It is proved that Epstein's zeta-function $\zeta_Q(s)$, related to a positive definite integral binary quadratic form, has a zero $ 1/2 +i\gamma $ with $T \leq \gamma \leq T+T^{5/11+\varepsilon }$ for sufficiently large positive numbers $T$. This improves a classical result of H. S. A. Potter and E. C. Titchmarsh (Proc. London Math. Soc. (2) 39 (1935) 372–384).

In this paper we use the Hausdorff metric to prove that two compact metric spaces are homeomorphic if and only if their canonical complements are uniformly homeomorphic. We take one of the two steps needed to prove that the difference between the homotopical and topological classifications of compact connected ANRs depends only on the difference between continuity and uniform continuity of homeomorphisms in their canonical complements, which are totally bounded metric spaces. The more important step was provided by the Chapman complement and the Curtis–Schori–West theorems. We also improve the multivalued description of shape theory given by J. M. R. Sanjurjo but only in the class of locally connected compacta.

The question of which Riemannian manifolds admit simple closed geodesics is still a mystery. It is not known whether all closed Riemannian manifolds contain simple closed geodesics. For closed manifolds with nontrivial fundamental group, a simple closed geodesic can always be found by taking the shortest homotopically nontrivial closed geodesic. When the manifold is closed but simply connected, the question is open for dimensions three and above. In dimension two, it is known by the theorem of Lusternik and Schnirelmann [6] (see also [3] and [4]) that the 2-sphere equipped with any smooth Riemannian metric contains at least three distinct simple closed geodesics.

Non-compact manifolds do not necessarily contain closed geodesics, Euclidean space being an obvious example. Even if the manifold is not simply connected, it may not contain any simple closed geodesics, as with the hyperbolic thrice-punctured sphere. However, among the orientable, finite area, complete hyperbolic 2-manifolds, the thrice-punctured sphere is the only example that contains no simple closed geodesic. In this paper, we shall determine which orientable hyperbolic 3-manifolds do and do not contain simple closed geodesics. We shall prove that the Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite, or even that it has finitely generated fundamental group. The simple closed geodesic which we produce arises from an interesting class of elements of the fundamental group. It is the shortest closed geodesic corresponding to a screw motion induced by the action of the fundamental group on hyperbolic 3-space.

In proving our result, we use geometric methods to obtain results about isometries of hyperbolic 3-space. These results apply to elliptic, as well as to parabolic and loxodromic isometries, and we state them in full generality.

A related question is whether a hyperbolic 3-manifold always contains a non-simple closed geodesic. In [2], Alan Reid and Ted Chinburg utilized arithmetic hyperbolic 3-manifold theory to construct examples of closed hyperbolic 3-manifolds in which every closed geodesic is simple.

We shall be working with orientable 3-manifolds, so all of the isometries discussed will be orientation preserving.

Let X be a smooth complex projective curve of genus g [ges ] 1. If g [ges ] 2, then assume further that X is either bielliptic or with general moduli. Fix integers r, s, a, b with r > 1, s > 1 and as [les ] br. Here we prove the existence of an exact sequence

formula here

of semistable vector bundles on X with rk(H) = r, rk(Q) = s, deg(H) = a and deg(Q) = b.

By a Kleinian group we mean a discrete subgroup of PSL(2, [Copf ]). We prove that abelian subgroups of finitely generated Kleinian groups are separable. In other words, if M=ℍ3/Γ is a hyperbolic 3-orbifold, with Γ finitely generated, then abelian subgroups of Γ are separable in Γ.

Let $X$ be the Shimura curve corresponding to the quaternion algebra over $\Q$ ramified only at 3 and 13. B. Jordan showed that $X_{\Q(\sqrt{-13})}$ is a counterexample to the Hasse principle. Using an equation of $X$ found by A. Kurihara, it is shown here, by elementary means, that $X$ has no $\Q(\sqrt{-13})$-rational divisor classes of odd degree. A corollary of this is the fact that this counterexample is explained by the Manin obstruction.

Throughout this paper, all 3-manifolds are closed and orientable. Our aim is to give some new examples of 3-manifolds which are virtual bundles, that is, have finite sheeted covers which are surface bundles over the circle. Thurston has raised the question as to whether all irreducible atoroidal 3-manifolds might have this property. The geometrisation conjecture would imply that all such 3-manifolds have hyperbolic metrics, and the virtual bundle question is then equivalent to showing the existence of geometrically infinite incompressible surfaces (compare [14]).

Not much progress has been made on this problem. Recently, Reid [9] has given an explicit example using an arithmetic hyperbolic 3-manifold, and Cooper, Long and Reid [3] have given methods of constructing immersed incompressible surface in surfaces bundles over the circle, which are sometimes geometrically infinite. In [10], examples are given of horizontal immersed incompressible surfaces in graph manifolds which do not lift to embeddings in any finite sheeted covering spaces. Similar examples are given in toroidal manifolds with cubings of non-positive curvatures in [9]. So the assumption that the manifold has the atoroidal property is essential in Thurston's question. Our aim is to give several simple constructions of large classes of examples using different polyhedral metrics of non-positive curvature on 3-manifolds (compare [2]).

We follow closely the construction of Thurston (compare [13]), where he observes that the right-angled regular hyperbolic dodecahedron can be viewed as a cube with arcs drawn on each face through midpoints of a pair of opposite sides so that no two such arcs share a common vertex. Then the foliation of the cube by planes orthogonal to a diagonal gives an induced foliation of any 3-manifold arising from a torsion-free finite index subgroup of the symmetry group of the tessellation of hyperbolic 3-space by the right-angled dodecahedron. Clearly, any leaf of this foliation is then an immersed geometrically infinite surface. One explicit example is then the 4-fold cyclic branched cover of the Borromean rings, since the Borromean rings has a flat 2-fold branched cover which is obtained by gluing two cubes together. This example is formed by a subgroup of index 4 in the above symmetry group.

Our examples come directly from this observation of Thurston, using different ways of dividing a 3-manifold into simple polyhedra so that there is an overall metric of non-positive curvature. We use some well-known tessellations of Euclidean 3-dimensional space, described in Coxeter [4], by truncated octahedra and tetrahedra, plus flying saucers (compare [2]), as well as the easy case of cubes.

In the final section we show that our classes of cubed examples and flying saucers admit geometric decompositions in the sense of Thurston, since they are virtual bundles. We would like to thank the referee for a number of helpful comments which improved the exposition.

We prove that Yuzvinskii's entropy addition formula cannot be extended for the action of certain finitely generated nonamenable groups.

The Borcherds isomorphism is proved to be Hecke equivariant if one considers multiplicative Hecke operators acting on the integral weight meromorphic modular forms. This answers a part of a question of Borcherds (see ‘Automorphic forms on ${\rm O}_{s+2,2}(R)$ and infinite products’, Invent. Math. 120 (1995) 161–213, 17.10), using his suggestion to define the multiplicative Hecke operators.

A result is proved which implies the following conjecture of Osgood and Yang from 1976: if $f$ and $g$ are non-constant entire functions, such that $ T(r, f) = O(T(r, g))$ as $ r \to \infty $ and such that $g(z) \in {\mathbb Z}$ implies that $f(z) \in {\mathbb Z}$, then there exists a polynomial $G$ with coefficients in ${\mathbb Q}$, such that $G({\mathbb Z}) \subseteq {\mathbb Z}$ and $f = G \circ g$.

