Related papers: On Gardner's conjecture

We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the $G$-shift $(\Lambda^G,\sigma)$, a typical measure has entropy $c$ and is Bernoulli.…

We consider a type of distance-regular graph $\Gamma=(X, \mathcal R)$ called a bilinear forms graph. We assume that the diameter $D$ of $\Gamma$ is at least $3$. Fix adjacent vertices $x,y \in X$. In our first main result, we introduce an…

We consider a connected graph $\Gamma$ as a coarse space and prove that $\Gamma$ admits a 2-selector if and only if $\Gamma$ is either bounded or coarsely equivalent to $\mathbb{N}$ or $\mathbb{Z}$. We apply this result to geodesic metric…

Let $\Gamma$ denote a finite, connected graph with vertex set $X$. Fix $x \in X$ and let $\varepsilon \ge 3$ denote the eccentricity of $x$. For mutually distinct scalars $\{\theta^*_i\}_{i=0}^\varepsilon$ define a diagonal matrix…

We prove that if $G$ is a countably infinite group and $(L, \lambda)$ and $(K, \kappa)$ are probability spaces having equal Shannon entropy, then the Bernoulli shifts $G \curvearrowright (L^G, \lambda^G)$ and $G \curvearrowright (K^G,…

A matching $M$ in a graph $\Gamma$ is positive if $\Gamma$ has a vertex-labeling such that $M$ coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of $\Gamma$ is an edge-partition $M_1,\ldots,M_p$…

Ab\'ert-Weiss have shown that the Bernoulli shift s of a countably infinite group \Gamma is weakly contained in any free measure preserving action (mpa) b of \Gamma. We establish a strong version of this result, conjectured by Ioana, by…

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $\Gamma$-symmetric theory. Depending on how anomalous $\Gamma$ is, we find that the symmetry of…

The evenness conjecture for the equivariant unitary bordism groups states that these bordism groups are free modules over the unitary bordism ring on even-dimensional generators. In this paper we review the cases in which the conjecture is…

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner's questions have been generalized and some of them are already solved. In the…

We undertake a comprehensive study of measure equivalence between general locally compact, second countable groups, providing operator algebraic and ergodic theoretic reformulations, and complete the classification of amenable groups within…

We give a short geometric proof of a result of Soardi & Woess and Salvatori that a quasitransitive graph is amenable if and only if its automorphism group is amenable and unimodular. We also strengthen one direction of that result by…

We define the notion of admissible pair for an algebra $A$, consisting on a couple $(\Gamma,R)$, where $\Gamma$ is a quiver and $R$ a unital, splitted and factorizable representation of $\Gamma$, and prove that the set of admissible pairs…

The Gardner method, traditionally used to generate conservation laws of integrable equations, is generalized to generate symmetries. The method is demonstrated for the KdV, Camassa-Holm and Sine-Gordon equations. The method involves…

For an extension $1\rightarrow N \rightarrow \Gamma \xrightarrow{q} \Gamma / N \rightarrow 1$ of discrete countable groups, it is known that the Baum-Connes conjecture with coefficients holds for $\Gamma$ if it holds for $\Gamma / N$ and…

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer in $G$ is trivial. By $b(G)$ we denote the size of the smallest base of $G$. Every permutation group with $b(G)=2$ contains…

It was proved by Mineyev and Yaman that, if $(\Gamma, \Gamma')$ is a relatively hyperbolic pair, the comparison map $$ H_b^k(\Gamma, \Gamma'; V) \to H^k(\Gamma, \Gamma'; V) $$ is surjective for every $k \ge 2$, and any bounded…

Let $\Gamma$ be a countable discrete amenable group, and let $A=l^\infty(\Gamma) \rtimes \Gamma$ or $A = \mathrm{C}(M) \rtimes \Gamma$, where $(M, \Gamma)$ is the universal minimal set of $\Gamma$. It is shown that if $a, b \in A \otimes…

A relation algebra is called measurable when its identity is the sum of measurable atoms, and an atom is called measurable if its square is the sum of functional elements. In this paper we show that atomic measurable relation algebras have…

This paper investigates the Hausdorff measure of certain sets of generics in computability theory. Let $\Gamma$ be the Turing ideal in which we take the dense open sets. The set of $\Gamma$-Cohen generics has measure positive if and only if…