A generalization of classical theorems on the existence of sections of real, complex and quaternionic Stiefel manifolds over spheres is proved. We obtain a complete list of Lie group homomorphisms $\rho : G \to G_n$, where $G_n$ is one of the groups $SO(n)$, $SU(n)$ or $Sp(n)$ and $G$ is one of the groups $SO(k)$, $SU(k)$ or $Sp(k)$, which reduce the structure group $G_n$ in the fibre bundle $G_n \to G_{n + 1} \to G_{n + 1} / G_n$.

We study the relationship between the cohomology of $q$-Schur algebras and Hecke algebras in type $A$. A key tool, which has independent interest, involves a specific Hecke algebra resolution, introduced in a special case by Deodhar. This resolution is somewhat akin to the Koszul resolution for Lie algebra cohomology, in that it has an explicit description yet applies for all modules. An application of the properties of this resolution establishes a ‘quantum Weyl reciprocity’ connecting, in a range of degrees, the cohomology of the $q$-Schur algebra with that of the Hecke algebra. This work also provides, independently of the resolution, cohomological inequalities which apply in all degrees and which sometimes can be related to Kazhdan–Lusztig polynomials. Finally, we extend to cohomology a decomposition number method of Erdmann, and illustrate its usefulness in computations. This method enables us to exhibit some properties of symmetric group cohomology which, if true, would imply the $SL_n$-version of Lusztig's characteristic $p$ character conjecture.

We obtain a number of results regarding the freeness of subgroups of Coxeter groups, Artin groups and one-relator groups with torsion. In the case of Coxeter groups, we also obtain results on quasiconvexity and subgroup separability.

Given a generalized octagon which satisfies certain geometric conditions, we show that it must be isomorphic to the Ree--Tits octagon arisingnaturally from a Ree group of type $^2F_4$ over a perfect field of characteristic 2, in which the Frobenius automorphism has a square root.

1991 Mathematics Subject Classification: 51E12.

We consider Hilbert transforms along curves $\Gamma$ on the Heisenberg group. In particular, necessary and sufficient conditions for the $L^2$-boundedness are given for $\Gamma(t)=(t,\gamma(t),0)$ when $\gamma$ is even or odd and convex on ${\Bbb R}^{+}$. 1991 Mathematics Subject Classification: 42B20,42B25.

For any non-empty subset $\mathcal I$ of the natural numbers, let $\Lambda_{\mathcal I}$ denote those numbers in the unit interval whose continued fraction digits all lie in $\mathcal I$. Define the corresponding transfer operator

$$ \mathcal L_{\mathcal I, \beta} f(z) = \sum_{n \in \mathcal I} \left( \frac{1}{n + z} \right)^{2 \beta} f\left( \frac{1}{n + z} \right) $$

for $\text{Re} (\beta) > \max (0, \theta_{\mathcal I})$, where $\text{Re} (\beta) = \theta_{\mathcal I}$ is the abscissa of convergence of the series $\sum_{n \in \mathcal I}n^{-2 \beta}$.

When acting on a certain Hilbert space $\mathcal H_{\mathcal I, \beta}$, we show that the operator $\mathcal L_{\mathcal I, \beta}$ is conjugate to an integral operator $\mathcal K_{\mathcal I, \beta}$. If furthermore $\beta$ is real, then $\mathcal K_{\mathcal I, \beta}$ is selfadjoint, so that $\mathcal L_{\mathcal I, \beta} : \mathcal H_{\mathcal I, \beta} \to \mathcal H_{\mathcal I, \beta}$ has purely real spectrum. It is proved that $\mathcal L_{\mathcal I, \beta}$ also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space $C^\omega [0, 1]$, and on the Fréchet space $C^\infty [0, 1]$.

The analytic properties of the map $\beta \mapsto \mathcal L_{\mathcal I, \beta}$ are investigated. For certain alphabets $\mathcal I$ of an arithmetic nature (for example, $\mathcal I = \{\text{primes}\}$, $\mathcal I = \{\text{squares}\}$, $\mathcal I$ an arithmetic progression, $\mathcal I$ the set of sums of two squares) it is shown that $\beta \mapsto \mathcal L_{\mathcal I, \beta}$ admits an analytic continuation beyond the half-plane $\text{Re} (\beta) > \theta_{\mathcal I}$.

A function $f$ from the symmetric group $S_n$ into $\Bbb R$ is called a class function if $f(\sigma ) = f(\tau )$ whenever $\tau$ is conjugate to $\sigma$. Let $d_f$ be thegeneralized matrix function associated with $f$, mapping the $n$-by-$n$ positive semidefinite Hermitian matricesto $\Bbb R$. For example, if $f(\sigma ) = \mbox{sgn}(\sigma )$, then $d_f(A) = \det A$.

