In this paper we conduct a systematic computerized search for groups generated by small, but highly symmetric, sets of elements of order 3. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. Firstly, we introduce monomial modular representations as these will prove useful later in the paper. Then the techniques of symmetric generation developed elsewhere are described afresh. The results we obtain are presented in a convenient tabular form, together with relevant character tables.

The McLaughlin sporadic simple group McL is the flag-transitive automorphism group of a Petersen-type geometry [Gscr ] = [Gscr ](McL) with the diagram

diagram here

where the edge in the middle indicates the geometry of vertices and edges of the Petersen graph. The elements corresponding to the nodes from the left to the right on the diagram P33 are called points, lines, triangles and planes, respectively. The residue in [Gscr ] of a point is the P3-geometry [Gscr ](Mat22) of the Mathieu group of degree 22 and the residue of a plane is the P3-geometry [Gscr ](Alt7) of the alternating group of degree 7. The geometries [Gscr ](Mat22) and [Gscr ](Alt7) possess 3-fold covers [Gscr ](3Mat22) and [Gscr ](3Alt7) which are known to be universal. In this paper we show that [Gscr ] is simply connected and construct a geometry [Gscr ]˜ which possesses a 2-covering onto [Gscr ]. The automorphism group of [Gscr ]˜ is of the form 323McL; the residues of a point and a plane are isomorphic to [Gscr ](3Mat22) and [Gscr ](3Alt7), respectively. Moreover, we reduce the classification problem of all flag-transitive Pmn-geometries with n, m [ges ] 3 to the calculation of the universal cover of [Gscr ]˜.

Let G be a connected real Lie group and let [gfr ] be its Lie algebra. We shall denote by [qfr ] ⊂ [gfr ] the radical of [gfr ]. Let [gfr ] = [qfr ] [ltimes ] [sfr ] (where [sfr ] is semisimple or 0) be a Levi decomposition of [gfr ] (cf. [11]). When [sfr ] ≠ 0 we can apply the Iwasawa decomposition on [sfr ] (cf. [8]) [sfr ] = [nfr ] [oplus ] [afr ] [oplus ] [kfr ], where [nfr ] is nilpotent and [afr ] is abelian and normalizes [nfr ] so that [nfr ] [oplus ] [afr ] is a soluble algebra. Since [nfr ] [oplus ] [afr ] normalizes [qfr ] it is clear that [rfr ] = [qfr ] [oplus ] [nfr ] [oplus ] [afr ] is a soluble Lie algebra of [gfr ]. By Lie's theorem (cf. [11]) we can find a basis on [rfr ]c = [rfr ] [otimes ] C for which the adjoint action of [rfr ] on [rfr ]c takes a triangular form. Let us denote by λ1(x); λ2(x), …, λn(x), x ∈ [rfr ] the corresponding eigenvalues. The λj's can be identified with elements of Homℝ([rfr ], C) and are called the roots of the adjoint action of [rfr ]. Let us denote by [Lscr ] = {L1, …, Lk} the set of the non zero real parts of the λj's. We shall say that the group G is a B-group if [Lscr ] ≠ &0slash; and if there exist α1, …, αk [ges ] 0, [sum ]kj=1 αj = 1, such that [sum ]kj=1 αjLj = 0. Otherwise we say that G is an NB-group. It can be shown that the above definition is independent of the particular choice of the Levi and Iwasawa decompositions that are used (cf. [13]).

We shall denote by dlg = dg (resp. drg) the left (resp. right) Haar measure on G and by m(g) = drg/dlg the modular function.

Let [Xscr ] = {X1, X2, …, Xn} be left invariant fields on G that verify the Hörmander condition (cf. [15]) and let Δ = −[sum ]X2j be the corresponding sub-Laplacian. Δ is formally self adjoint on the Hilbert space L2(G, drg) and the spectral gap of Δ is defined by

formula here

We show that for an arbitrary locally compact group G and for E in a certain class of closed subsets of the primitive ideal space of L1(G), the kernel k(E) has a bounded approximate unit. This generalizes some well-known previous results.

