This paper addresses the notion of time change equivalence for Borel R^{d}-flows. We show that all free R^{d}-flows are time change equivalent up to a compressible set. An appropriate version of this result for non-free flows is also given..
Given a set of positive reals we provide a necessary and sufficient condition for a free Borel flow to admit a cross section with all distances between adjacent points coming from this set.
Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.
The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue Orbit Equivalence, by which we understand an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non smooth Euclidean flows are shown to be Lebesgue Orbit Equivalence if and only if they admit the same number of invariant ergodic probability measures.
We construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups G and H can be embedded into a Polish group T in such a way that the subgroup of T generated by G and H is isomorphic to the free product G*H.
We give a construction of two-sided invariant metrics on free products (possibly with amalgamation) of groups with two-sided invariant metrics and, under certain conditions, on HNN extensions of such groups. Our approach is similar to the Graev's construction of metrics on free groups over pointed metric spaces.
A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.
We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the randomly ordered rational Urysohn space.
We reexamine the Riemann Rearrangement Theorem for different types of convergence and classify possible sum ranges for statistically convergent series and for series that converge along the 2n-filter.
This lecture notes form an eight week introductory course to topological groups of Cantor systems. Some of the topics covered by the notes include: Verhsik maps and Bratteli diagrams, flip conjugacy, commutator subgroup, amenability of full groups.
These are the notes from a semester long course taught at UIC in the Fall of 2015. They cover compressible and hyperfinite relations, including Nadkarni's characterization of compressibility, classfification of hyperfinite relations by Dougherty, Jackson, and Kechris, and some examples of groups that give rise to hyperfinite relations only, after Jackson, Kechris, and Louveau.