Related papers: Convergence of group actions in metric measure geo…

We study several properties of expansive group actions on metric spaces and obtain relation between expansivity for subgroup and group actions. Through counter examples necessity of hypothesis are justified. We also study expansivity of…

In this paper the metric on the set of mixing actions of a countable infinite group is introduced so that the corresponding space is complete and separable. Keywords and phrases. Monotilable group, measure preserving transformations, mixing…

The title refers to the area of research which studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on ergodic theory of their actions. The paper is a survey of recent developments…

Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several…

We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with…

We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat…

We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry,…

Let (G, X) be a second-countable transformation group with G acting freely on X. It is shown that measure-theoretic accumulation of the action and topological strength of convergence in the orbit space X/G provide equivalent ways of…

We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum…

In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program…

Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation…

Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a…

We construct a universal action of a countable locally finite group (the Hall's group) on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces…

An action of a compact quantum group on a compact metric space $(X,d)$ is (D)-isometric if the distance function is preserved by a diagonal action on $X\times X$. We show that an isometric action in this sense has the following additional…

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize…

We investigate the geometry of median metric spaces. The group-theoretic applications are towards Kazhdan's property (T) and Haagerup's property.

In this paper, from the viewpoint of the concentration theory of maps, we study a compact group and a L\'{e}vy group action to a large class of metric spaces, such as R-trees, doubling spaces, metric graphs, and Hadamard manifolds.

In this paper, generalized metrics mean metrics taking values in general linearly ordered Abelian groups. Using the Hahn fields, we first prove that for every generalized metric space, if the set of the Archimedean equivalence classes of…

We study concentration of measure in Lie group actions. We define the notion of concentration locus of a flag sequence of Lie groups. Some examples of infinite group action on an infinite dimensional compact and non compact manifold show…

Certain notions of convergence of sequences functions such as pointwise convergence and (uniform) convergence on compact or bounded sets come from suitable topological function spaces; see [1]. Under certain conditions these topologies…