Countable Short Recursively Saturated Models of Arithmetic

Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated

Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update

On groups and initial segments in nonstandard models of Peano Arithmetic

This thesis concerns M-finite groups and a notion of discrete measure in models of Peano Arithmetic. First we look at a measure construction for arbitrary non-M-finite sets via suprema and infima of appropriate M-finite sets. The basic properties of the measures are covered, along with non-measurable sets and the use of end-extensions. Next we look at nonstandard finite permutations, introducing nonstandard symmetric and alternating groups. We show that the standard cut being strong is necessary and sufficient for coding of the cycle shape in the standard system to be equivalent to the cycle being contained within the external normal closure of the nonstandard symmetric group. Subsequently the normal subgroup structure of nonstandard symmetric and alternating groups is given as a result analogous to the result of Baer, Schreier and Ulam for infinite symmetric groups. The external structure of nonstandard cyclic groups of prime order is identified as that of infinite dimensional rational vector spaces and the normal subgroup structure of nonstandard projective special linear groups is given for models elementarily extending the standard model. Finally we discuss some applications of our measure to nonstandard finite groups

Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture

Sofic and hyperlinear groups are the countable discrete groups that can be approximated in a suitable sense by finite symmetric groups and groups of unitary matrices. These notions turned out to be very deep and fruitful, and stimulated in the last 15 years an impressive amount of research touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and more recently even quantum information theory. Several longstanding conjectures that are still open for arbitrary groups were settled in the case of sofic or hyperlinear groups. These achievements aroused the interest of an increasing number of researchers into some fundamental questions about the nature of these approximation properties. Many of such problems are to this day still open such as, outstandingly: Is there any countable discrete group that is not sofic or hyperlinear? A similar pattern can be found in the study of II_1 factors. In this case the famous conjecture due to Connes (commonly known as the Connes embedding conjecture) that any II_1 factor can be approximated in a suitable sense by matrix algebras inspired several breakthroughs in the understanding of II_1 factors, and stands out today as one of the major open problems in the field. The aim of these notes is to present in a uniform and accessible way some cornerstone results in the study of sofic and hyperlinear groups and the Connes embedding conjecture. The presentation is nonetheless self contained and accessible to any student or researcher with a graduate level mathematical background. An appendix by V. Pestov provides a pedagogically new introduction to the concepts of ultrafilters, ultralimits, and ultraproducts for those mathematicians who are not familiar with them, and aiming to make these concepts appear very natural.Comment: 157 pages, with an appendix by Vladimir Pesto

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic