Arlan Ramsay

Topologies on measured groupoids

Abstract If an analytic Borel group G has a quasiinvariant measure, it is known that G is actually a locally compact group with the original Borel structure being generated by the topology and the original measure being equivalent to Haar measure. In this paper a variation is given on the known proof which then extends to show that an analytic measured groupoid has a σ-compact, and also a locally compact, inessential reduction which is a topological groupoid. In the σ-compact case, it is proved that every “almost” homomorphism agrees a.e. with a (strict) homomorphism. Also, the topology is used to show that every measured groupoid has a complete countable section ¦7¦ and that every locally compact equivalence relation has a complete transversal ¦3¦. These are further used to show that some results of Feldman et al. ¦7¦ apply in general and that a locally compact groupoid with (continuous) Haar system has sufficiently many non-singular Borel G -sets provided that the orbit measures are atom-free ¦23¦.

The Mackey-Glimm Dichotomy for Foliations and Other Polish Groupoids

Abstract For Polish groups acting on Polish spaces, E. Effros gave a condition under which 14 properties of the orbits or the orbit space of an anti-ergodic character are equivalent. He also raised the question of extending these results to Polish groupoids, and this paper gives an affirmative answer to that question. The paper also includes results about totally disconnected quotient spaces and an application to indecomposable continua.

Nontransitive quasi-orbits in Mackey's analysis of group extensions

G. W. Mackey developed a general method for analyzing the dual of a locally compact group G (always second countable) in terms of the dual of a closed normal subgroup N and the cocycle duals of subgroups of G/1V, provided tha t the action of G on the dual of N is sufficiently regular [9]. Ill this case regularity means tha t every ergodic quasi-invariant measure under the action of G is concentrated on an orbit, which means tha t the associated quasi-orbit is transitive. The theory of virtual groups was introduced by Mackey for the purpose of dealing with the less regular case [11, 12]. Section 9 of this paper gives proofs tha t the theorems of section 8 of [9] remain valid in the more general setting. I t should be remarked tha t this leaves work yet to be done before a complete understanding of the general ease is achieved. For instance, one of the theorems establishes a one-one correspondence between part of the dual of G and the co-dual of a certain virtual group for a certain eoeyele co, but an example due to C. C. Moore shows tha t the lat ter can be empty [1]. This example is discussed in section 10 of this paper, and shows tha t representations of virtual groups need not decompose into pr imary representations. The organization of the paper is as follows. The first six sections deal with the machinery of inducing representations from one group action to another. More particularly, sections 1 and 2 give preliminary material on 1-Iilbert bundles and bundle representations of groupoids. In section 3 this is used to define induced representations, and it is proved tha t the definition given is an extension of the definition for subgroups. One novelty here is the proof of Proposition 3.4, which uses no special choice of Radon-Nikodym derivatives. In section 4 a lemma needed in later sections is proved, concerning intertwining operators.

Boolean duals of virtual groups

Abstract The purpose of this paper is to formulate the notion of virtual group and that of a homomorphism between two in such a way that no null sets are involved. That is, we replace the point set by the measure algebra of Borel sets modulo null sets, and the various functions involved must be described by objects associated to these measure algebras, such as σ-homomorphisms. In the process it is necessary to redefine functions satisfying algebraic identities almost everywhere to get new ones satisfying the identities in a stricter sense. Similar refinements on quasiinvariance of measures are also required.

Probability measures on solenoids corresponding to fractal wavelets

The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen in (12) is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in R d and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized Baumslag-Solitar group. In contrast, the fiber measures for filters on inflated fractal spaces cannot be atomic, and thus can support neither MSF wavelets nor systems of imprimitivity.

Non-monomial Multiplier Representations of Abelian Groups

Abstract It has been proved by K. C. Hannabuss and by Roger Howe that if G is locally compact and nilpotent and σ is a type I multiplier on G , then ( G , σ ) is a monomial group: every irreducible σ-representation of G is induced by a one-dimensional σ-representation of a closed subgroup. The converse of this was proved in the special case in which G is discrete and abelian by A. Carey and W. Moran. In this paper we give a different proof for the case in which G is abelian but not necessarily discrete. The method is basically that of virtual little groups. If ( G , σ ) is not type I, we find quasiorbits in the dual of a certain closed subgroup of G that are associated with irreducible σ-representations of G , but not with irreducible σ-representations that are monomial.

Lacunary sections for locally compact groupoids

It is proved that every second countable locally Hausdorff and locally compact continuous groupoid has a Borel set of units that meets every orbit and is what is called ‘lacunary’, a property that implies that the intersection with every orbit is countable.

Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models

We take as primary model of the hyperbolic plane an abstract surface S in the sense of Section 5.3, whose geometry is determined by the methods of differential geometry in such a way that all the axioms of the hyperbolic plane are satisfied. We consider several different coordinate systems, each of which covers the entire surface S, some of which are more useful than others for certain purposes. Each coordinate system leads to one of the classical models of the hyperbolic plane based on Euclidean geometry. Differential geometry is based on analysis, which is based on the real number system ℝ. It follows that the hyperbolic axioms are consistent, if the axioms of ℝ are consistent. It is proved that the axiom system is categorical, in the that any model of the hyperbolic plane is isomorphic to any other model. Lastly, as an amusement, we describe a hyperbolic model of the Euclidean plane.

Some Neutral Theorems of Plane Geometry

We discuss some of those theorems of Euclidean plane geometry that are independent of the parallel axiom. They will be needed in the development of hyperbolic geometry. We assume they are more or less known, so that our treatment is not as complete as a full treatment of Euclidean geometry would be. Some of the theorems are weaker than the corresponding Euclidean ones, because the more complete form would require the parallel axiom. Some of them go a little beyond Euclid in that they use the notion of continuity as it appears in calculus. Results in the last two sections go beyond Euclid in that the ideas in them are more recent, as in the Jordan Curve Theorem or the study of isometries.

ℍ3 and Euclidean Approximations in ℍ2

To investigate the fine structure of hyperbolic geometry, we use a method due essentially to Lobachevski. It turns out that the geometry of the horosphere, a certain surface in three-dimensional hyperbolic space ℍ3, is Euclidean if the definitions are made as follows. The “straight lines” of that geometry are the horocycles in that surface, the “distances” are the arclengths between points along horocycles lying in that surface, and the “angle” between two intersecting horocycles is the angle between the tangent vectors. In that sense, the horosphere, together with the geometry that it inherits from the ℍ3 in which it is embedded, is a model of the Euclidean plane. In it, parallel horocycles are equidistant throughout their entire lengths; there are Cartesian coordinates; and so on. All the formulas and relations of Euclidean geometry apply. Therefore, by projecting small figures in that surface onto a tangent plane in ℍ3, we find that in that plane, for small figures near the point of tangency, the Euclidean formulas hold approximately, with relative error terms that tend to zero as the sizes of the figures tend to zero. In order to carry out this approach, we first develop three-dimensional hyperbolic geometry, which, however, has independent interest. We give a set of axioms, which, like those of the hyperbolic plane, are user friendly. The first seven axioms are almost identical with those of the hyperbolic plane and the last three deal with planes in ℍ3. We then develop a set of theorems of the geometry of ℍ3, and we prove that the geometry inherited by a horosphere is Euclidean.