A point λ in the complex plane is said to be free if the group generated by the matrices

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is free. The paper gives an infinite family of polynomials whose roots are the non-free points. The main idea in the paper is to employ a symmetry relation.

Asymptotic cones were first used by Gromov in [6], where he constructed limit spaces of nilpotent groups in order to prove that groups with polynomial growth are virtually nilpotent. Gromov does not use the term asymptotic cone, which was introduced later by Van den Dries and Wilkie in [12], when they gave a nonstandard interpretation of Gromov's results. Ultrafilters appear in [12] for the first time in this context. Later, Gromov gave an extensive treatment of asymptotic cones in [7]. Since then several authors have used asymptotic cones to obtain interesting results, for instance, in identifying quasi-isometry classes of 3-manifolds [8] or relating asymptotic cones with Dehn functions of finitely presented groups (see [2] and [11], the results of which are stated in Section 2).

The purpose of this paper is to develop some of the results stated in [7], in particular those describing the asymptotic cone of the Baumslag–Solitar groups and of Sol. According to [2], these spaces are not simply connected, since their Dehn functions are exponential, so our primary goal is to study their fundamental groups. It will be proved that these fundamental groups are uncountable and nonfree (Section 9), by constructing subgroups isomorphic to the fundamental group of the Hawaiian earring. These subgroups are constructed by finding subspaces in the asymptotic cones which are homotopically equivalent to the Hawaiian earring, and which induce injections in the fundamental group level (Section 8). Crucial to the proof of these facts is the computation of the covering dimension of these asymptotic cones (Section 7), which is done using a more general theorem on dimensions of spaces which admit certain maps into well-known spaces (Section 6).

Throughout this paper, r will denote a prime. Our object in this paper and another is to describe the finite irreducible subgroups of L(r):=SL(r, [Copf ]), where [Copf ] denotes the field of complex numbers. The cases for small degrees are known (see [14] for r=2 and r=3; [1] and [23–25] for r=5 and r=7; [13, 18]). The imprimitive irreducible subgroups of L(r) are necessarily monomial because r is a prime, and we describe these in another paper [4]. In this paper, we show how the classification of finite simple groups can be used to classify the finite primitive groups of prime degree. Much of the necessary work has already been done by V. Landazuri and G. M. Seitz (see Lemma 1.3 below).

We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre's Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including SL$(n,\mathbb{Z})$ for $n\geq 3$. We also give a way of thinking about Property (QFA) by breaking it down into statements about particular classes of trees.

The classical transmission problem for the scattering of time-harmonic electromagnetic waves by a dielectric object is solved using layer potential techniques. The surface of the object is only assumed to be a Lipschitz manifold. Unlike in the case of smoother surfaces, neither the calculus of pseudodifferential operators nor the compactness of certain boundary operators are available tools. Nevertheless, the particular representation of the solutions and the choice of the space of boundary data used in the paper make the problem tractable via harmonic analysis techniques for elliptical partial differential equations in nonsmooth domains.

A study is made of groups with finitely many derived groups of subgroups or of infinite subgroups. These groups are classified completely in the locally graded case. In the general case, detailed structural information about groups in each class is found.

A variation on the unique product property of groups is described which seems natural from a geometric point of view. It is stronger than the unique product property, and hence implies, for example, that the group rings have no zero divisors. Some of its closure properties are described. It is shown that most surface groups satisfy this condition, and various other examples are given. It is explained how these ideas can give a more geometric interpretation of Promislow's example of a non-unique-product group.

For an irrational rotation α of the circle group T=R/Z and a piecewise absolutely continuous function f@[ratio ]T→R, the unitary operator Vh(x)=e2πif(x)h(x+α) on L2(T) is studied. It is shown that if f has a single discontinuity with non-integer jump then V is κ-weakly mixing for some κ with 0<[mid ]κ[mid ]<1. In particular V has continuous singular spectrum. The property of κ-weak mixing (with possible change of the value of κ, 0<[mid ]κ[mid ]<1) holds for all irrational rotations and, given α, is stable under perturbations of f by functions with sufficiently small O(1/n)-norm. On the other hand, there exists a piecewise linear function f with two non-integer jumps such that the spectrum of V is continuous singular for one value of α and Lebesgue for another.

An interpretation is given of the Shintani descent from the points of a connected algebraic group over a finite field to its group of points over a subfield in terms of a Shintani descent between two different rational structures over the same finite field, using restriction of scalars. From this a generalization is deduced of a result of Gyoja's on the Shintani descent of Deligne–Lusztig characters of a finite group of Lie type.

The norm problem is considered for elementary operators of the form $U_{a,b}\,{:}\,{\cal A}\,{\longrightarrow}\,{\cal A},\;x\longmapsto axb\,{+}\,bxa (a,\,b\,{\in}\,{\cal A})$ in the special case when ${\cal A}$ is a subalgebra of the algebra of bounded operators on a Banach space. In particular, the lower estimate $\|U_{a,b}\|\geq\|a\|\|b\|$ is established when the Banach space is a Hilbert space and ${\cal A}$ is the algebra of all bounded linear operators.

It is shown that there exists a non-separable reflexive Banach space on which every bounded linear operator is the sum of a scalar multiple of the identity operator and an operator of separable range. There is a strong sense that such a Banach space has as few operators as its linear and topological properties allow.