Recently Calabi–Yau threefolds have been studied intensively by physicists and mathematicians. They are used as physical models of superstring theory [Y] and they are one of the building blocks in the classification of complex threefolds [KMM]. These are three dimensional analogues of K3 surfaces. However, there is a fundamental difference as is to be expected. For K3 surfaces, the moduli space N of K3 surfaces is irreducible of dimension 20, inside which a countable number of families Ng with g [ges ] 2 of algebraic K3 surfaces of dimension 19 lie as a dense subset. More explicitly, an element in Ng is (S, H), where S is a K3 surface and H is a primitive ample divisor on S with H2 = 2g − 2. For a generic (S, H), Pic (S) is generated by H, so that the rank of the Picard group of S is 1. A generic surface S in N is not algebraic and it has Pic (S) = 0, but dim N = h1(S, TS) = 20 [BPV]. It is quite an interesting problem whether or not the moduli space M of all Calabi–Yau threefolds is irreducible in some sense [R]. A Calabi–Yau threefold is algebraic if and only if it is Kaehler, while every non-algebraic K3 surface is still Kaehler. Inspired by the K3 case, we define Mh,d to be {(X, H)[mid ]H3 = h, c2(X) · H = d}, where H is a primitive ample divisor on a smooth Calabi–Yau threefold X. There are two parameters h, d for algebraic Calabi–Yau threefolds, while there is only one parameter g for algebraic K3 surfaces. (Note that c2(S) = 24 for every K3 surface.) We know that Ng is of dimension 19 for every g and is irreducible but we do not know the dimension of Mh,d and whether or not Mh,d is irreducible. In fact, the dimension of Mh,d = h1(X, TX), where (X, H) ∈ Mh,d. Furthermore, it is well known that χ(X) = 2 (rank of Pic (X) − h1(X, TX)), where χ(X) is the topological Euler characteristic of X. Calabi–Yau threefolds with Picard rank one are primitive [G] and play an important role in the moduli spaces of all Calabi–Yau threefolds. In this paper we give a bound on c3 of Calabi–Yau threefolds with Picard rank 1.

The Kähler case of Riemannian homogeneous structures [3, 15, 18] has been studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a classification of Kähler homogeneous structures, which has four primitive classes [Kscr ]1, …, [Kscr ]4 (see [6, theorem 5·1] for another proof and Section 2 below for the result). The purpose of the present paper is to prove the following result:

THEOREM 1·1. A simply connected irreducible homogeneous Kähler manifold admits a nonvanishing Kähler homogeneous structure in Abbena–Garbiero's class [Kscr ]2 [oplus ] [Kscr ]4if and only if it is the complex hyperbolic space equipped with the Bergman metric of negative constant holomorphic sectional curvature.

From the viewpoint of Birman and Menasco, a particular type of singular foliation on a surface with boundary induces an embedding of the surface in 3-space, such that the boundary of the surface is braided relative to the z-axis; hence the foliation determines a conjugacy class in the braid group. Birman and Hirsch describe an explicit algorithm to find a braid word representing this conjugacy class, given the foliation. A braid word β ∈ Bn is said to be irreducible if it is not conjugate to a braid of the form bσ±1n−1, with b ∈ Bn−1. We exhibit a foliation of a disk and show that the corresponding braid word is an irreducible element of B4. We give an explicit geometric description of the embedding induced by the foliation and describe a particularly nice form of symmetry possessed by this example.

Let M be a complete hyperbolic n-manifold of finite volume. By a systole of M we mean a shortest closed geodesic in M. By the systole length of M we mean the length of a systole. We denote this by sl (M). In the case when M is closed, the systole length is simply twice the injectivity radius of M. In the presence of cusps, injectivity radius becomes arbitrarily small and it is for this reason we use the language of ‘systole length’.

In the context of hyperbolic surfaces of finite volume, much work has been done on systoles; we refer the reader to [2, 10–12] for some results. In dimension 3, little seems known about systoles. The main result in this paper is the following (see below for definitions):

A family of sets {Fd}d is said to be ‘represented by the measure μ’ if, for each d, the set Fd comprises those points at which the local dimension of μ takes some specific value (depending on d). Finding the Hausdorff dimension of these sets may then be thought of as finding the dimension spectrum, or multifractal spectrum, of μ. This situation pertains surprisingly often, with many familiar families of sets representable by measures which have simple dimension spectra. Examples are given from Diophantine approximation, Kleinian groups and hyperbolic dynamical systems.

In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension.

We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire's theorem. We also give a constructive proof, avoiding Baire's theorem, of the existence of universal Taylor series in any arbitrary simply connected domain.

Mourre's commutator method is used together with Besov spaces, in this paper, to settle the question of uniqueness of solutions to the scattering problem for the acoustic wave propagation in perturbed stratified fluids in Rn with n [ges ] 2: The uniqueness of solutions is established by introducing a radiation condition in the framework of Besov spaces, and is then used to prove resolvent estimates in the Besov space setting for the acoustic propagator in perturbed stratified fluids. These estimates are ‘sharp’ in a sense made precise in the text and are important in establishing existence of solutions to the scattering problem.