Dedicated to the memory of Candida Silveira. Abstract. We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of Brock-Farb’s Rank Conjecture which asserts that MCG has quasi-flats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasiflats in Teichmuller space with the Weil-Petersson metric. The coarse geometric structure of a finitely generated group can be studied by passage to its asymptotic cone, which is a space obtained by a limiting process from sequences of rescalings of the group. This has played an important role in the quasi-isometric rigidity results of [DS], [KL1] [KL2], and others. In this paper we study the asymptotic cone M ω (S) of the mapping

We introduce observation frames as an extension of ordinary frames. The aim is to give an abstract representation of a mapping from observable predicates to all predicates of a specific system. A full subcategory of the category of observation frames is shown to be dual to the category of T 0 topological spaces. The notions we use generalize those in the adjunction between frames and topological spaces in the sense that we generalize finite meets to infinite ones. We also give a predicate logic of observation frames with both infinite conjunctions and disjunctions, just like there is a geometric logic for (ordinary) frames with infinite disjunctions but only finite conjunctions. This theory is then applied to two situations: firstly to upper power spaces, and secondly we restrict the adjunction between the categories of topological spaces and of observation frames in order to obtain dualities for various subcategories of T 0 spaces. These involve non sober spaces. Contents 1 Introduct...

analytic spaces

One emerging approach to reducing the labour and costs of software development favours the specialisation of techniques to particular application domains. The rationale is that programs within a given domain often share enough common features and assumptions to enable the incorporation of substantial support mechanisms into domain-specific programming languages and associated tools. Instead of being machine-oriented, algorithmic implementations, programs in many domain-specific languages (DSLs) are rather user-level, problem-oriented specifications of solutions. Taken further, this view suggests that the most appropriate representation of programs in many domains is diagrammatic, in a way which derives from existing design notations in the domain. This thesis conducts an investigation, using mathematical techniques and supported by case studies, of issues arising from the use of diagrammatic representations in DSLs. Its structure is conceptually divided into two parts: the first is co...

A topological group G is h-complete if every continuous homomorphic image of G is (Raĭkov-)complete; we say that G is hereditarily h-complete if every closed subgroup of G is h-complete. In this paper, we establish open-map properties of hereditarily h-complete groups with respect to large classes of groups, and prove a theorem on the (total) minimality of subdirectly represented groups. Numerous applications are presented, among them: 1. Every hereditarily h-complete group with quasi-invariant basis is the projective limit of its metrizable quotients; 2. If every countable discrete hereditarily h-complete group is finite, then every locally compact hereditarily h-complete group that has small invariant neighborhoods is compact. In the sequel, several open problems are formulated.

Abstract. This article continues the study of computable elementary topology started in [7]. We introduce a number of computable versions of the topological T0 to T3 separation axioms and solve their logical relation completely. In particular, it turns out that computable T1 is equivalent to computable T2. The strongest axiom SCT3 is used in [2] to construct a computable metric. 1

I gratefully acknowledge the generous financial support received from the Killam Trusts and Dalhousie

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we study connectedness and disconnectedness properties of groups G with the property that every closed subgroup of G is locally pseudocompact. We show that the completion of the component G0 of G contains every connected compact subgroup of the completion of G. We also prove that the question of whether G/G0 is zero-dimensional (or equivalently, whether G0 is dense in the component of the completion of G) can be reduced to the case where G is a dense subgroup of a group of the form N ×R, where N is zero-dimensional and compact. 1.

Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications Tim Austin We study a generalized notion of a homogeneous skew-product extension of a probability-preserving base system in which the homogeneous space fibres can vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of ‘direct integral ’ for a ‘measurable family’ of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions, including non-ergodic versions of the results of Mackey describing ergodic components of such extensions [29], of the Furstenberg-Zimmer Structure Theory [45, 44, 18] and of results of Mentzen [32] describing the structure of automorphisms of relatively ergodic such extensions.