and which induces an isomorphism on H_{1} and an epimorphism on H_{2} but where length(C) > ω and length(B) = ω.

Proof. Let B = ⟨u, s, t | [s, t] = e, utu^{−1} = t^{−1}⟩, C = ⟨u, s, t | [s, t^{3}] = e, utu^{−1} = t^{−1}⟩ and define f(u) = u, f(s) = s and f(t) = t^{3}. Since H_{1}(B) and H_{1}(C) are Z×Z×Z_{2} where t is an element of order 2, f induces an isomorphism on H_{1}. We claim that H_{2}(C) ≅ Z generated by [s, t^{3}]. To see this let D be the normal subgroup generated by {s, t}. Then D has presentation ⟨s_{i}, t | [s_{i}, t^{3ϵ_{i}}]⟩ where i ∈ Z, s_{i} = u^{i}su^{−i} and ϵ_{i} = (−1)^{i}, so H_{2}(D) is free abelian on the set {α_{i} | i ∈ Z} where α_{i} = [s_{i}, t^{3ϵ_{i}}]. There is an exact sequence as below

⟶ H_{2}(D) ⟶^{u_{∗}−id} H_{2}(D) ⟶ H_{2}(C) ⟶ H_{1}(D) ⟶^{u_{∗}−id} H_{1}(D) ⟶ .

Since u_{∗}(α_{i}) = u[s_{i}, t^{3ϵ_{i}}]u^{−1} = α_{i+1}, the cokernel of u_{∗} − id on H_{2} is Z, generated by α_{0} = [s, t^{3}]. It is easily seen that u_{∗} − id is injective on H_{1}(D) so H_{2}(C) ≅ Z generated by [s, t^{3}]. By an entirely analogous argument, H_{2}(B) ≅ Z generated by [s, t]. Since f([s, t]) = [s, t^{3}], f induces an isomorphism on H_{2}.

But note that B is isomorphic to the quotient of C by the normal subgroup generated by [s, t]. We claim that the kernel of this map π : C ⟶ B is C_{ω}. First, [s, t] ∈ C_{ω} since, assuming [s, t] ∈ C_{n}, we see that [s, t]^{3} ≡ [s, t^{3}] ≡ 0 modulo C_{n+1} and [s, t]^{2^{n}} ≡ [s, t^{2^{n}}] ≡ 0 modulo C_{n+1} since t is a generalized 2-element. Moreover B is residually nilpotent since it has a semidirect product decomposition 1 ⟶ Z ⟶ B ⟶ Z ∗ Z ⟶ 1 where Z is generated by t. It follows that B_{ω} = {e} just as in the proof of 8.3. Therefore the length of B is ω (B is not nilpotent since it maps onto Z ∗ Z).

Finally, by Stallings’ theorem, C_{ω}/C_{ω+1} is isomorphic to the cokernel of H_{2}(C) ⟶^{π_{∗}} H_{2}(B). But π_{∗}([s, t^{3}]) = [s, t^{3}] ≡ [s, t]^{3} in H_{2}(B) so C_{ω}/C_{ω+1} ≅ Z_{3}. Hence the length of C is at least ω + 1. □

It would take more work to find an example which could be realized by an homology cobordism of 3-manifolds but this should not be difficult.

§10. Questions and Problems.

1. Prove that any planar discontinuous group has length at most ω. (see 5.6 and 5.9)

2. Prove that the fundamental group of a compact Seifert fibered 3-manifold has length at most ω. (see 6.10).

3. What lengths are possible for fundamental groups of compact 3-manifolds? Are

there restrictions on G_{α}/G_{α+1}?

4. Does the fundamental group of the exterior of a ribbon link in S^{3} have length ω?

(see 1.7).

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Tim D. Cochran Kent E. Orr Mathematics Department Mathematics Department Rice University P.O.Box 1892 Indiana University Houston, Texas Bloomington, Indiana 77251–1892 47405