There is a pairing between two Borel subalgebras of a quantized Kac–Moody algebra, which plays the rôle of R-matrix. Over the field ℚ(q) this pairing is non-degenerate. We show the existence of a braiding in some categories of representations of a quantized Kac-Moody algebra.

A profinite topological group is compact and therefore possesses a unique invariant probability measure (Haar measure). We shall see that it is possible to define a fractional dimension on such a group in a canonical way, making use of Haar measure and a natural choice of invariant metric. This fractional dimension is analogous to Hausdorff dimension in ℝ.

It is therefore natural to ask to what extent known results concerning Hausdorff dimension in ℝ carry over to the profinite setting. In this paper, following a line of thought initiated by B. Volkmann in [12], we consider rings of a-adic integers and investigate the possible dimensions of their subgroups and subrings. We will find that for each prime p the ring of p-adic integers possesses subgroups of arbitrary dimension. This should cause little surprise since a similar result is known to hold in ℝ. However, we will also find that there exists a ring of a-adic integers possessing Borel subrings of arbitrary dimension. This is in contrast with the situation in ℝ, where the analogous statement is known to be false.

Let p be an odd prime, and h*( − ) a multiplicative mod p cohomology theory with a complex orientation satisfying the 2(p − 1)-sparseness condition. A formula concerning the formal group law for such h*( − ) is used to investigate the structure of h*(BG), G being a finite group whose p-Sylow subgroups are cyclic.

Let A be an artin algebra over a commutative artin ring R, mod A be the category of finitely generated right A-modules, and rad∞ (modA) be the infinite power of the Jacobson radical rad(modA) of modA. Recall that A is said to be representation-finite if mod A admits only finitely many non-isomorphic indecomposable modules. It is known that A is representation-finite if and only if rad∞ (mod A) = 0. Moreover, from the validity of the First Brauer–Thrall Conjecture [26, 2] we know that A is representation-finite if and only if there is a common bound on the length of indecomposable modules in mod A.

Several years ago the authors, together with Dave Benson, conducted an investigation into the vanishing of cohomology for modules over group algebras [2]. It was mostly in the context of kG-modules where k is a field of finite characteristic p and G is a finite group whose order is divisible by p. Aside from some general considerations, the main results of [2] related the existence of kG-modules M with H*(G, M) = 0 to the structure of the centralizers of the p-elements in G. Specifically it was shown that there exists a non-projective module M in the principal block of kG with H*(G, M) = 0 whenever the centralizer of some p-element of G is not p-nilpotent. The converse was proved in the special case that the prime p is an odd integer (p > 2). In addition there was some suspicion and much speculation about the structure of the varieties of such modules. However, proofs seemed to be waiting for a new idea.

Let d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.

The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.

We use computational methods to find the irreducible 2-modular characters of Conway's second group. Three of these are produced by probabilistic methods, which means that we have very strong evidence that they are correct, but we do not yet have a complete proof.

We determine the stable decompositions of the classifying spaces of metacyclic 2-groups into wedges of indecomposable spectra. The stable decompositions of classifying spaces of finite groups with metacyclic 2-Sylow subgroups are also determined.

Let (A, G, α, τ) be a twisted covariant system such that G/Nτ is abelian, and let Nτ ⊆ M ⊆ H be closed subgroups of G. We show that inducing covariant representations from A ⋊α, τM to A ⋊α, τH is in a certain sense dual to restricting representations from (A⋊α, τG)⋊άM┴ to (A⋊α, τG)⋊άH┴, and that, similarly, restricting representations from A⋊α, τH to A⋊α, τM is dual to inducing representations from (A⋊τG)⋊άH┴ to (A ⋊α, τG)⋊άM┴.

It is well-known that the structure of βℕ, the Stone—Čech compactification of the discrete semigroup (ℕ, +), is very complex. For example, it has 2c minimal left ideals and 2c minimal right ideals, its minimal ideal contains 2c copies of the free group on 2c generators, see [5], and Lisan[8] proved the existence of 2c copies of the same group outside the closure of the minimal ideal.

Connected completely 0-simple semigroups are defined by a number of equivalent conditions, and a formula for the rank of these semigroups is proved. As a consequence an alternative proof of the result from [11] is given. In the case of a Rees matrix semigroup M0 [G, I, Λ, P] the rank is expressed in terms of |I|, |Λ|, G and a certain subgroup of G depending on P. At the end the minimal rank of all semigroups M0[G, I, Λ, P] is found for a given group G. Since every completely simple semigroup is connected, every result has a corollary for these semigroups.

Chinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

In this paper we study the uniform boundedness of oscillatory singular integral operators with degenerate phase functions on the Hardy space H1. The H1 boundedness was previously known when the phase function is nondegenerate. Here we obtain a sufficient condition for H1 boundedness which allows the phase function vanishing to infinite order.

It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2i for each integer i ≥ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (P)i ≥ 0 with per (Pi) = 2i which form a period-doubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many ‘zero-entropy’ cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point).

Let Σ be a closed surface of genus ≥ 1, G a compact Lie group, not necessarily connected with Lie algebra g, ξ,: P → Σ a principal G-bundle, and suppose Σ equipped with a Riemannian metric and g with an invariant scalar product so that the Yang—Mills equations on ξ are defined. Further, let

be the universal central extension of the fundamental group π of Σ and ΓR the group obtained from Γ when its centre Z is extended to the additive group R of the reals. We show that there are bijective correspondences between various spaces of classes of Yang—Mills connections on ξ and spaces of representations of Γ and ΓR (as appropriate) in G. In particular, we show that the holonomy establishes a homeomorphism between the moduli space N(ξ) of central Yang–Mills connections on ξ and the space Repξ(Γ, G) of representations of Γ in G determined by ξ. Our results rely on a detailed study of the holonomy of a central Yang–Mills connection and extend corresponding ones of Atiyah and Bott for the case where G is connected.