We investigate the structure of transitive, untwisted, superlinked finite covers whose kernels are central in their automorphism groups. We introduce the concept of an $n$-conjugate system for a pair $(W,K)$, where $W$ is a permutation structure, $K$ is a finite abelian group and $n\in\omega+1$. This concept allows us to characterize the given class of finite covers for structures $W$ which satisfy a certain connectedness condition; further, the irreducibility of such a cover is equivalent to a simple condition on a corresponding $n$-conjugate system. Finally, we consider structures with strong types, for which there is a much simpler characterization.

1991 Mathematics Subject Classification: 03C35, 20B27.

Let $p_n$ be the $n$th prime. Then this paper is concerned with proving the following result on the distribution of consecutive primes.

Theorem.}\begin{equation}\sum_{p_{n+1}-p_n>x^{\frac 12},\ x \leq p_n \leq 2x} (p_{n+1}-p_n) \llx^{\frac{25}{36}+\epsilon}.\end{equation}

The exponent of $x$ in this theorem improves on the work of Heath-Brown who proved $(1)$ with exponent $\frac 34$. Under the Riemann hypothesis one can prove$(1)$ with exponent $\frac 12$.The proof of the theorem starts with the Heath-Brown--Linnik identity which leads to a formula giving the number of primes in an interval in terms of coefficients of certain Dirichlet series. I then estimate the coefficients by using, among other things, the information which can be gained from Montgomery's mean value theorem and Huxley's version of the Hal\' asz lemma. Furthermore, by using familiar sieve arguments I am able to discard some of the coefficients allowing us to gain an improvement over the previous result of Heath-Brown.

1991 Mathematics Subject Classification: 11N05.

Suppose a group $G$ acts on a Gromov-hyperbolic space $X$ properly discontinuously. If the limit set $L(G)$ of the action has at least three points, then the second bounded cohomology group of $G$,$H^2_b(G;{\Bbb R})$ is infinite dimensional. For example, if $M$ is a complete, pinched negatively curved Riemannian manifold with finite volume, then $H_b^2(\pi _1(M); {\Bbb R})$ is infinite dimensional. As an application, we show that if $G$ is a knot group with $G \not\simeq{\Bbb Z}$, then $H^2_b(G;{\Bbb R})$ is infinite dimensional.

1991 Mathematics Subject Classification: primary 20F32; secondary 53C20, 57M25.

This paper is concerned with rational representations of reductive algebraic groups over fields of positive characteristic $p$. Let $G$ be a simple algebraic group of rank $\ell$. It is shown that a rational representation of $G$ is semisimple provided that its dimension does not exceed $\ell p$. Furthermore, this result is improved by introducing a certain quantity $\mathcal{C}$ which is a quadratic function of $\ell$. Roughly speaking, it is shown that any rational $G$ module of dimension less than $\mathcal{C} p$ is either semisimple or involves a subquotient from a finite list of exceptional modules.

Suppose that $L_1$ and $L_2$ are irreducible representations of $G$. The essential problem is to study the possible extensions between $L_1$ and $L_2$ provided $\dim L_1 + \dim L_2$ is smaller than $\mathcal{C} p$. In this paper, all relevant simple modules $L_i$ are characterized, the restricted Lie algebra cohomology with coefficients in $L_i$ is determined, and the decomposition of the corresponding Weyl modules is analysed. These data are then exploited to obtain the needed control of the extension theory.

1991 Mathematics Subject Classification: 20G05.

A notion of $L^2$-determinant is established generalizing $L^2$-torsion. It is shown that classical regularized determinants of elliptic operators converge to $L^2$-determinants in a tower of coverings. A product formula is proved, which provides an Euler-product expansion to regularized determinants, having generalized $L^2$-determinants as Euler-factors.

1991 Mathematics Subject Classification: 58G26, 58F20.

Much is known about the connection between the growth and decay of subharmonic functions. The results indicate that there is a general principle: {\em asubharmonic function cannot decay ‘too fast’ relative to its growth}.Three theorems are proved which, together with work previously published elsewhere, give a fairly complete account of how this principle works out for a subharmonic function having extremal decay along a ray.

1991 Mathematics Subject Classification: 30D20, 31A05.

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.