Related papers: Isometric orbit equivalence for probability-measur…

We prove that orbit equivalence of measure preserving ergodic a.e. free actions of a countable group with the relative property (T) is a complete analytic equivalence relation.

Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group…

We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation…

We propose a notion of isometric coaction of a compact quantum group on a compact quantum metric space in the framework of Rieffel where the metric structure is given by a Lipnorm. We prove the existence of a quantum isometry group for…

First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to…

Given a closed orientable surface (\Sigma) of genus at least two, we establish an affine isomorphism between the convex compact set of isotopy-invariant topological measures on (\Sigma) and the convex compact set of additive functions on…

In this paper we study geodesic Ptolemy metric spaces $X$ which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that $X$ is equivariantly rough isometric to a Euclidean…

Inspired by constructions of Kova\v{c}evi\'{c}, we introduce the amalgamated free product of circle actions, obtained by blowing up two actions along prescribed orbits and rearranging the inserted intervals. Under natural orbit and index…

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$\kappa$-to-one in a natural sense for some $\kappa>0$. For non-amenable graphs, all quasi-isometries are quasi-$\kappa$-to-one for any…

We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to…

We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere…

A graph $\G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $\Aut(\G)$, we say that $\G$ is {\em normal} with respect to…

In classical mechanics, the Kepler potential and the Harmonic potential share the following remarkable property: in either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum.…

We introduce an algebraic invariant for aperiodic inclusions of probability measure preserving equivalence relations. We use this invariant to prove that every stable orbit equivalence between free pmp actions of direct products of…

Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic…

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed…

We obtain a partial converse of Vershik's description of ergodic probability measures on a compact metric space with respect to an isometric action by an inductively compact group. This allows us to identify, in this setting, the set of…

We construct affine uniformly Lipschitz actions on $\ell^1$ and $L^1$ for certain groups with hyperbolic features. For acylindrically hyperbolic groups, our actions have unbounded orbits, while for residually finite hyperbolic groups and…

In this paper we study topological rigidity of affine actions on compact connected metrizable abelian groups. We also classify one-parameter flows of translations upto orbit equivalence and discrete group actions by translations upto…

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_\infty$-invariant and concentrated on a single…