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 withproving 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 workof Heath-Brown who proved $(1)$ with exponent $\frac 34$. Under the Riemann hypothesis one can prove$(1)$ withexponent $\frac 12$.The proof of the theorem startswith the Heath-Brown--Linnik identity which leads to aformula giving the number of primes in an interval in terms of coefficients of certain Dirichlet series. Ithen estimate the coefficients by using, among other things, the information which can be gained fromMontgomery's mean value theorem and Huxley's version of the Hal\' asz lemma. Furthermore, by using familiarsieve arguments I am able to discard some of the coefficients allowing us to gain an improvement over theprevious result of Heath-Brown.

1991 Mathematics Subject Classification: 11N05.

Suppose a group $G$ acts on a Gromov-hyperbolic space $X$ properly discontinuously. If the limitset $L(G)$ 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 curvedRiemannian manifold with finite volume, then $H_b^2(\pi _1(M); {\Bbb R})$ is infinite dimensional. As anapplication, we show that if $G$ is a knot group with $G \not\simeq{\Bbb Z}$, then $H^2_b(G;{\Bbb R})$ isinfinite dimensional.

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

This paper is concerned with rationalrepresentations of reductive algebraic groups over fields of positive characteristic $p$.Let $G$ be a simplealgebraic group of rank $\ell$.It is shown that a rational representation of $G$ is semisimple provided thatits 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 finitelist of exceptional modules.

Suppose that $L_1$ and $L_2$ are irreducible representations of $G$. Theessential 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, therestricted Lie algebra cohomology with coefficients in $L_i$ is determined, and the decomposition of thecorresponding Weyl modules is analysed. These data are then exploited to obtain the needed control of theextension 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 ofelliptic operators converge to $L^2$-determinants in a tower of coverings. A product formula is proved, which provides anEuler-product expansion to regularized determinants, having generalized $L^2$-determinants asEuler-factors.

1991 Mathematics Subject Classification: 58G26, 58F20.

Much is knownabout the connection between the growth and decay of subharmonic functions. Theresults indicate that there is a general principle: {\em asubharmonic functioncannot decay ‘too fast’ relative to its growth}.Three theorems areproved which, together with work previously published elsewhere, give a fairlycomplete account of howthis principle works out for a subharmonic functionhaving extremal decay along a ray.

1991 Mathematics SubjectClassification: 30D20, 31A05.

Given a generalized octagonwhich satisfies certain geometric conditions, we show that it must be isomorphic to the Ree--Tits octagonarisingnaturally from a Ree group of type $^2F_4$ over a perfect field of characteristic 2, in which theFrobenius automorphism has a square root.

1991 Mathematics Subject Classification: 51E12.