Let G be a finite soluble group with Fitting subgroup F(G). The Fitting series of G is defined, as usual, by F0(G) = 1 and Fi(G)/Fi−1(G) = F(G/Fi−1(G)) for i ≥ 1, and the Fitting height h = h(G) of G is the least integer such that Fn(G) = G. Suppose now that a finite soluble group A acts on G. Let k be the composition length of A, that is, the number of prime divisors (counting multiplicities) of |A|. There is a certain amount of evidence in favour of the

CONJECTURE. |G:Fk(G)| is bounded by a number depending only on |A| and |CG(A)|.

This paper is concerned with group presentations satisfying the small cancellation condition T(6). The definition of this condition is given in §1·2, together with some examples. Before giving the definition, however, we describe (in §1·1) some material which, to a certain extent, motivated our paper. In § 1·3 we state our main theorem, which provides new solutions to the word and conjugacy problems for T(6)-groups.

The Adams operations {ψk} considered as self-maps of BU have the property that . Id2n on H2n(BU; ℚ). Sullivan in his M.I.T. notes [12] constructed ψk-type self-maps of BU(n) in the case in which k is prime to n!. Later work by Friedlander[5] and Wilkerson[14] gave constructions for all other compact connected Lie groups G, subject to the condition that k be prime to the order of the Weyl group W(G).

Suppose a group H is given as a free product with amalgamation

determined by groups A0, A1, A2 and homomorphisms α1: A0 → A1, α2: A0 → A2. Thus H may be described as the quotient of the free product A * A2 by the relations i1 α1 (α0) = i2α2 (α0) for all α0 ∈ A0, where i1, i2 are the two injections of A1, A2 into A1 * A2. We do not assume that α1, α2 are injective, so the canonical homomorphisms α′i: Ai → H, i = 0,1,2, also need not be injective.

Let C be a complex algebraic curve in the projective space ℙ2. The purpose of this paper is to calculate the fundamental group G of the complement of C in the case when C = X ∩ H1 ∩ … ∩ Hn−2, where

and Hi are generic hyperplanes (i = 1, … n − 2).

The fundamental group G of a μ-component link has the properties:

(1) G is finitely presented, with deficiency 1;

(2) G/G′ is free abelian on μ distinguished generators, say {t1 …, tμ}.

Let ψ: G → G/G′ → 〈t〉 be the composition of the canonical projection G → G/G′ and the epimorphism defined by . Then the Z〈t〉-module M = Ker ψ/(Ker ψ)′ (the so-called module of the link) has a square (say n × n) relation matrix N. We write [N] for the Z〈t〉-equivalence class of N (Fox [3], p. 199) and N′̅ for the conjugate transpose of N.

The topology most often used on a totally ordered group (G, <) is the interval topology. There are usually many ways to totally order G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology.

A link or knot in S3 is universal if it serves as common branching set for all closed, oriented 3-manifolds. A knot is simple if its exterior space is simple, i.e. any incompressible torus or annulus is parallel to the boundary. No iterated torus knot or link is universal, but we know of many knots and links that are universal. Thurston gave the first examples of universal links [8], and subsequently we proved that all 2-bridge knots and links that can be universal (no torus knots or links) are in fact universal [3]. Some other universal knots are described in [1] and [2], together with a general procedure for constructing such knots. For a general reference to knots see [9].

In 1973 Ronnie Lee introduced the notion of semicharacteristic classes, which are invariants of the bordism group ℜ*(Bπ) of closed manifolds equipped with a free action of a finite group π. In this paper we relativize his theory. Associated to a homomorphism G → π of finite groups, there is the relative bordism group ℜ*(BG → Bπ), which is the bordism group of compact manifolds M with a free π-action, so that the action on ∂M is induced from a free G-action, i.e. ∂M = π xGN for some manifold N with a free G-action. The invariants defined here are invariants of this relative group.

In [4], A. Connes has defined the convolution algebra associated to a foliation ℱ of the compact manifold M. Here is the graph or holonomy groupoid of the foliation ℱ (Winkelnkemper [15]). By forming the completion of in its regular representation, he obtains the C*-algebra C*{M, ℱ) associated to the foliation. The completeness of C*(M, ℱ) makes it easier to handle in some analytical contexts, but in others it seems to be too big, and it is necessary to consider instead some carefully selected dense subalgebra (cf. [6]). The purpose of this note is to show that certain spectral functions of leafwise elliptic operators, which might a priori be expected to belong to C*(M, ℱ), in fact belong to the more controllable dense subalgebra . We give a couple of applications, including a proof not passing through C*-algebras of Connes' index theorem for measured foliations [4]. It should be emphasized that the proof of that result offered here is essentially Connes' one, but the presentation may perhaps be more congenial to those who are not C*-algebra specialists.

A fundamental and interesting question in knot theory is:

Question 1. How many prime knots of n crossings are there ?