We consider the cone $K_n$ of those $f$ for which $d_f(A) \geq 0$ for all $n$-by-$n$ positive semidefinite Hermitian matrices. For $n = 4$ we show that $K_n$ is polyhedral, and explicitly find the extreme rays. Equivalently, $f$ belongs to $K_n$ if $d_f(A) \geq 0$ for a finite minimal set of ‘test matrices’. This solves a problem posed by Gordon James. As a corollary, we characterize all inequalities involving linear combinations of immanants on the positive semidefinite Hermitian matrices of order $n = 4$.

We obtain similar results for 4-by-4 real positive semidefinite matrices.

The functional Ito formula, in the form $df({\bf \Lambda})=f({\bf \Lambda} +d{\bf \Lambda})-f({\bf \Lambda} )$, is formulated and proved in the context of a Lie algebra $\mathcal{L}$ associated with a quantum (non-commutative) stochastic calculus. Here $f$ is an element of the universal enveloping algebra $\mathcal{U}$ of $\mathcal{L}$, and $f({\bf \Lambda} + d{\bf \Lambda})-f({\bf \Lambda} )$ is given a meaning using the coproduct structure of $\mathcal{U}$ even though the individual terms of this expression have no meaning. The Ito formula is equivalent to a chaotic expansion formula for $f({\bf \Lambda} )$ which is found explicitly.

1991 Mathematics Subject Classification: primary 81S25; secondary 60H05; tertiary 18B25.

A function f : Rn → R is a connectivity function if the graph of its restriction f|C to any connected C ⊂ Rn is connected in Rn x R. The main goal of this paper is to prove that every function f : Rn → R is a sum of n + 1 connectivity functions (Corollary 2.2). We will also show that if n > 1, then every function g: Rn → R which is a sum of n connectivity functions is continuous on some perfect set (see Theorem 2.5) which implies that the number n + 1 in our theorem is best possible (Corollary 2.6).

To prove the above results, we establish and then apply the following theorems which are of interest on their own.

For every dense Gδ-subset G of Rn there are homeomorphisms h1...,hn of Rn such that Rn=G∪h1(G)∪...∪hn(G) (Proposition 2.4).

For every n >1 and any connectivity function f : Rn → R, if x ∈ Rn and ε >0 then there exists an open set U⊂Rn such that x ∈ U⊂Bn(x,ε), f|bd(U) is continuous, and |(x)-f(y)| < ε for every y∈bd(U) (Proposition 2.7).

It is shown that Teichmüller distance for analytically finite surfaces is C2. This extends Earle's classical result of 1977 that the distance is C1. Earle showed that the first derivative is given by a quadratic differential. In order to obtain the C2 result, first, a formula is derived for the second derivative for a generic choice of quadratic differential, and with respect to certain local coordinates on Teichmüller space. Then this formula is interpreted so that it can be seen that the limit exists at non-generic points.

2000 Mathematical Subject Classification: 30F60, 32G15.

We use Nielsen methods to study generating sets of subgroups of groups that act on simplicial trees and give several applications. In particular, we exhibit an explicit bound for the complexity of acylindrical splittings of a finitely generated group in terms of its rank. This is applied to JSJ-splittings of word-hyperbolic groups and 3-manifolds. As a last application we construct examples of amalgamated products that show that there exists no non-trivial rank formula for amalgamated products.

2000 Mathematical Subject Classification: 20E06, 20E08, 57M27.

Given a spectrum $X$, we construct a spectral sequence of $BP_{*}BP$-comodules that converges to $BP_{*}(L_{n}X)$, where $L_{n}X$ is the Bousfield localization of $X$ with respect to the Johnson–Wilson theory $E(n)_{*}$. The $E_{2}$-term of this spectral sequence consists of the derived functors of an algebraic version of $L_{n}$. We show how to calculate these derived functors, which are closely related to local cohomology of $BP_{*}$-modules with respect to the ideal $I_{n + 1}$.

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which $G$ is the Galois group of an extension of global fields and the resulting annihilator/Fitting ideal relation is closely connected to Stickelberger's Theorem and to the conjectures of Coates and Sinnott, and Brumer. Higher Stickelberger ideals are defined in terms of special values of L-functions; when these vanish we show how to define fractional ideals, generalising the Stickelberger ideals, with similar annihilator properties. The fractional ideal is constructed from the Borel regulator and the leading term in the Taylor series for the L-function. En route, our methods yield new proofs, in the case of abelian number fields, of formulae predicted by Lichtenbaum for the orders of K-groups and étale cohomology groups of rings of algebraic integers.

