Related papers: On universal continuous actions on the Cantor set

A function of two variables F(x,y)is universal iff for every other function G(x,y) there exists functions h(x) and k(y) with G(x,y) = F(h(x),k(y)) Sierpinski showed that assuming the continuum hypothesis there exists a Borel function F(x,y)…

Grigorchuk and Medynets recently announced that the topological full group of a minimal Cantor Z-action is amenable. They asked whether the statement holds for all minimal Cantor actions of general amenable groups as well. We answer in the…

We consider the following conjecture (from Huang, et al): Let $\Delta^+$ denote the upper half disc in $\mathbb{C}$ and let $\gamma = ( - 1, 1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic…

Every endofunctor of the category of classes is proved to be set-based in the sense of Aczel and Mendler, therefore, it has a final coalgebra. Other basic properties of these endofunctors are proved, e.g. the existence of a free completely…

We prove that a C$^*$-algebra $A$ has uniform property $\Gamma$ if the set of extremal tracial states, $\partial_e T(A)$, is a non-empty compact space of finite covering dimension and for each $\tau \in \partial_e T(A)$, the von Neumann…

We consider a fixed free and proper action of a locally compact group $G$ on a space $T$, and actions $\alpha:G\to \Aut A$ on $C^*$-algebras for which there is an equivariant embedding of $(C_0(T),\rt)$ in $(M(A),\alpha)$. A recent theorem…

We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a…

We show that every finitely-generated non-amenable linear group over a field of characteristic zero admits an ergodic action which is rigid in the sense of Popa. If this group has trivial solvable radical, we prove that these actions can be…

Let $\Gamma$ be a discrete group satisfying the approximation property (AP). Let $X$, $Y$ be $\Gamma$-spaces and $\pi \colon Y \to X$ be a proper factor map which is injective on the non-free part. We prove the one-to-one correspondence…

We prove that for every finitely generated group $\Gamma$, at least one of the following holds: (1) $\Gamma$ has fixed price; (2) each of its Cayley graphs $G$ has infinitely many infinite clusters for some Bernoulli percolation on $G$.

We show that in the theory ZF + DC + for every cardinal {\lambda}, the set of infinite subsets of {\lambda} is well-ordered (i.e., Shelah's AX4), the {\theta}-function measuring the surjective size of the powersets P({\kappa}) can take…

We prove the following well known conjecture: let $\Sigma$ be an oriented surface of finite type whose fundamental group is a nonabelian free group. Let $\phi \in \textup{Mod}(\Sigma)$ be a an infinite order mapping class. Then there exists…

Let $\Gamma$ be a finitely generated group, and let $\mu$ be a nondegenerate, finitely supported probability measure on $\Gamma$. We show that every co-compact $\Gamma$ action on a locally compact Hausdorff space admits a nonzero…

For a continuous action of a countable discrete group $G$ on a Polish space $X$, a countable Borel partition $P$ of $X$ is called a generator if $G \cdot P := \{ gC : g \in G, C \in P \}$ generates the Borel $\sigma$-algebra of $X$. For $G…

In this two part work we prove that for every finitely generated subgroup $\Gamma < \text{Out}(F_n)$, either $\Gamma$ is virtually abelian or $H^2_b(\Gamma;\mathbb{R})$ contains an embedding of $\ell^1$. The method uses actions on…

In this paper, we consider the positional numeration system, called the Cantor real expansion, on the unit interval $[\gamma, \gamma+1]$, where $\gamma \in \mathbb{R}$, with respect to an alternate base (i.e., a base which is a purely…

By a Cantor group we mean a topological group homeomorphic to the Cantor set. We show that a compact metric space of rational cohomological dimension $n$ can be obtained as the orbit space of a Cantor group action on a metric compact space…

If $G$ is a semisimple Lie group of real rank at least 2 and $\Gamma$ is an irreducible lattice in $G$, then every homomorphism from $\Gamma$ to the outer automorphism group of a finitely generated free group has finite image.

After Voronin proved the universality theorem of the Riemann zeta function in the 1970s, universality theorems have been proposed for various zeta and L-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universality theorem of…

For an action of a discrete group $\Gamma$ on a set $X$, we show that the Schreier graph on $X$ has property A if and only if the permutation representation on $\ell_2X$ generates an exact $\mathrm{C}^*$-algebra. This is well known in the…