Over time, knot theorists have answered this question for n ≤ 13 by the method of exhaustion: one writes down a list of all possible knots of n crossings, and then works hard to eliminate duplications from the list [12]. A perhaps easier question is the following:

Let K be a knot in S3, and let K(a/b) denote the closed oriented 3-manifold obtained by a/b-Dehn surgery on K. (We parametrize the Dehn surgeries on K by ℚ ∩ {1/0} as in [9]; in particular, K(1/0) = S3.) If K is a (p, q)-cable knot, then K(pq) is always reducible (see Section 3), and it is conjectured by González-Acuña and Short in [5] that these are the only examples where Dehn surgery on a knot in S3 yields a reducible manifold. One of the results in this direction proved in [5] is that if π1 (K(a/b)) is a non-trivial free product then |b| ≤ 5. We show that this may be sharpened to the assertion in the title.

After the characterization of 1-point topological compactifications by Alexandroff in 1924, n-point topological compactifications by Magill [4] in 1965, and 1-point order compactifications by McCallion [5] in 1971, spaces that admit an n -point order compactification are characterized in Section 2. If X* and X** are finite-point order compactifications of X, sup{X*, X**} is given explicitly in terms of X* and X** in § 3. In § 4 it is shown that if an ordered topological space X has an m-point and an n-point order compactification, then X has a k-point order compactification for each integer k between m and n. The author is indebted to Professor Darrell C. Kent, who provided assistance and encouragement during the preparation of this paper.

Let L be an alternating link and be its reduced (or proper) alternating diagram. Let w() denote the writhe of [3], i.e. the number of positive crossings minus the number of negative crossings. Let VL(t) be the Jones polynomial of L [2]. Let dmaxVL(t) and dminVL(t) denote the maximal and minimal degrees of VL(t), respectively. Furthermore, let σ(L) be the signature of L [5].

Let Mn be a smooth closed n-dimensional manifold and let DIFF (Mn) be the group of diffeomorphisms of Mn. Two diffeomorphisms f0, f1 ∈ DIFF(Mn) are said to be concordant (pseudo-isotopic) if there is a diffeomorphism F ∈ DIFF (Mn × I), where I = [0, 1], such that F(x, 0) = f0(x) and F(x, 1) = f1(x) for all x ∈ Mn.

In the understanding of hyperbolic 3-manifolds, some of the important open questions revolve around the behaviour of a surface upon the passage to a finite covering space; for example one would like to know if it is possible to replace an immersion by an embedding, or a separating surface by a non-separating one.

A (connected) Riemannian manifold M is geodesically complete (i.e. each geodesic may be extended to a geodesic with domain ( −∞, + ∞)) iff, as a metric space under the induced Riemannian distance function, M is Cauchy complete ([12], p. 138). Furthermore, all compact Riemannian manifolds are complete. On the other hand, compact pseudo-Riemannian manifolds exist which are not geodesically complete. For example Fierz and Jost[8] have constructed incomplete metrics of the form 2dx dy + h dy2 on T2. Furthermore, Williams [16] has shown that geodesic completeness may fail to be stable for pseudo-Riemannian manifolds. Here a property is said to be stable if the set of metrics for M with this property is open. This failure of stability may occur for both compact and non-compact manifolds. In particular, Williams has found that M = S1 x S1 = ℝ/(2πZ x 2πZ), which may be given the complete Lorentzian metric g = 2dx dy, also admits Lorentzian metrics gn = 2dx dy+(sin(x)/n) dy2 which are geodesically incomplete yet which for large n are arbitrarily close to g in the Whitney fine Cr topologies. In this example the set S = {(0, y)|0 ≤ y ≤ 2π} represents the image of a closed null geodesic for (M, g) and also for all (M, gn). However, for the metrics (M, gn) this closed null geodesic has affine parametrizations which are incomplete because the velocity vector of this null geodesic returns to a scalar multiple of itself after each complete circuit of S. In general, a closed null geodesic will be complete iff its velocity vector returns exactly to itself after each trip around its image.

In a previous paper [6] the Bergman projection in D = A ∪ B, the union of domains A, B ⊂ CN, was described by alternating projections procedure. It was shown that when N = 1, and A, B are two halfplanes, the procedure can be carried out by explicit analytic calculations. In the present paper we show that the same is true for A = {|Re z| < π} and B = {Re z > 0}. In this case the calculations are more involved, and an essential use is made of L2-theorems of Paley–Wiener type due to T. Genchev, M. Dzrbasjan and W. Martirosjan. Of central importance is an operator W which maps f(z) onto . We show that for every g ∈ L2H(B) the sequence Wng (n = 1, 2, …) converges to zero locally uniformly in B.

A few years ago, the authors considered briefly Banach algebras whose duals could be identified ‘naturally’ with their multiplier algebras [17]. In this context, naturalness can be interpreted as meaning that, for each element b of the algebra B and each pair of elements u, v of the dual B′,

where 〈, 〉 denotes the dual pairing and the products are of elements of B′ regarded as left or right multipliers on B. In the present paper we return to the same circle of ideas but begin with a more general situation. We assume only that the algebra B is injectively embedded in its algebra of left, and in its algebra of right, multipliers and that its dual B′ can be injectively embedded in the algebra M(B) of double multipliers on B (definition below) in such a way that the above relation holds. From these assumptions we shall prove that there is a normed algebra A such that M(B) is the dual of A and is the algebra of continuous left multipliers on A (or, equally, right multipliers).

We give a quick proof of a result of A. Nagel and E. M. Stein concerning the boundedness of certain generalizations of the Hardy-Littlewood maximal operator.