A surprising relationship is established in this paper, between the behaviour modulo a prime $p$ of the number $s_n({\cal G})$ of index $n$ subgroups in a group ${\cal G}$ , and that of the corresponding subgroup numbers for a normal subgroup in ${\cal G}$ with cyclic quotient of $p$ –power order. The proof relies, among other things, on a twisted version due to Philip Hall of Frobenius' theorem concerning the equation $x^m=1$ in finite groups. One of the applications of this result, presented here, concerns the explicit determination modulo $p$ of $s_n({\cal G})$ in the case when ${\cal G}$ is the fundamental group of a tree of groups all of whose vertex groups are cyclic of $p$ –power order. Furthermore, a criterion is established (by a different technique) for the function $S_n({\cal G})$ to be periodic modulo $p$ .

The problem of classifying finitely generated groups up to virtual isomorphism is strictly harder than that of classifying them up to isomorphism, in the sense that there is a Borel reduction from the isomorphism problem to the virtual isomorphism problem, but not vice versa.Research partially supported by NSF Grants.

Let X be a space that has the homotopy type of a finite simply connected CW complex. We denote by [Escr ](X) the group of homotopy classes of self-homotopy equivalences of X. This group has been extensively studied (see [1] for a survey). In this paper we consider the subgroup [Escr ]n#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on the homotopy groups πi(X) for i[les ]n, and the subgroup [Escr ]#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on all the homotopy groups. Our first result is as follows.

Let $V$ be a commutative valuation domain of arbitrary Krull-dimension, with quotient field $F$, let $K$ be a finite Galois extension of $F$ with group $G$, and let $S$ be the integral closure of $V$ in $K$. Suppose that one has a 2-cocycle on $G$ that takes values in the group of units of $S$. Then one can form the crossed product of $G$ over $S$, $S\ast G$, which is a $V$-order in the central simple $F$-algebra $K\ast G$. If $S\ast G$ is assumed to be a Dubrovin valuation ring of $K\ast G$, then the main result of this paper is that, given a suitable definition of tameness for central simple algebras, $K\ast G$ is tamely ramified and defectless over $F$ if and only if $K$ is tamely ramified and defectless over $F$. The residue structure of $S\ast G$ is also considered in the paper, as well as its behaviour upon passage to Henselization.

Let $G$ be a finitely generated, discrete group of polynomial growth. This paper provides a proof of the boundedness in $L_p(G)$, $1<p<\infty$, of some higher-order Riesz transform operators associated with a probability density on $G$.

Let f[ratio ]C→Ĉ be a function which is either transcendental meromorphic or rational with degree at least 2. We discuss the uniform perfectness of the attracting or parabolic cycle of stable domains of f(z), and include a proof that the Julia set of a meromorphic function of finite type is uniformly perfect.

We prove for a wide class of structures that if [Fscr ] is a family of substructures which are pairwise uniformly commensurable, then there is a commensurable substructure invariant under automorphisms stabilizing [Fscr ] setwise.

Let G be a finite simple group. A conjecture of J. D. Dixon, which is now a theorem (see [2, 5, 9]), states that the probability that two randomly chosen elements x, y of G generate G tends to 1 as [mid ]G[mid ]→∞. Geoff Robinson asked whether the conclusion still holds if we require further that x, y are conjugate in G. In this note we study the probability Pc(G) that 〈x, xy〉=G, where x, y∈G are chosen at random (with uniform distribution on G×G). We shall show that Pc(G)→1 as [mid ]G[mid ]→∞ if G is an alternating group, or a projective special linear group, or a classical group of bounded dimension. In fact, some (but not all) of the exceptional groups of Lie type will also be dealt with.

The authors prove in this paper that an involutory Hopf algebra with non-zero integral over a field of characteristic zero is cosemisimple.

Let (R, [mfr ]) be a local Noetherian ring. We show that if R is complete, then an R-module M satisfies local duality if and only if the Bass numbers μi([mfr ], M) are finite for all i. The class of modules with finite Bass numbers includes all finitely generated, all Artinian, and all Matlis reflexive R-modules. If the ring R is not complete, we show by example that modules with finite Bass numbers need not satisfy local duality. We prove that Matlis reflexive modules satisfy local duality in general, where R is any local ring with a dualizing complex.