The role played by the classical braid groups in the interplay between geometry, algebra and topology (see [Ca]) derives, in part, from their definition as the fundamental groups of configuration spaces of points in the plane. Seeking to generalize these groups and to understand them better, one is led to ask: are there other discrete groups whose topological invariants arise from configuration spaces?

The groups of marked homeomorphisms (1·1) provide a positive response which is in some sense banal; the realization problem (1·5) is to find non-banal examples.

The Harada-Norton group is one of the twenty-six sporadic simple groups. It has order 273, 030, 912, 000, 000 = 214.36.56.7.11.19. In this paper our main objective is:

Theorem 1. The Harada-Norton group acts as a group of linear automorphisms of a 133-dimensional commutative, non-associative algebra defined over F5.

The Scott Conjecture, proven by Bestvina and Handel [2] says that an automorphism of a free group of rank n has fixed subgroup of rank at most n. We characterise in Theorem A below those automorphisms that realise this maximum. It follows from this characterisation, for example, that any such automorphism has linear growth. In our paper [3], we generalised the Scott Conjecture to arbitrary free products, using Kuros rank (see Section 2 below) in place of free rank; in Theorem B, we characterise those automorphisms of a free product realising the maximum. We show that in this case the growth rate is also linear. These results extend those of [4].

The intuition behind the notion of a cycle-free partial order (CFPO) is that it should be a partial ordering (X, ≤ ) in which for any sequence of points (x0, x1;…, xn–1) with n ≤ 4 such that xi is comparable with xi+1 for each i (indices taken modulo n) there are i and j with j ╪ i, i + 1 such that xj lies between xi and xi+1. As its turn out however this fails to capture the intended class, and a more involved definition, in terms of the ‘Dedekind–MacNeille completion’ of X was given by Warren[5]. An alternative definition involving the idea of a betweenness relation was proposed by P. M. Neumann [1]. It is the purpose of this paper to clarify the connections between these definitions, and indeed between the ideas of semi-linear order (or ‘tree’), CFPO, and the betweenness relations described in [1]. In addition I shall tackle the issue of the axiomatizability of the class of CFPOs.

Topological circle actions on 4-manifolds are studied using modifications of known techniques for smooth actions. This yields topological versions of some previously known restrictions on the fundamental groups of 4-manifolds admitting smooth circle actions.

Let G be a locally compact group and L1(G) be the group algebra of G. We show that G is abelian or compact if every continuous automorphism of L1(G)** maps L1(G) onto L1(G) This characterizes all groups with this property and answers a question raised by F. Ghahramani and A. T. Lau in [7]. We also show that if G is a compact group and θ is any (algebra) isomorphism from L1(G)** onto L1(H)**, then H is compact and θ maps L1(G) onto L1(H).

Let D be a bounded region in ℝm, m ≥ 2. We say that a function u defined in D has asymptotic value α if there is a boundary path Γ:x(t), 0≤t<1, in D (that is, dist (x(t), ∂D)→0 as t→1), such that u(x(t))→α as t→1. If in addition, x(t)→ξ as t→1, then u has asymptotic value α at ξ.

We give an example of a univalent orientation-preserving harmonic mapping f = h + g¯ defined on the unit disc U which is real on the real axis, satisfies and is not typically real. Furthermore, we give a geometric characterization for univalent, harmonic and typically real mappings.

Our main result can be stated as follows:

Theorem.

for all λ ≥ λ0, where λ0 is the unique root in (0, 1) of the equation

andis the Gegenbauer polynomial of degree 2k and order λ. The only cases of equality in (1·1) are when θ = 0 and n is odd.

We show that for a probability measure μ on ℝn

for almost all m–dimensional subspaces V, provided dimH μ≤m. Here projv denotes orthogonal projection onto V, and dimH and dimp denote the Hausdorff and packing dimension of a measure. In the case dimH μ > m we show that at μ-almost all points x the slices of μ by almost all (n − m)-planes Vx through x satisfy

We give examples to show that these inequalities are sharp.

We show that for arbitrary metric spaces X and Y the following dimension inequalities hold:

where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the proof is to use modified constructions of the Hausdorff and packing measure to deduce appropriate inequalities for the measure of X × Y.

We prove some elementary intuitive estimates on moving boundaries hitting times by one-dimensional Brownian motion (in ℝ and on the circle). These results give an alternative approach to Beurling's radial projection theorem on harmonic measure in a disc.

The Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded.