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Yair Hartman

Publications and source records attributed to Yair Hartman.

At least 19 recordsLinked to original sources
arXiv2026

The No-Core Principle for Stationary Actions and Ends of Stationary Random Subgroups

We prove a No-Core Principle for stationary actions of countable groups. Namely, if a Borel set intersects almost every orbit in finitely many points and has positive measure, then it is supported, modulo null sets, on the finite-orbit part of the action. This extends to stationary actions a basic regularity phenomenon known for measure-preserving actions. We apply this principle to the geometry of Stationary Random Subgroups. For a finitely generated group, we prove that the Schreier graph of a stationary random subgroup has almost surely 0,1,2, or infinitely many ends. Finally, we contrast this probabilistic regularity with the topological notion of Boomerang subgroups: for every $k\geq 3$, including $k=\aleph_0$, we construct a Boomerang subgroup of $\mathbb{F}_3$ whose Schreier graph has exactly $k$ ends.

math.DS↗
arXiv2026

Stabilizer Subgroups and the Simplicity of Reduced Crossed Products

Given a minimal action $G\curvearrowright X$ of a countable group $G$ on a compact space $X$, we prove that if the reduced crossed product $G\ltimes_rC(X)$ is simple, then there exists a point whose stabilizer subgroup has trivial amenable radical. As a consequence, we give a complete characterization of the simplicity of the reduced crossed product of minimal actions of countable linear groups, hyperbolic groups, and, more generally, for groups with countably many amenable subgroups. This answers a question of Ozawa (2014) for these classes of groups. Furthermore, in the case of an infinite uniformly recurrent subgroup of a $C^*$-simple group, we prove that almost every subgroup has a trivial amenable radical, with respect to a fully supported, atomless probability measure.

math.OA↗
arXiv2026

On the inclusion of bounded harmonic functions of random walks

We investigate the conditions under which the space of bounded harmonic functions of a probability measure $\mu$ on a group $G$ is contained in that of another measure $\theta$. We establish that asymptotic commutativity, defined by the condition $\|\mu^{*t}*\theta - \theta*\mu^{*t}\|_{TV} \to 0$ as $t \to \infty$, is sufficient to guarantee the inclusion $H^\infty(G, \mu) \subseteq H^\infty(G, \theta)$, provided $\theta$ is absolutely continuous with respect to a convex combination of convolution powers of $\mu$. By employing martingale convergence techniques rather than ergodic-theoretic arguments, we demonstrate that this result holds without topological assumptions on $G$ (such as local compactness) and extends to general Markov chains. Furthermore, utilizing hitting models for the Poisson boundary, we characterise the inclusion $H^\infty(G, \mu) \subseteq H^\infty(G, \theta)$ as equivalent to the asymptotic invariance of $\theta$ under $\mu$ in the weak* topology. We apply these results to provide a probabilistic proof of the Choquet-Deny theorem for nilpotent groups, among other applications.

math.PR↗
arXiv2025

Retraction Theorems for Group Compactifications

We characterize group compactifications of discrete groups for which there exists an equivariant retraction onto the boundary. In particular, we prove an equivariant analogue of Brouwer's No-Retraction theorem for large classes of group compactifications, which includes actions of hyperbolic groups on their Gromov boundary.

math.GR↗
arXiv2024

On the amenable subalgebras of group von Neumann algebras

We approach the study of sub-von Neumann algebras of the group von Neumann algebra $LΓ$ for countable groups $Γ$ from a dynamical perspective. It is shown that $L(Γ)$ admits a maximal invariant amenable subalgebra. The notion of invariant probability measures (IRAs) on the space of sub-algebras is introduced, analogous to the concept of Invariant Random Subgroups. And it is shown that amenable IRAs are supported on the maximal amenable invariant sub-algebra.

math.OA↗
arXiv2023

Subalgebras, subgroups, and singularity

This paper concerns the non-commutative analog of the Normal Subgroup Theorem for certain groups. Inspired by Kalantar-Panagopoulos, we show that all $Γ$-invariant subalgebras of $LΓ$ and $C^*_r(Γ)$ are ($Γ$-) co-amenable. The groups we work with satisfy a singularity phenomenon described in Bader-Boutonnet-Houdayer-Peterson. The setup of singularity allows us to obtain a description of $Γ$-invariant intermediate von Neumann subalgebras $L^{\infty}(X,ξ)\subset\mathcal{M}\subset L^{\infty}(X,ξ)\rtimesΓ$ in terms of the normal subgroups of $Γ$.

