Let X be an infinite, locally finite, almost transitive graph with polynomial growth. We show that such a graph X is the inverse limit of an infinite sequence of finite graphs satisfying growth conditions which are closely related to growth properties of the infinite graph X.

1991 Mathematics Subject Classification. Primary 05C25, Secondary 20F8.

We obtain two refinements of the so called local duality of ultrapowers, that is, the ultrapower version of the well-known principle of local reflexivity.

1991 Mathematics Subject Classification. 46B07, 46B08.

In this paper we deduce the existence of analytic structure in a neighbourhood of a maximal ideal M in the spectrum of a commutative Banach algebra, A, from homological assumptions. We assume properties of certain of the cohomology groups H^n(A,A/M), rather than the stronger conditions on the homological dimension of the maximal ideal the first author has considered in previous papers. The conclusion is correspondingly weaker: in the previous work one deduces the existence of a Gel'fand neighbourhood with analytic structure, here we deduce only the existence of a metric neighbourhood with analytic structure. The main method is to consider products of certain co-cycles to deduce facts about the symmetric second cohomology, which is known to be related to the deformation theory of algebras.

1991 Mathematics Subject Classification. 46J20, 46M20.

We prove that exponentially harmonic morphisms are precisely the Riemannian submersions with minimal fibres.

1991 Mathematics Subject Classification 58E20.

Let L^2_a (D, d\sigma d\theta /2\pi ) be a complete weighted Bergman space on the open unit disc D, where d\sigma is a positive finite Borel measure on [0, 1). We show the following : when \phi is a continuous function on the closed unit disc \bar {D}, T_\phi is compact if and only if \phi = 0 on \partial D.

1991 Mathematics Subject Classification 47B35, 47B07.

In this paper, groups are investigated in which all subgroups, all normal subgroups, or all characteristic subgroups have a proper supplement. This supplement can be either an arbitrary subgroup, a normal or a characteristic subgroup, resulting in nine classes of groups. Properties of these classes are studied such as containment and closure properties, and characterizations for several of these classes are given.

1991 Mathematics Subject Classification Primary 20E34, Secondary 20E15.

Let A be a noetherian local ring, x a non-unit element of A, B=A/(x). Let E be the Koszul complex associated to an arbitrary set of generators of the ideal (x) of A. Assume that H1(E) is a free B-module. Then A is Gorenstein if and only if B is also.

1991 Mathematics Subject Classification 13H10, 13D03.

We extend splitting theorems due to Zaicev and Duan proving the following result. Let G be a locally soluble FC-hypercentral group and let A be a periodic artinian ℤG-module. If A has no finite ℤG-submodules then any extension E of A by G splits conjugately over A.

1991 Mathematics Subject Classification 20F19.

For each m≥1, u_{m}(G) is defined to be the intersection of the normalizers of all the subnormal subgroups of defect at most m in G. An ascending chain of subgroups u_{m,i}(G) is defined by setting u_{m,i}(G)/u_{m,i−1}(G)=u_{m}(G/u_{m,i−1}(G)). If u_{m,n}(G)=G, for some integer n, the m-Wielandt length of G is the minimal of such n.

In [3], Bryce examined the structure of a finite soluble group with given m-Wielandt length, in terms of its polynilpotent structure. In this paper we extend his results to infinite soluble groups.

1991 Mathematics Subject Classification. 20E15, 20F22.

We define the notion of lightcone Gauss maps, lightcone pedal curves and lightcone developables of spacelike curves in Minkowski 3-space and establish the relationships between singularities of these objects and geometric invariants of curves under the action of the Lorentz group.

It is well known that a compact embedded hypersurface of the Euclidean space without boundary is a round sphere if one of mean curvature functions is constant. In this note, we show that a compact embedded hypersurface of the Euclidean space (and other constant curvature spaces) without boundary is a round sphere if the ratio of some two mean curvature functions is constant.

1991 Mathematics Subject Classification 53C40, 53C20.

Given a full subcategory [Fscr ] of a category [Ascr ], the existence of left [Fscr ]-approximations (or [Fscr ]-preenvelopes) completing diagrams in a unique way is equivalent to the fact that [Fscr ] is reflective in [Ascr ], in the classical terminology of category theory.

In the first part of the paper we establish, for a rather general [Ascr ], the relationship between reflectivity and covariant finiteness of [Fscr ] in [Ascr ], and generalize Freyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories.

In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of {\rm Mod}_{R}, and has some connections with the “pure-semisimple conjecture”.

1991 Mathematics Subject Classification 18A40, 16D90, 16E70.

In this paper we study the problem of the existence on non-inner automorphisms for the class of torsion-free supersolvable groups, answering a question raised by Robinson.

1991 Mathematics Subject Classification 20F16, 20F28.

The univalent functions in the diagonal Besov space A_{p}, where 1<p<\infty , are characterized in terms of the distance from the boundary of a point in the image domain. Here A_{2} is the Dirichlet space. A consequence is that there exist functions in A_{p},\ p>2, for which the area of the complement of the image of the unit disc is zero.

1991 Mathematics Subject Classification 30C99, 46E35.

In this paper, we show new results on slant submanifolds of an almost contact metric manifold. We study and characterize slant submanifolds of K-contact and Sasakian manifolds. We also study the special class of three-dimensional slant submanifolds. We give several examples of slant submanifolds.

1991 Mathematics Subject Classification 53C15, 53C40.

In this paper, we prove that if M^2 is a complete maximal spacelike surface of an anti-de Sitter space {\bf H}^{4}_{2}(c) with constant scalar curvature, then S=0, S={-10c\over 11}, S={-4c\over 3} or S=-2c, where S is the squared norm of the second fundamental form of M^{2}. Also

(1) S=0 if and only if M^2 is the totally geodesic surface {\bf H}^2(c);

(2) S={-4c\over 3} if and only if M^2 is the hyperbolic Veronese surface;

(3) S=-2c if and only if M^2 is the hyperbolic cylinder of the totally geodesic

surface {\bf H}^{3}_{1}(c) of {\bf H}^{4}_{2}(c).

1991 Mathematics Subject Classifaction 53C40, 53C42.