In this paper, we prove that the modular curve $X(11)$ over a field of characteristic 3 admits the Mathieu group $M_{11}$ as an automorphism group. We also examine some aspects of the geometry of the curve $X(11)$ in characteristic 3. In particular, we show that every point of the curve is a point of inflection, the curve has 110 hyperflexes and there are no inflectional triangles and 11232 inflectional pentagons, of which 144 are self-conjugate. The hyperflexes correspond to the supersingular elliptic curves. We comment on the relationship of Ward's quadrilinear invariant for $M_{12}$ to our work and announce for the first time the equations for Klein's A-curve of level 11. We also comment on the relation of our work to some unpublished work of Bott and Tate.

1991 Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10, 14N10, 20B25, 20C34.

This paper concerns a Markov operator $T$ on a space $L_1$, and a Markov process $P$ which defines a Markov operator on a space $M$ of finite signed measures. For $T$, the paper presents necessary and sufficient conditions for: \begin{enumerate}\item [(a)] the existence of invariant probability densities (IPDs)\item [(b)] the existence of strictly positive IPDs, and\item [(c)] the existence and uniqueness of IPDs.\end{enumerate} Similar results on invariant probability measures for $P$ are presented. The basic approach is to pose a fixed-point problem as the problem of solving a certain linear equation in a suitable Banach space, and then obtain necessary and sufficient conditions for this equation to have a solution.

1991 Mathematics Subject Classification: 60J05, 47B65, 47N30.

Engel-4 groups of exponent 5 are now known to be locally finite. In this paper we determine the orders of the largest finite groups of this kind on $m$ generators.

In the process we show that such a group has a subdirect decomposition into a group which is centre-by-Engel-3 and one which is nilpotent of class at most 10. We work via associated Lie algebras and Lie superalgebras and make use of computer calculations.

1991 Mathematics Subject Classification: 20F45.

We construct a sequence of concordance invariants for classical links, which depend on the peripheral isomorphism type of the nilpotent quotients of the link fundamental group. The terminology stems from the fact that we replace the Magnus expansion in the definition of Milnor's μ-invariants by the similar Campbell-Hausdorff expansion. The main point is that we introduce a new universal indeterminacy, which depends only on the number of components of the link. The Campbell-Hausdorff invariants are new, effectively computable and can efficiently distinguish (unordered and unoriented) isotypes of links, as we indicate on several families of closed braid examples. They also satisfy certain natural dependence relations, which generalize well-known symmetries of the μ-invariants.

Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over ${\Bbb P}^2$ and the Prym variety of a naturally defined étale double cover of the discriminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold, and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over ${\Bbb P}^2$. We further generalize the isomorphism to the primitive {\em cohomology} of a smooth cubic hypersurface in ${\Bbb P}^n$. We give two applications of our construction: one is a special case of the generalized Hodge conjectures and the other is an Abel-Jacobi isomorphism.

1991 Mathematics Subject Classification: primary 14J70; secondary 14J45.

A {\em convex corner} is a compact convex down-set of full dimension in ${\mathbb R}_+^n$. Convex corners arise in graph theory, for instance as stable set polytopes of graphs. They are also natural objects of study in geometry, as they correspond to 1-unconditional norms in an obvious way. In this paper, we study a parameter of convex corners, which we call the {\em content}, that is related to the volume. This parameter has appeared implicitly before: both in geometry, chiefly in a paper of Meyer ({\em Israel J.\ Math.} 55 (1986) 317--327) effectively using content to give a proof of Saint-Raymond's Inequality on the volume product of a convex corner, and in combinatorics, especially in a paper of Sidorenko ({\em Order} 8 (1991) 331--340) relating content to the number of linear extensions of a partial order. One of our main aims is to expose connections between work in these two areas. We prove many new results, giving in particular various generalizations of Saint-Raymond's Inequality. Content also behaves well under the operation of {\em pointwise product} of two convex corners; our results enable us to give counter-examples to two conjectures of Bollob\'as and Leader ({\em Oper.\ Theory Adv.\ Appl.} 77 (1995) 13--24) on pointwise products. 1991 Mathematics Subject Classification: 52C07, 51M25, 52B11, 05C60, 06A07.

The Teichmüller space of a finite-type surface is considered. It is shown that Teichmüller distance is not $C^{2 + \epsilon }$ for any $\epsilon > 0$. Furthermore, Teichmüller distance is not $C^{2 + g}$ for any gauge function $g$ with $\liminf_{u \to 0} g(u)\log (1 / u)=0$.