We investigate the structure of groups satisfying a positi¨e law, that is, an identity of the form u '¨, where u and¨are positive words. The main question here is whether all such groups are nilpotent-by-finite exponent. We answer this question affirmatively for a large class C C of groups including soluble and residually finite groups, showing that moreover the nilpotency class and the finite exponent in question are bounded solely in terms of the length of the positive law. It follows, in particular, that if a variety of groups is locally nilpotent-by-finite, then it must in fact be contained in the product of a nilpotent variety by a locally finite variety of finite exponent. We deduce various other corollaries, for instance, that a torsionfree, residually finite, n-Engel group is nilpotent of class bounded in terms of n. We also consider incidentally a question of Bergman as to whether a positive law holding in a generating subsemigroup of a group must in fact be a law in the whole group, showing that it has an affirmative answer for soluble groups. ᮊ 1997 Academic Press 510

This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class <& including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class <€ is both finite-of-exponent-*?(n)-bynilpotent-of-class< c(n) and nilpotent-of-class< c(«)-by-finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

If w = 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class S* of groups including all residually or locally soluble-or-finite groups. In fact the groups of 5? satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of w alone. This yields a dichotomy for words. Finally, if the law w = 1 satisfies a certain additional condition-obtaining in particular for any monoidal or Engel law-then the conclusion extends to the much larger class consisting of all 'locally graded' groups.2000 Mathematics subject classification: primary 20F19, 20E10, 20F45.

A case of cerebellar hemangioblastoma and coexistent arteriovenous malformation (AVM) is presented. Angiography displayed the AVM, but histological examination revealed a coexisting hemangioblastoma. Various theories concerning the etiology of this condition are discussed.