Related papers: Sharper changes in topologies

For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then…

Let $X$ be a K3 or Enriques surface with good reduction. Let $G$ be a finite group acting (not necessarily linearly) on $X$. We give a criterion for this group action to extend to a smooth model of $X$ in terms of the action of $G$ on the…

We prove some "universality" results for topological dynamical systems. In particular, we show that for any continuous self-map $T$ of a perfect Polish space, one can find a dense, $T$-invariant set homeomorphic to the Baire space ${\mathbb…

We define and study Poissonian actions of Polish groups as a framework to Poisson suspensions, characterize them spectrally, and provide a complete characterization of their ergodicity. We further construct 'spatial' Poissonian actions,…

In this paper we study the Buchstaber invariant of simplicial complexes, which comes from toric topology. With each simplicial complex $K$ on $m$ vertices we can associate a moment-angle complex $\mathcal Z_K$ with a canonical action of the…

In this note we extend the concept of topological stability from homeomorphisms to group actions on compact metric spaces, and prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property then…

Let $X, Y$ be two complex manifolds, let $D\subset X,$ $ G\subset Y$ be two nonempty open sets, let $A$ (resp. $B$) be an open subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup…

We study continuous actions of Polish groups on Polish spaces. We develop Scott analysis introduced by Hjorth for studying orbit equivalence relations. We define eventually open actions and prove that this property characterizes the actions…

Let X be an irreducible variety and Bir(X) its group of birational transformations. We show that the group structure of Bir(X) determines whether X is rational and whether X is ruled. Additionally, we prove that any Borel subgroup of Bir(X)…

Suppose that a group $G$ acts non-elementarily on a hyperbolic space $S$ and does not fix any point of $\partial S$. A subgroup $H\le G$ is said to be geometrically dense in $G$ if the limit sets of $H$ and $G$ coincide and $H$ does not fix…

The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group $G$, let $E(G)$ be the…

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\mu$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group…

In these lectures we consider how algebraic properties of discrete subgroups of Lie groups restrict the possible actions of those groups on surfaces. The results show a strong parallel between the possible actions of such a group on the…

Let $\Gamma$ be a finitely generated group acting by probability measure preserving maps on the standard Borel space $(X,\mu)$. We show that if $H\leq\Gamma$ is a subgroup with relative spectral radius greater than the global spectral…

When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of $G$. We introduce a class of groups, the CAP groups, which provides a neat…

We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an…

We associate a root system to a finite set in a free abelian group and prove that its irreducible subsystem is of type A, B or D. We apply this general result to a torus manifold, where a torus manifold is a $2n$-dimensional connected…

We prove that for any countable group G there exists a free minimal continuous action of G on the Cantor set admitting an invariant Borel probability measure.

We show that systems with some specification properties are topologically or almost Borel universal, in the sense that any aperiodic subshift with lower entropy may be topologically or almost Borel embedded.

We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant…