If H, K are subgroups of a group G, then HK is a subgroup of G if and only if HK = KH. This condition certainly holds if H ≤ NG(K) or K ≤ NG(H). But the majority of groups can also be expressed as HK, where neither H nor K is normal. In this paper we consider groups G for which no subgroup G1 can be expressed as the product of non-normal subgroups of G1. Such a group is said to be equilibrated. Thus G is equilibrated if and only if either H ≤ NG(K) or K ≤ NG(H) whenever H, K and HK are subgroups of G.

In the modular representation theory of finite groups, much recent effort has gone into describing cohomological properties of the category of finitely generated modules. In recent joint work of the author with Jon Carlson and Jeremy Rickard[3], it has become clear that for some purposes the finiteness restriction is undesirable. In particular, in the quotient category of kG-modules by the subcategory of modules of less than maximal complexity, it turns out that finitely generated modules can have infinitely generated summands, and that including these summands in the category repairs the lack of Krull–Schmidt property.

It has now been almost twenty years since Alperin introduced the idea of the complexity of a finitely generated kG-module, when G is a finite group and k is a field of characteristic p > 0. In proving one of the first major results in the area [1], Alperin and Evens demonstrated the connection of the study of complexity for modules to the group cohomology. That connection eventually led to the categorization of modules according to their associated varieties in the maximal ideal spectrum of the cohomology ring H*(G, k). In all of the work that has followed, two principles have proved to be extremely important. The first is that the associated variety of a module is directly related to the structure of the module through the rank variety which is defined by the matrix representation of the module. The second major result is the tensor product theorem which says that the variety associated to a tensor product M ⊗kN is the intersection of the varieties associated to the modules M and N. In this paper we generalize these results to infinitely generated kG-modules.

The class of nilpotent semigroups was introduced, via a semigroup identity, independently in [8] and [9], cf. [11], Every nilpotent cancellative semigroup S was shown to have a group of classical fractions G which is nilpotent and of the same nilpotency class as S. Groups, and linear groups in particular, satisfying certain related semigroup identities, introduced in [19], have been recently studied in [1], [15] and [18]. In particular, finitely generated residually finite groups satisfying a semigroup identity must be almost nilpotent, [18]. On the other hand, it was recently shown in [14] that a finitely generated linear semigroup S ⊆ Mn(K), over a field K, with no free non-commutative subsemigroups satisfies an identity and for every maximal subgroup H of Mn(K) the subgroup gp(S ∩ H) of H generated by S ∩ H is almost nilpotent. A natural question that arises here is to decide which of these semigroups are nilpotent. Because of the powerful classical theory of nilpotent linear groups, cf. [21], one can also ask whether such semigroups can be approached via group theoretical methods. We note that the very special case of nilpotent connected algebraic monoids has been recently considered in [4]. In [5] the structure of semigroup algebras of nilpotent semigroups was studied, in particular via prime Goldie homomorphic images, leading naturally to nilpotent subsemigroups of the matrix monoids Mn(D) over division rings D.

In this paper, a distance formula to a Douglas algebra is established. We use this distance formula to show that the distance of an L∞ function to the intersection of arbitrary Douglas algebras is equivalent to the supremum of the distance of that function to these algebras. As an application, we prove that the Bourgain algebra of the intersection of two Douglas algebras is equal to the intersection of the Bourgain algebras of these two algebras.

Here we study (in a more general setting) the following problem. Let C be a smooth projective curve, E and F vector bundles on C and V ⊆ H0 (C, E) (resp. W ⊆ H0 (C, F)) vector spaces generically spanning E (resp. F); find lower bounds for the dimension of the image of the multiplication map V ⊗ W → H0 (C, E ⊗ F) generalizing the case rank(E) = rank(F) = 1.

1·1. Groups of the first kind. In [11], Patterson proved a hyperbolic space analogue of Khintchine's theorem on simultaneous Diophantine approximation. In order to state Patterson's theorem, some notation and terminology are needed. Let ‖x‖ denote the usual Euclidean norm of a vector x in k+1, k + 1-dimensional Euclidean space, and let be the unit ball model of k + 1-dimensional hyperbolic space with Poincaré metric ρ. A non-elementary geometrically finite group G acting on Bk + 1 is a discrete subgroup of Möb (Bk+l), the group of orientation preserving Mobius transformations preserving Bk + 1, for which there exists some convex fundamental polyhedron with finitely many faces. Since G is non-elementary, the limit set L(G) of G – the set of limit points in the unit sphere Sk of any orbit of G in Bk+1 – is uncountable. The group G is said to be of the first kind if L(G) = Sk and of the second kind otherwise.

Let Γ be a discrete group. Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torsion-free subgroup Γ' of Γ such that Γ' has finite cohomological dimension over . Examples of such groups include the fundamental group of a finite graph of finite groups, arithmetic groups, mapping class groups and outer automorphism groups of free groups. One of the fundamental problems in topology is to understand the cohomology of these finite vcd-groups.

Given a knot K in a 3-manifold M, we use N(K) to denote a regular neighbourhood of K. Suppose γ is a slope (i.e. an isotopy class of essential simple closed curves) on ∂N(K). The surgered manifold along γ is denoted by (H, K; γ), which by definition is the manifold obtained by gluing a solid torus to H – Int N(K) so that γ bounds a meridional disc. We say that M is ∂-reducible if ∂M is compressible in M, and we call γ a ∂-reducing slope of K if (H, K; γ) is ∂-reducible. Since incompressible surfaces play an important rôle in 3-manifold theory, it is important to know what slopes of a given knot are ∂-reducing. In the generic case there are at most three ∂-reducing slopes for a given knot [12], but there is no known algorithm to find these slopes. An exceptional case is when M is a solid torus, which has been well studied by Berge, Gabai and Scharlemann [1, 4, 5, 10]. It is now known that a knot in a solid torus has ∂-reducing slopes only if it is a 1-bridge braid. Moreover, all such knots and their corresponding ∂-reducing slopes are classified in [1]. For 1-bridge braids with small bridge width, a geometric method of detecting ∂-reducing slopes has also been given in [5]. It was conjectured that a similar result holds for handlebodies, i.e. if K is a knot in a handlebody with H – K ∂-irreducible, then K has ∂-reducing slopes only if K is a 1-bridge knot (see below for definitions). One is referred to [13] for some discussion of this conjecture and related problems.

The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, ‖ ‖E) is 2-convex, then there is another Banach sequence space (F, ‖ ‖F) such that ‖x;‖ = ‖x2‖F for all x ε E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.

The long open problem raised by I. Kaplansky if, for an infinite compact Hausdorff space X, there is a discontinuous homomorphism from (X) into a Banach algebra was settled in the 1970s, independently, by H. G. Dales and J. Esterle. If the continuum hypothesis is assumed, then there is a discontinuous homomorphism from (X) (see [8] for a survey of both approaches and [9] for a unified exposition). The techniques developed by Dales and Esterle are powerful enough to yield discontinuous homomorphisms from commutative Banach algebras other than (X). In fact, every commutative Banach algebra with infinitely many characters is the domain of a discontinuous homomorphism ([7]).

We study the multifractal structure of product measures. for a Borel probability measure μ and q, t Є , let and denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [O11] Let μ be a Borel probability merasure on k and let v be a Borel probability measure on t. Fix q, s, t Є . We prove that there exists a number c > 0 such that for E ⊆k, F ⊆l and Hk+l provided that μ and ν satisfy the so-called Federer condition.

Using these inequalities we give upper and lower bounds for the multifractal spectrum of μ × ν in terms of the multifractal spectra of μ and ν

Let (pn) be a sequence of real numbers with pn ~ R(n), R(.) a regulary varying function with index greater than −1/2. We prove the Hartman–Wintner law of the iterated logarithm for the corresponding (Jp) power series transform and generalized Nörlund transforms (Nβp) of sequences (Xn) of i.i.d. random variables with mean-zero and variance 1. We also identify the cluster sets.