A famous Diophantine equation is given by

formula here

For integers k[ges ]2 and m[ges ]2, this equation only has the solutions x = −j (j = 1, …, m), y = 0 by a remarkable result of Erdős and Selfridge [9] in 1975. This put an end to the old question of whether the product of consecutive positive integers could ever be a perfect power (except for the obviously trivial cases). In a letter to D. Bernoulli in 1724, Goldbach (see [7, p. 679]) showed that (1) has no solution with x[ges ]0 in the case k = 2 and m = 3. In 1857, Liouville [18] derived from Bertrand's postulate that for general k[ges ]2 and m[ges ]2, there is no solution with x[ges ]0 if one of the factors on the right-hand side of (1) is prime. By use of the Thue–Siegel theorem, Erdős and Siegel [10] proved in 1940 that (1) has only trivial solutions for all sufficiently large k[ges ]k0 and all m. This was closely related to Siegel's earlier result [30] from 1929 that the superelliptic equation

formula here

has at most finitely many integer solutions x, y under appropriate conditions on the polynomial f(x). The ineffectiveness of k0 was overcome by Baker's method [1] in 1969 (see also [2]).

In 1955, Erdős [8] managed to re-prove the result jointly obtained with Siegel by elementary methods. A refinement of Erdős' ideas finally led to the above-mentioned theorem as follows.

Ideas from string theory and quantum field theory have been the motivation for new invariants of knots and 3-dimensional manifolds which have been constructed from complex algebraic structures such as Hopf algebras [17, 22], monoidal categories with additional structure [24], and modular functors [14, 23]. These constructions are closely related. Here we take a unifying categorical approach based on a natural 2-dimensional generalisation of a topological field theory in the sense of Atiyah [1], and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces.

The main objective of this paper is to establish a large deviation principle for heat kernel measures on loop spaces. It gives an extension of Fang and Zhang's results on loop groups. For the proof, we use the continuity theorem of Lyons' rough path theory.

In 1998, Zalesskii proposed to classify all instances of irreducible characters of quasi- simple groups which are of prime power degree. In joint work, Malle and Zalesskii then dealt with all quasi-simple groups with the exception of the alternating groups and their double covers [5]. In an earlier article [1] we have classified all the irreducible characters of $S_n$ of prime power degree and have deduced from this also the corresponding classification for the alternating groups. In the present article we complete Zalesskii's programme by dealing with the final case left open in [5], the double covers of the alternating groups. We derive this result from a corresponding result on the double covers of the symmetric groups. If one is only interested in spin characters, one easily sees that only 2-powers can occur as prime power degrees. However from a combinatorial point of view it is natural to ask more generally: when is the number of shifted standard tableaux of a given shape a prime power? Since our method is independent of the prime, we will answer this question, showing that apart from a few accidental cases for small $n$ only the ‘obvious’ partitions satisfy the prime power condition. Thus in turn for the spin characters, apart from exceptions for small $n$ , the ‘obvious’ spin characters of 2-power degree are indeed the only ones (this confirms the conjecture stated in [5]).

It is proved that there exists a scheme that represents the functor of line modules over a graded algebra, and it is called the line scheme of the algebra. Its properties and its relationship to the point scheme are studied. If the line scheme of a quadratic, Auslander-regular algebra of global dimension 4 has dimension 1, then it determines the defining relations of the algebra.

Moreover, the following counter-intuitive result is proved. If the zero locus of the defining relations of a quadratic (not necessarily regular) algebra on four generators with six defining relations is finite, then it determines the defining relations of the algebra. Although this result is non-commutative in nature, its proof uses only commutative theory.

The structure of the line scheme and the point scheme of a 4-dimensional regular algebra is also used to determine basic incidence relations between line modules and point modules.

A theorem of Hall extends the sylow theory to π-subgroups of a finite solvable group for any set of primes π. This theorem has been generalized to solvable periodic linear groups [21] and to other similar settings [19]. We consider the same problem in a model-theoretic context.

In model theory, the study of ω-stable groups generalizes the classical theory of algebraic groups in the case of a algebraically closed base field, and is closely connected with certain aspects of the general structure theory developed by Shelah, particularly in connection with the more geometric theory of Zilber, Buechler, Hrushovski, Pillay, and others. In this connection, the primary concern is with the structure of simple ω-stable groups of finite rank. One way to approach the study of ω-stable groups is by generalizing some of the methods of finite group theory to this setting. One anticipates that character theory and counting arguments will not be available, though one would hope to recover much of the ‘local analysis’. Unfortunately there is as yet no very satisfactory sylow theory for ω-stable groups, except for the prime 2 [4]. One of us (Nesin) noticed initially that there is a sylow theorem for connected solvable ω-stable groups of finite Morley rank; this prompted us to look for a generalization along the following lines.

To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites $f\,{\circ}\,F$ with isolated singularities is studied, where $f\,{:}\,Y\,{\longrightarrow}\,\mathbb{C}$ is a function with (possibly non-isolated) singularity and $F:X\,{\longrightarrow}\, Y$ is a map into the domain of $f$, and $F$ only is deformed. The corresponding $T^1(F)$ is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that \[\tau=\mu(f\circ F)-\beta_0+\beta_1\], where $\tau$ is the length of $T^1(F)$ and $\beta_i$ is the length of $\mbox{Tor}_i^{\cal{O}_Y}({\cal O}_Y{/}J_f,\cal{O}_X)$. This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When $f$ has Cohen–Macaulay singular locus (for example when $f$ is the determinant function), relations between $\tau$ and the rank of the vanishing homology of the zero locus of $f\circ F$ are obtained.

Localization and dévissage theorems are proved for the hermitian $K$-theory of rings that are analogous to well-known theorems in algebraic $K$-theory. The proofs rely on, among other things, a study of derived categories, a generalization of a theorem of Pedersen and Weibel to the hermitian setting, and a cofinality result for triangular Witt groups. Applications include a proof of a conjecture of Karoubi and algebraic Bott periodicity.

A construction of the Monster simple group is described implicitly as $196882\,{\times}\,196882$ matrices over the field of 3 elements.