Volume 11, Issue 3

The purpose of this paper is to show that the automorphism groups of many of the ‘cycle-free’ partial orders studied in [Warren, R.: The structure of k-C S -transitive cycle-free partial orders. Memoirs of the American Mathematical Society (1997), to appear] and [Creed, P., Truss, J. K. and Warren, R.: The structure of k-C S -transitive cycle-free partial orders having infinite chains, to appear] are simple. This contrasts strongly with the situation for trees, of which they form a natural generalization. It was shown in [Droste, M., Holland, W.C. and Macpherson, H.D.: Automorphism groups of infinite semilinear orders (I) and (II). Proc. London Math. Soc. 58 (1989), 454–478 and 479–494] that the automorphism group of any sufficiently transitive tree has at least normal subgroups. All the infinite chain cycle-free partial orders studied in [Creed, P., Truss, J. K. and Warren, R.: The structure of k-C S -transitive cycle-free partial orders having infinite chains, to appear] have simple automorphism groups. The finite chain case is more involved; where the ordering on chains of the Dedekind-MacNeille completion can be expressed as a lexicographic product by a non-trivial discrete (transitive) ordering (respected by the group), the automorphism group is not simple. For both finite and infinite chain cases the simple automorphism groups split into two classes: those where there is a bound (≤ 2) on the number of conjugates required to express one non-identity element in terms of another, and those in which there is no such bound.

A set of Beurling generalized integers consists of the unit n 0 = 1 plus the set n 1 ≤ n 2 ≤ … of all power products of a set of generalized primes 1 < g 1 ≤ g 2 ≤ g 3 ≤ … with g i → ∞, with these power products arranged in increasing order and counted with multiplicity. We say that has the Delone property if there are positive constants r , R such that R ≥ n i + 1 – n i ≥ r for all i ≥ 1. Any set with the Delone property has unique factorization into irreducible elements and is therefore a subsemigroup of ℝ + . We classify all such semigroups which are contained in the integers . The set of generalized primes of any such consists of all but finitely many primes, plus finitely many other composites.

We use geometric algebra and the theory of automorphic forms to realize theta series attached to an indefinite quadratic form as the sum of a specific Eisenstein series and an L 2 -function. From this we obtain explicit formulas for the measure of the representation of an integer by an indefinite quadratic form.

A polar space with point set P is laxly embedded in the projective space PG( d , q ), d ≥ 2, if the following conditions are satisfied: (i) P is a point set of PG( d , q ) which generates PG( d , q ), and (ii) each line L of is a subset of a line L ′ of ( d , q ), and distinct lines L 1 , L 2 of define distinct lines L ′ 1 , L ′ 2 of PG( d , q ). In this paper we determine all polar spaces of rank at least three which are laxly embedded in PG( d , q ), where d ≥ 4 if is isomorphic to the polar space W (2 m + 1, s), m ≥ 2 and s odd, arising from a symplcctic polarity in PG(2 m + 1, s), and where d ≥ 3 in all other cases. Laxly embedded generalized quadrangles were considered in a foregoing paper.

Let X be a discrete variety of ℂ n . Berenstein and Li [Berenstein, C.A. and Li, B.Q.: Interpolating varieties for weighted spaces of entire functions in ℂ n . Publicacions. Malemàtiqucs 38 (1994), 157–173] obtained a necessary and sufficient analytic condition for X to be interpolating for A p . In this paper, we extend it to the case when X is a disjoint union of complex affine subspaces of ℂ n . As a corollary, we solve an open problem posed by Berenstein and Taylor [Berenstein, C.A. and Taylor, B. A.: On the geometry of interpolating varieties. In: Seminaire Lelong-Skoda (1980/1981). (ed. by P. Lelong and H. Skoda), pp. 1–25. Lecture Notes in Math. 919. Springer-Verlag, New York–Berlin–Heidelberg 1982] on the interpolation, when the subvariety under consideration is a disjoint union of complex affine subspaces.

We construct a class of normal projective embeddings of PSL (2, k ), for k = ℝ and ℂ, which we call boundary compactifications of SL (2, k ). These arise essentially as the Zariski closures of orbits in under the diagonal action of SL (2, k ). In addition, we determine precisely when our examples can be SL (2, k )-homeomorphic, showing that the resulting deformation space is a countable union of positive-dimensional families.