The $\Sigma^3$-conjecture for metabelian groups is proved in the split extension case.

Removable singularities for Hardy spaces $H^p(\Omega) = \{f \in \hbox{Hol}(\Omega):\vert f\vert^p \le u \hbox{ in } \Omega \hbox{ for some harmonic }u\}, 0 < p < \infty$ are studied. A set $E \subset \Omega$ is a weakly removable singularity for $H^p(\Omega\backslash E)$ if $H^p(\Omega\backslash E) \subset \hbox{Hol}(\Omega)$ , and a strongly removable singularity for $H^p(\Omega\backslash E)$ if $H^p(\Omega\backslash E) = H^p(\Omega)$ . The two types of singularities coincide for compact $E$ , and weak removability is independent of the domain $\Omega$ .

The paper looks at differences between weak and strong removability, the domain dependence of strong removability, and when removability is preserved under unions. In particular, a domain $\Omega$ and a set $E \subset \Omega$ that is weakly removable for all $H^p$ , but not strongly removable for any $H^p (\Omega\backslash E), 0 < p < \infty$ , are found.

It is easy to show that if $E$ is weakly removable for $H^p(\Omega\backslash E)$ and $q > p$ , then $E$ is also weakly removable for $H^q(\Omega\backslash E)$ . It is shown that the corresponding implication for strong removability holds if and only if $q/p$ is an integer.

Finally, the theory of Hardy space capacities is extended, and a comparison is made with the similar situation for weighted Bergman spaces.

The paper obtains the limiting behaviour of the expectations of the zeros of holomorphic sections of the canonical line bundles over a tower of covers of a compact complex manifold.

In [17, 18, 19], we began to investigate the continuity properties of homomorphisms from (non-abelian) group algebras. Already in [19], we worked with general intertwining maps [3, 12]. These maps not only provide a unified approach to both homomorphisms and derivations, but also have some significance in their own right in connection with the cohomology comparison problem [4].

The present paper is a continuation of [17, 18, 19]; this time we focus on groups which are connected or factorizable in the sense of [26]. In [26], G. A. Willis showed that if G is a connected or factorizable, locally compact group, then every derivation from L1(G) into a Banach L1(G)-module is automatically continuous. For general intertwining maps from L1(G), this conclusion is false: if G is connected and, for some n∈ℕ, has an infinite number of inequivalent, n-dimensional, irreducible unitary representations, then there is a discontinuous homomorphism from L1(G into a Banach algebra by [18, Theorem 2.2] (provided that the continuum hypothesis is assumed). Hence, for an arbitrary intertwining map θ from L1(G), the best we can reasonably hope for is a result asserting the continuity of θ on a ‘large’, preferably dense subspace of L1(G). Even if the target space of θ is a Banach module (which implies that the continuity ideal [Iscr ](θ) of θ is closed), it is not a priori evident that θ is automatically continuous: the proofs of the automatic continuity theorems in [26] rely on the fact that we can always confine ourselves to restrictions to L1(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4]. It is not clear if this strategy still works for an arbitrary intertwining map from L1(G) into a Banach LL1(G)-module.

We investigate the limit functions of iterates of a function belonging to a convergence group or of a uniformly quasiregular mapping. We show that it is not possible for a subsequence of iterates to tend to a non-constant limit function, and for another subsequence of iterates to tend to a constant limit function. It follows that the closure of the stabiliser of a Siegel domain for a uniformly quasiregular mapping is a compact abelian Lie group, which we further conjecture to be infinite. This result concerning possible limits of convergent subsequences of iterates for holomorphic rational functions on the Riemann sphere is known, and the only known method of proof involves universal covering surfaces and Möbius groups. Hence, our method yields a new and perhaps more elementary proof also in that case.

