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

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

We obtain two refinements of the socalled local duality of ultrapowers, that is, the ultrapower version of thewell-known principle of local reflexivity.

1991 Mathematics Subject Classification. 46B07, 46B08.

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

1991 Mathematics Subject Classification. 46J20, 46M20.

Weprove that exponentially harmonic morphisms are precisely the Riemanniansubmersions with minimal fibres.

1991 Mathematics Subject Classification 58E20.

Let L^2_a (D, d\sigmad\theta /2\pi ) be a complete weighted Bergman space on the open unitdisc D, where d\sigma is a positive finiteBorel measure on [0, 1). We show the following : when \phi isa 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.

Inthis paper, groups are investigated in which all subgroups, all normal subgroups,or all characteristic subgroups have a proper supplement. This supplement can beeither an arbitrary subgroup, a normal or a characteristic subgroup, resulting innine classes of groups. Properties of these classes are studied such ascontainment and closure properties, and characterizations for several of theseclasses are given.

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

Let A be anoetherian local ring, x a non-unit element of A,B=A/(x). Let E be the Koszul complex associated to anarbitrary set of generators of the ideal (x) of A. Assume thatH1(E) is a free B-module. Then Ais Gorenstein if and only if B isalso.

1991 Mathematics Subject Classification 13H10, 13D03.

We extendsplitting theorems due to Zaicev and Duan proving the following result. LetG be a locally soluble FC-hypercentral group and letA be a periodic artinian ℤG-module. IfA has no finite ℤG-submodules then anyextension E of A by Gsplits conjugately over A.

1991 Mathematics Subject Classification 20F19.

For eachm≥1, u_{m}(G) is defined to be theintersection of the normalizers of all the subnormal subgroups of defect at mostm in G. An ascending chain of subgroupsu_{m,i}(G) is defined by settingu_{m,i}(G)/u_{m,i−1}(G)=u_{m}(G/u_{m,i−1}(G)). Ifu_{m,n}(G)=G, for some integer n, them-Wielandt length of G is theminimal of such n.

In [3], Bryce examined thestructure of a finite soluble group with given m-Wielandtlength, in terms of its polynilpotent structure. In this paper we extend hisresults to infinite soluble groups.

1991 Mathematics SubjectClassification. 20E15, 20F22.

We define the notion of lightcone Gauss maps,lightcone pedal curves and lightcone developables of spacelike curves inMinkowski 3-space and establish the relationships between singularities of theseobjects and geometric invariants of curves under the action of the Lorentzgroup.

It is wellknown that a compact embedded hypersurface of the Euclidean space withoutboundary is a round sphere if one of mean curvature functions is constant. Inthis note, we show that a compact embedded hypersurface of the Euclidean space(and other constant curvature spaces) without boundary is a round sphere if theratio of some two mean curvature functions isconstant.

1991 Mathematics Subject Classification 53C40, 53C20.

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

In the first part of the paper we establish, for a rather general[Ascr], the relationship between reflectivity and covariantfiniteness of [Fscr] in [Ascr], andgeneralize Freyd's adjoint functor theorem (for inclusion functors) to notnecessarily complete categories. Also, we study the good behaviour of reflectionswith respect to direct limits. Most results in this part are dualizable, thusproviding corresponding versions for coreflective subcategories.

In thesecond half of the paper we give several examples of reflective subcategories ofabelian and module categories, mainly of subcategories of the form Copres(M) and Add (M). The second case covers thestudy 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 thispaper we study the problem of the existence on non-inner automorphisms for theclass of torsion-free supersolvable groups, answering a question raised byRobinson.

1991 Mathematics Subject Classification 20F16, 20F28.

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

1991 Mathematics Subject Classification 30C99, 46E35.

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

1991 Mathematics Subject Classification 53C15, 53C40.

Inthis paper, we prove that if M^2 is a complete maximalspacelike surface of an anti-de Sitter space {\bfH}^{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 ofM^{2}. Also

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

(2) S={-4c\over3} if and only if M^2 is the hyperbolic Veronesesurface;

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

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

1991 Mathematics Subject Classifaction 53C40, 53C42.