math.OA↗
arXiv2021

Tight inclusions of C*-dynamical systems

We study a notion of tight inclusions of C*- and W*-dynamical systems which is meant to capture a tension between topological and measurable rigidity of boundary actions. An important case of such inclusions are $C(X)\subset L^\infty(X, ν)$ for measurable boundaries with unique stationary compact models. We discuss the implications of this phenomenon in the description of Zimmer amenable intermediate factors. Furthermore, we prove applications in the problem of maximal injectivity of von Neumann algebras.

math.OA↗
arXiv2020

Stationary C*-dynamical systems

We introduce the notion of stationary actions in the context of C*-algebras. We develop the basics of the theory, and provide applications to several ergodic theoretical and operator algebraic rigidity problems.

math.OA↗
arXiv2020

Random walks on dense subgroups of locally compact groups

Let $Γ$ be a countable discrete group, $H$ a lcsc totally disconnected group and $ρ: Γ\rightarrow H$ a homomorphism with dense image. We develop a general and explicit technique which provides, for every compact open subgroup $L < H$ and bi-$L$-invariant probability measure $θ$ on $H$, a Furstenberg discretization $τ$ of $θ$ such that the Poisson boundary of $(H,θ)$ is a $τ$-boundary. Among other things, this technique allows us to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not $L^p$-irreducible for any $p \geq 1$, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we give an example of a countable discrete group $Γ$ and two spread-out probability measures $τ_1$ and $τ_2$ on $Γ$ such that the boundary entropy spectrum of $(Γ,τ_1)$ is an interval, while the boundary entropy spectrum of $(Γ,τ_2)$ is a Cantor set.

math.DS↗
arXiv2020

Kudo-Continuity Of Entropy Functionals

We study in this paper real-valued functions on the space of all sub-$σ$-algebras of a probability measure space, and introduce the notion of Kudo-continuity, which is an a priori strengthening of continuity with respect to strong convergence. We show that a large class of entropy functionals are Kudo-continuous. On the way, we establish upper and lower continuity of various entropy functions with respect to asymptotic second order stochastic domination, which should be of independent interest. An application to the study of entropy spectra of $μ$-boundaries associated to random walks on locally compact groups is given.

math.PR↗
arXiv2020

The stabilized automorphism group of a subshift

For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple, and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group.

math.DS↗
arXiv2019

Choquet-Deny groups and the infinite conjugacy class property

A countable discrete group $G$ is called Choquet-Deny if for every non-degenerate probability measure $μ$ on $G$ it holds that all bounded $μ$-harmonic functions are constant. We show that a finitely generated group $G$ is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that $G$ is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when $G$ is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.

math.GR↗
arXiv2017

Furstenberg Entropy of Intersectional Invariant Random Subgroups

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a-priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.

math.DS↗
arXiv2016

Stabilizer Rigidity in Irreducible Group Actions

We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader-Shalom and Stuck-Zimmer.

math.DS↗
arXiv2015

Generic Stationary Measures and Actions

Let $G$ be a countably infinite group, and let $μ$ be a generating probability measure on $G$. We study the space of $μ$-stationary Borel probability measures on a topological $G$ space, and in particular on $Z^G$, where $Z$ is any perfect Polish space. We also study the space of $μ$-stationary, measurable $G$-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When $μ$ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of $(G,μ)$. When $Z$ is compact, this implies that the simplex of $μ$-stationary measures on $Z^G$ is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on $\{0,1\}^G$. We furthermore show that if the action of $G$ on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when $G$ has property (T), the ergodic actions are meager. We also construct a group $G$ without property (T) such that the ergodic actions are not dense, for some $μ$. Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual.

math.GR↗
arXiv2015

Furstenberg entropy realizations for virtually free groups and lamplighter groups

Let $(G,μ)$ be a discrete group with a generating probability measure. Nevo shows that if $G$ has property (T) then there exists an $ε>0$ such that the Furstenberg entropy of any $(G,μ)$-stationary ergodic space is either zero or larger than $ε$. Virtually free groups, such as $SL_2(\mathbb{Z})$, do not have property (T), and neither do their extensions, such as surface groups. For these, we construct stationary actions with arbitrarily small, positive entropy. This construction involves building and lifting spaces of lamplighter groups. For some classical lamplighters, these spaces realize a dense set of entropies between zero and the Poisson boundary entropy.

math.DS↗
arXiv2014

Property (T) and the Furstenberg Entropy of Nonsingular Actions

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure $μ$ on a countable group $G$, A. Nevo showed that a necessary condition for $G$ to have property (T) is that the Furstenberg $μ$-entropy values of the ergodic, properly nonsingular $G$-actions are bounded away from zero. We show that this is also a sufficient condition.

math.GR↗