Related papers: M\"obius disjointness for homogeneous dynamics

Let $\mathbb{U}$ be a Banach Lie group and $S\subseteq \mathbb{U}$ an ad-bounded subset thereof, in the sense that there is a uniform bound on the adjoint operators induced by elements of $S$ on the Lie algebra of $\mathbb{U}$. We prove…

The celebrated Erd\H{o}s-Hajnal conjecture says that any graph without a fixed induced subgraph $H$ contains a very large homogeneous set. A direct analog of this conjecture is not true for hypergraphs. In this paper we present two natural…

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the…

We prove absence of absolutely continuous spectrum for discrete one-dimensional Schr\"odinger operators on the whole line with certain ergodic potentials, $V_\omega(n) = f(T^n(\omega))$, where $T$ is an ergodic transformation acting on a…

Recently, Glasner, Tsankov, Weiss, and Zucker showed that if $\Gamma$ is an infinite discrete group, then every minimal $\Gamma$-flow is disjoint from the Bernoulli shift $2^\Gamma$. Their proof is somewhat involved; in particular, it…

For a fixed odd prime $\ell$, we define a variant of the classical M\"{o}bius function on the poset of isomorphism classes of finite abelian $\ell$-groups, then we prove an analog of Hall's theorem on the vanishing of the M\"{o}bius…

We show that if a non-amenable, quasi-transitive, unimodular graph $G$ has all degrees even then it has a factor-of-iid balanced orientation, meaning each vertex has equal in- and outdegree. This result involves extending earlier…

Let $G$ be a linear Lie group that acts on it's Lie algebra $\mathfrak{g}$ by the adjoint action: $\mathrm{Ad}(g)X=gXg^{-1}$. An element $X\in \mathfrak {g}$ is called $\mathrm{Ad}_G$-real if $-X = \mathrm{Ad}(g)X $ for some $g\in G$. An…

Let $X=G/H$ be a homogeneous space of a Lie group $G$. When the isotropy subgroup $H$ is non-compact, a discrete subgroup $\Gamma$ may fail to act properly discontinuously on $X$. In this article, we address the following question: in the…

We establish a connection between analytic number theory and computational learning theory by showing that the M\"obius function belongs to a class of functions that is statistically hard to learn from random samples. Let $\mu_R$ denote the…

Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_G\Gamma$ has the same Zariski closure as $\Ad G$. If $\alpha \colon \Gamma \to \GL_n(\real)$ is any finite-dimensional representation…

Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group $G$ acts transitively and isometrically on $M$. In this situation, the metric on $M$ induces a bilinear form…

We prove that for a homogeneous linear partial differential operator $\mathcal A$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation…

A coadjoint orbit $O_\lambda \subseteq {\mathfrak g}^*$ of a Lie group $G$ is said to carry a Gibbs ensemble if the set of all $x \in {\mathfrak g}$, for which the function $\alpha \mapsto e^{-\alpha(x)}$ on the orbit is integrable with…

We establish a general spectral gap theorem for actions of products of groups which may replace Kazhdan's property (T) in various situations. As a main application, we prove that a confined subgroup of an irreducible lattice in a higher…

We study the homotopy type of spaces of commuting elements in connected nilpotent Lie groups, via almost commuting elements in their Lie algebras. We give a necessary and sufficient condition on the fundamental group of such a Lie group $G$…

Let $(Z,\omega)$ be a \Keler manifold and let $U$ be a compact connected Lie group with Lie algebra $\mathfrak{u}$ acting on $Z$ and preserving $\omega$. We assume that the $U$-action extends holomorphically to an action of the complexified…

Let $X = G/\Gamma$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\operatorname{Ad}$-diagonalizable over $\mathbb{C}$ and whose action on $(X,m_X)$ is mixing. In this…

Let $\gamma_{d}$ be the $d$-dimensional standard Gaussian measure and $\{Q_{t}\}_{t\ge 0}$ the Ornstein-Uhlenbeck semigroup acting on $L^{1}(\gamma_{d})$. We show that the hypercontractivity of $\{Q_{t}\}_{t\ge 0}$ is equivalent to the…

This article provides an account of the functorial correspondence between irreducible singular $G$-monopoles on $S^1\times \Sigma$ and $\vec{t}$-stable meromorphic pairs on $\Sigma$. The main theorem of [1] is thus generalized here from…