Let Kn ={x∈ℝn[ratio ]xi[ges ]0 for 1[les ]i[les ]n} and suppose that f[ratio ]Kn→Kn is nonexpansive with respect to the [lscr ]1-norm and f(0)=0. It is known that for every x∈Kn there exists a periodic point ξ=ξx∈Kn (so fp(ξ)=ξ for some minimal positive integer p=pξ) and fk(x) approaches {fj(ξ)[ratio ]0[les ]j<p} as k approaches infinity. What can be said about P*(n), the set of positive integers p for which there exists a map f as above and a periodic point ξ∈Kn of f of minimal period p? If f is linear (so that f is a nonnegative, column stochastic matrix) and ξ∈Kn is a periodic point of f of minimal period p, then, by using the Perron–Frobenius theory of nonnegative matrices, one can prove that p is the least common multiple of a set S of positive integers the sum of which equals n. Thus the paper considers a nonlinear generalization of Perron–Frobenius theory. It lays the groundwork for a precise description of the set P*(n). The idea of admissible arrays on n symbols is introduced, and these arrays are used to define, for each positive integer n, a set of positive integers Q(n) determined solely by arithmetical and combinatorial constraints. The paper also defines by induction a natural sequence of sets P(n), and it is proved that P(n)⊂P*(n)⊂Q(n). The computation of Q(n) is highly nontrivial in general, but in a sequel to the paper Q(n) and P(n) are explicitly computed for 1[les ]n[les ]50, and it is proved that P(n)=P*(n)=Q(n) for n[les ]50, although in general P(n)≠Q(n). A further sequel to the paper (with Sjoerd Verduyn Lunel) proves that P*(n)=Q(n) for all n. The results in the paper generalize earlier work by Nussbaum and Scheutzow and place it in a coherent framework.

Until recently, the known symmetric nets were class-regular and therefore satisfied a certain geometric condition that defines the class of nets known as tactical symmetric nets. Thus the known symmetric nets were tactical. This paper gives a general method of construction for non-tactical symmetric nets, which are therefore not class-regular. It is not known whether there are tactical symmetric nets which are not class regular.

In 1940 Nisnevič published the following theorem [3]. Let (Gα)α∈Λ be a family of groups indexed by some set Λ and (Fα)α∈Λ a family of fields of the same characteristic p[ges ]0. If for each α the group Gα has a faithful representation of degree n over Fα then the free product *α∈ΛGα has a faithful representation of degree n+1 over some field of characteristic p. In [6] Wehrfritz extended this idea. If (Gα)α∈Λ [les ]GL(n, F) is a family of subgroups for which there exists Z[les ]GL(n, F) such that for all α the intersection Gα∩F.1n=Z, then the free product of the groups *ZGα with Z amalgamated via the identity map is isomorphic to a linear group of degree n over some purely transcendental extension of F.

Initially, the purpose of this paper was to generalize these results from the linear to the skew-linear case, that is, to groups isomorphic to subgroups of GL(n, Dα) where the Dα are division rings. In fact, many of the results can be generalized to rings which, although not necessarily commutative, contain no zero-divisors. We have the following.

This research was partially supported by the National Security Agency. It is well known from work of Bruhat and Tits that an affine building has a spherical building at infinity. The paper studies the structure at infinity for an affine {\em twin} building. It is shown that the twinning restricts the structure at infinity in a natural way, producing two smaller spherical buildings that are canonically isomorphic to one another. In the process it is shown that to each wall and panel of these buildings at infinity is attached a twin tree.

Let U be a domain, convex in $x$ and symmetric about the $y$-axis, which is contained in a centered and oriented rectangle $S$. It is proved that $H_t{(U^+)}/H_t{(U)}\,{\leq}\, H_t{(S^+)}/H_t{(S)}$ where $H_t$ stands for heat content, that is, the remaining heat in the domain at time $t$ if it initially has uniform temperature 1, with Dirichlet boundary conditions, where $A^+\,{=}\,A\,{\cap}\, \{(x,y)\,{:}\,x\,{>}\,0\}$. It is also shown that the analog of this inequality holds for some other Schrödinger operators.

The paper first proves a result on the upper semi-continuity of the entropy map of random bundle maps, and then proves some large deviation results for random maps perturbations of an Axiom A diffeomorphism.

