Related papers: Structure theorems for actions of homeomorphism gr…

Let a group $G$ act properly discontinuously and cocompactly on a locally compact space $X$. A Hausdorff compact space $Z$ that contains $X$ as an open subspace has the perspectivity property if the action $G\curvearrowright X$ extends to…

Let M be a compact manifold. We show the identity component $\mathrm{Homeo}_0(M)$ of the group of self-homeomorphisms of M has a well-defined quasi-isometry type, and study its large scale geometry. Through examples, we relate this large…

Let $G$ be a compact Lie group acting effectively by isometries on a compact Riemannian manifold $M$ with nonempty fixed point set $Fix(M,G)$. We say that the action is \emph{fixed point homogeneous} if $G$ acts transitively on a normal…

On closed symplectically aspherical manifolds, Schwarz proved a classical result that the action function of a nontrivial Hamiltonian diffeomorphism is not constant by using Floer homology. In this article, we generalize Schwarz's theorem…

We give a new categorical approach to the Halmos-von Neumann theorem for actions of general topological groups. As a first step, we establish that the categories of topological and measure-preserving irreducible systems with discrete…

This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of…

For every compact, connected manifold $M$, we prove the existence of a sentence $\phi_M$ in the language of groups such that the homeomorphism group of another compact manifold $N$ satisfies $\phi_M$ if and only if $N$ is homeomorphic to…

In this paper, we consider semigroup actions of discrete countable semigroups on compact spaces by surjective local homeomorphisms. We introduce notions of continuous one-sided orbit equivalence and continuous orbit equivalence for…

This article surveys gauge theory for families and its applications to the comparison between the diffeomorphism group and the homeomorphism group of $4$-manifolds, up to 2021.

We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and…

If a finite p-group G acts continuously on a compact topological manifold M then, with some bound C depending on M alone, G has a subgroup H of index at most C such that the H-action on M has at most C stabilizer subgroups. This result…

Two smooth manifolds M and N are called R-diffeomorphic if their product with the real line are diffeomorphic. We consider the following simplification problem: does R-diffeomorphism imply diffeomorphism or homeomorphism? For compact…

Given a compact Lie group $G$, a reconstruction theorem for free $G$-manifolds is proved. As a by-product reconstruction results for locally trivial bundles are presented. Next, the main theorem is generalized to $G$-manifolds with one…

The special structures that arise in symplectic topology (particularly Gromov--Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This…

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a…

We prove a compactness result for classes of actions of many small categories on quantum compact metric spaces by Lipschitz linear maps, for the topology of the covariant Gromov-Hausdorff propinquity. In particular, our result applies to…

In a previous work it is shown that every finite group $G$ of diffeomorphisms of a connected smooth manifold $M$ of dimension $\geq 2$ equals, up to quotient by the flow, the centralizer of the group of smooth automorphisms of a…

Given a manifold $M$ and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of $M$ to the group of isotopy classes of…

We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.

We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an…