Related papers: Diffeomorphisms groups of tame Cantor sets and Tho…

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group, whose graded traces are weight 3/2 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to…

We generalize the construction of tensor categories of endomorphisms of a type III factor $M$ associated with a $G$-kernel, from the case of a discrete group $G$ to that of a compact second countable group. Our approach is based on the…

The braided Ptolemy-Thompson group $T^*$ is an extension of the Thompson group $T$ by the full braid group $B_{\infty}$ on infinitely many strands. This group is a simplified version of the acyclic extension considered by Greenberg and…

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and nondense forward orbits under the other is a dense, uncountable set. The pair of maps can be noncommuting. We…

One way to better understand the smooth mapping class group of the 4-sphere would be to give a list of generators in the form of explicit diffeomorphisms supported in neighborhoods of submanifolds, in analogy with Dehn twists on surfaces.…

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative…

We give a new very concrete description of the C*-envelope of the tensor algebra associated to multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism, and as a groupoid…

Implementing an idea due to John Baez and James Dolan we define new invariants of Whitney stratified manifolds by considering the homotopy theory of smooth transversal maps. To each Whitney stratified manifold we assign transversal homotopy…

The existence of quasimorphisms on groups of homeomorphisms of manifolds has been extensively studied under various regularity conditions, such as smooth, volume-preserving, and symplectic. However, in this context, nothing is known about…

This paper is a continuation of our 2005 paper on complex topology and its implication on invertibility (or non-invertibility). In this paper, we will try to classify the complexity of inversion into 3 different classes. We will use…

Let $M=I$ or $M=\mathbb{S}^1$ and let $k\geq 1$. We exhibit a new infinite class of Polish groups by showing that each group $\mathop{\rm Diff}_+^{k+AC}(M)$, consisting of those $C^k$ diffeomorphisms whose $k$-th derivative is absolutely…

Let \beta : S^n \to S^n, for n = 2k + 1, k \geq 1, be one of the known examples of a non-uniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S^n, \beta) there is a Cantor minimal…

Given a minimal action $\alpha$ of a countable group on the Cantor set, we show that the alternating full group $\mathsf{A}(\alpha)$ is non-amenable if and only if the topological full group $\mathsf{F}(\alpha)$ is $C^*$-simple. This…

We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of…

In this note we study countable subgroups of the full group of a measure preserving equivalence relation. We provide various constraints on the group structure, the nature of the action, and on the measure of fixed point sets, that imply…

In this short note, we show that R. Thompson's group $F$ admits a normalish amenable subgroup, and that the standard copy of $F$ in R. Thompson's group $T$ is normalish in $T$. We further conjecture that if $F$ is non-amenable, then $T$…

Real or complex tensor model observables, the backbone of the tensor theory space, are classical (unitary, orthogonal, symplectic) Lie group invariants. These observables represent as colored graphs, and that representation gives an handle…

Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable…

It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasov's question on…

For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for…