A metric space X has the unique midset property if there is a topology-preserving metric d on X such that for every pair of distinct points x, y there is one and only one point p such that d(x, p) = d(y, p). The following are proved. (1) The discrete space with cardinality [nfr ] has the unique midset property if and only if [nfr ] ≠ 2, 4 and [nfr ] [les ] [cfr ], where [cfr ] is the cardinality of the continuum. (2) If D is a discrete space with cardinality not greater than [cfr ], then the countable power DN of D has the unique midset property. In particular, the Cantor set and the space of irrational numbers have the unique midset property.

A finite discrete space with n points has the unique midset property if and only if there is an edge colouring ϕ of the complete graph Kn such that for every pair of distinct vertices x, y there is one and only one vertex p such that ϕ(xp) = ϕ(yp). Let ump(Kn) be the smallest number of colours necessary for such a colouring of Kn. The following are proved. (3) For each k [ges ] 0, ump(K2k+1) = k. (4) For each k [ges ] 3, k [les ] ump(K2k) [les ] 2k−1.

Symplectic groups are well known as the groups of isometries of a vector space with a non-singular bilinear alternating form. These notions can be extended by replacing the vector space by a module over a ring R, but if R is non-commutative, it will also have to have an involution. We shall here be concerned with symplectic groups over free associative algebras (with a suitably defined involution). It is known that the general linear group GLn over the free algebra is generated by the set of all elementary and diagonal matrices (see [1, Proposition 2.8.2, p. 124]). Our object here is to prove that the symplectic group over the free algebra is generated by the set of all elementary symplectic matrices. For the lowest order this result was obtained in [4]; the general case is rather more involved. It makes use of the notion of transduction (see [1, 2.4, p. 105]). When there is only a single variable over a field, the free algebra reduces to the polynomial ring and the weak algorithm becomes the familiar division algorithm. In that case the result has been proved in [3, Anhang 5].

Projections are constructed in the rotation algebra that are orthogonal to their Fourier transform and which are fixed under the flip automorphism. Such projections are expected in a construction of an inductive limit structure for the irrational rotation algebra that is invariant under the Fourier transform (namely, as two circle algebras of the same dimension, which are swapped by the Fourier transform, plus a few points). The calculation is based on Rieffel's construction of the Schwartz space as an equivalence bimodule of rotation algebras as well as on the theory of theta functions.

The paper considers nonlinear, probability measure-valued dynamical systems that generalise those classical Lotka–Volterra equations in which, to use ecological terminology, the total size of a finite number of populations of interacting species is conserved. In this generalisation there is a ‘different species’ at each point of an arbitrary measurable space.

Such infinitely-many-species analogues of the classical Lotka–Volterra equations appear as hydrodynamic-type limits of stochastic systems of randomly coalescing masses related to those that have been used to model physical and chemical processes of agglomeration, coagulation and condensation.

One natural instance of this generalisation has closed-form solutions, including a family of solutions that exhibit soliton-like behaviour. The large time asymptotics of other classes of examples can be completely described using analogues of Lyapunov function techniques. Moreover, there are conserved quantities in the form of relative entropies that generalise those found by Volterra in the classical case. Finally, each solution has a series expansion as a time-varying, geometric mixture of a fixed sequence of probability measures. The existence of this expansion is related to the fact that the system is in martingale problem duality with a function-valued Markov process.

The following result is well known and easy to prove (see [14, Theorem 2.2.6]).

Theorem 0. If A is a primitive associative Banach algebra, then there exists a Banach space X such that A can be seen as a subalgebra of the Banach algebra BL(X) of all bounded linear operators on X in such a way that A acts irreducibly on X and the inclusion A[rarrhk ]BL(X) is continuous.

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in Theorem 0 are satisfied and even the inclusion A[rarrhk ]BL(X) is contractive.

Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of Theorem 0.

The number (up to isomorphism) of positive-definite, even, unimodular lattices of rank 8r grows rapidly with r. However, Bannai [1] has shown that, when counted according to weight, those with non-trivial automorphisms make up a fraction of the whole, which goes rapidly to zero as r→∞. Therefore it is of some interest to produce families of positive-definite, even, unimodular lattices with large automorphism groups and unbounded ranks.

Suppose that G is a finite group and V is an irreducible ℚ[G]-module such that V[otimes ]ℝ is still irreducible. Then, as observed by Gross [8], the space of G-invariant symmetric bilinear forms on V is one-dimensional and is necessarily generated by a positive-definite form, unique up to scaling by non-zero positive rationals. Thompson [23] showed that, if V is also irreducible modp for all primes p, then it contains an invariant lattice (unique up to scaling) which is even and unimodular with appropriate scaling of the quadratic form. Examples arising in this manner are the E8-lattice of rank 8, the Leech lattice of rank 24 and the Thompson–Smith lattice of rank 248. Gow [6] has also constructed some examples associated with the basic spin representations of 2An and 2Sn.

Let G be a discrete group and (G, G+) be a quasi-ordered group. Set G0+ = G+∩(G+)−1 and G1 = (G+[setmn ]G0+)∪{e}. Let [Fscr ]G1(G) and [Fscr ]G+(G) be the corresponding Toeplitz algebras. In the paper, a necessary and sufficient condition for a representation of [Fscr ]G+(G) to be faithful is given. It is proved that when G is abelian, there exists a natural C*-algebra morphism from [Fscr ]G1(G) to [Fscr ]G+(G). As an application, it is shown that when G = ℤ2 and G+ = ℤ+ × ℤ, the K-groups K0([Fscr ]G1(G)) ≅ ℤ2, K1([Fscr ]G1(G)) ≅ ℤ and all Fredholm operators in [Fscr ]G1(G) are of index zero.

Let $G$ be a separable locally compact group and let $\hat{G}$ be its dual space with Fell's topology. It is well known that the set $P(G)$ of continuous positive-definite functions on $G$ can be identified with the set of positive linear functionals on the group $C^*$ -algebra $C^*(G)$ . We show that if $\pi$ is discrete in $\hat{G}$ , then there exists a nonzero positive-definite function $\phi_{\pi}$ associated with $\pi$ such that $\phi_{\pi}$ is a $w^*$ -strongly exposed point of $P(G)_0$ , where $P(G)_0=\{f\in p(G):f(e)\le 1\}$ . Conversely, if some nonzero positive-definite function $\phi_\pi$ associated with $\pi$ is a $w^*$ -strongly exposed point of $P(G)_0$ , then $\pi$ is isolated in $\hat{G}$ . Consequently, $G$ is compact if and only if, for every $\pi\in\hat{G}$ , there exists a nonzero positive-definite function associated with $\pi$ that is a $w^*$ -strongly exposed point of $P(G)_0$ . If, in addition, $G$ is unimodular and $\pi\in\hat{G}_\rho$ , then $\pi$ is isolated in $\hat{G}_{\rho}$ if and only if some nonzero positive-definite function associated with $\pi$ is a $w^*$ -strongly exposed point of $P_{\rho}(G)_0$ , where $\rho$ is the left regular representation of $G$ and $\hat{G}_{\rho}$ is the reduced dual space of $G$ . We prove that if $B_{\rho}(G)$ has the Radon–Nikodym property, then the set of isolated points of $\hat{G}_{\rho}$ (so square-integrable if $G$ is unimodular) is dense in $\hat{G}_{\rho}$ . It is also proved that if $G$ is a separable SIN-group, then $G$ is amenable if and only if there exists a closed point in $\hat{G}_{\rho}$ . In particular, for a countable discrete non-amenable group $G$ (for example the free group $F_2$ on two generators), there is no closed point in its reduced dual space $\hat{G}_{\rho}$ .

Mahler's measure of a polynomial can be written as a logarithmic integral over the torus. We propose a definition when the underlying group is an elliptic curve. Having reviewed some of the classical results in the toral case, we take some first steps towards realising elliptic analogues. In particular, we focus on elliptic analogues of Kronecker's theorem and Lehmer's problem. We wish to stress the fundamental role played by Jensen's formula in both the toral and elliptic formulations of these results.