arXiv subjects

Nachi Avraham-Re'em

Publications and source records attributed to Nachi Avraham-Re'em.

12 recordsLinked to original sources

arXiv2026

Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry

We study the two $\ell^{2}$-Dirichlet structures on a countable group $G$ arising from the left and right regular actions on $\mathbb{R}^{G}$. Although the two regular representations are unitarily equivalent, their $\ell^{2}$-Dirichlet subspaces of $\mathbb{R}^{G}$ need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: $$\mathcal{D}_{2}\left(G,λ\right)=\mathcal{D}_{2}\left(G,ρ\right)\quad\Longleftrightarrow\quad G \text{ is an FC-group}.$$ The proof is based on a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over $\mathbb{Z}$. We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.

math.GR↗

arXiv2026

Wasserstein stability and the nonsingular Borel lifting problem

The Borel lifting problem asks when a nonsingular near action of a Polish group can be represented by a genuine Borel action. For locally compact Polish groups, a classical theorem of Mackey and Ramsay gives an affirmative answer. At the opposite extreme, Glasner--Tsirelson--Weiss proved that every probability preserving Borel action of a Lévy group is trivial, and asked whether a Lévy group can admit a nontrivial nonsingular Borel action. We prove a fixed-point theorem which gives a negative answer for most of the standard Lévy groups in the literature: if $G$ is a locally Wasserstein group, then every quasi-invariant $σ$-finite Borel measure for a Borel action of $G$ is supported on the fixed points. The proof introduces Wasserstein stability, a local-to-global principle for compact measure metric groups that refines the Gromov--Milman framework for concentration of measure in topological groups. It asserts that for a compact measure metric group, on average, the total variation of locally distorted density kernels on the group controls their global Wasserstein distance from the Haar measure. We show that Wasserstein stability implies concentration, and prove a functional inequality using martingales in the spirit of Milman--Schechtman, which is governed by $\ell^{1}$-sums of martingale increments. This gives a geometric criterion by which most of the standard Lévy groups in the literature are Wasserstein or locally Wasserstein, including $L^{0}$-groups with compact targets, measure preserving and nonsingular automorphism groups, full groups of amenable equivalence relations, isometry groups of $L^{p}$-spaces for $p\neq2$, the unitary group of the hyperfinite $\mathrm{II}_{1}$-factor, the infinite dimensional unitary and orthogonal groups, the Cameron--Martin affine group, and the isometry group of the Urysohn space.

math.DS↗

arXiv2026

Ergodicity and weak mixing for group-indexed infinitely divisible stationary processes

We prove that for an arbitrary indexing group, every ergodic infinitely divisible stationary process that is separable in probability is weakly mixing. This shows that, as in the well-known case of Gaussian stationary processes, ergodicity implies weak mixing is intrinsic to infinite divisibility, removing all structural assumptions on the group from prior results. The main ingredient is a general construction of stochastically continuous extensions for separable in probability stationary processes, reducing the problem to stochastically continuous processes indexed by Polish groups and then to countable groups, where we combine the Maruyama representation with an ergodicity criterion for Poisson suspensions.

math.PR↗

arXiv2026

On stationary actions of locally compact groups and their Radon-Nikodym cocycles

We study stationary actions of locally compact measured groups through the structure and regularity of their Radon-Nikodym cocycles. We start with two dynamical consequences of stationarity. Extending a theorem of Furstenberg-Glasner from discrete groups to noncompact locally compact groups, we show that every stationary action is conservative. Thus stationary actions are never of type I. We then show that an ergodic stationary action admitting an absolutely continuous invariant sigma-finite measure is in fact probability preserving. Thus stationary actions are never of type II_infty. Using a construction of Katznelson-Weiss and Vaes-Verjans, we show that if a group admits a stationary action of type III_1, then it admits stationary actions of every type III_lambda. The second part concerns the regularity of the Radon-Nikodym cocycle. We introduce the harmonic majorant on normalized positive harmonic functions, which gives Harnack-type control of Radon-Nikodym derivatives of stationary actions. For compactly supported probability measures with an Lp-density for some p>1, we prove that the harmonic majorant is finite and locally bounded. As a consequence, such measured groups admit a universal compact Radon-Nikodym model: a single compact G-space with a continuous cocycle into which stationary actions can be realized, so that the ambient cocycle gives a version of its Radon-Nikodym cocycle. This strengthens the Mackey-Varadarajan compact model theorem by incorporating the Radon-Nikodym cocycle into the model. By contrast, we construct a random walk on the real affine group whose Poisson boundary fails Kaimanovich's SAT* property: the Poisson kernel is unbounded arbitrarily close to the identity. Therefore, Harnack's inequality already fails for positive harmonic functions. In particular, this Poisson boundary admits no topological model with continuous Poisson kernel.

math.DS↗

arXiv2026

Locally integrable cross sections and their intersection covolume

We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume.

math.DS↗

arXiv2026

The Hopf decomposition of locally compact group actions

We develop a unified approach to the classical Hopf Decomposition (also known as the conservative--dissipative decomposition) for actions of locally compact second countable groups. While the decomposition is well understood for free actions of countable groups, the extension to general actions requires new techniques and structural insights, particularly concerning recurrence and transience, cocycle behavior, and the structure of stabilizers. We establish several new characterizations and prove a structure theorem for totally dissipative actions, generalizing Krengel's classical result for flows.

math.DS↗

arXiv2025

Periodicity of point processes in abelian groups without lattices

We investigate the intersection covolume of cross sections for probability preserving actions of a class of abelian groups without lattices, including $p$-adic groups and the group of finite adeles. We show that for cross sections with a uniformly discrete return time set, the intersection covolume is bounded below by twice the intensity, revealing a strict gap compared to the lattice case. Our main theorem asserts that cut-and-project systems uniquely attain the minimal intersection covolume. As an application, we characterize the generalized Farey fractions in the finite adeles via their Banach density.

math.DS↗

arXiv2025

Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem

We show that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (à la Connes-Feldman-Weiss) precisely when it is 'hypercompact'. This implies an uncountable version of the Ornstein-Weiss Theorem and that every locally compact group admitting a hypercompact probability preserving free action is amenable. We also establish an uncountable version of Danilenko's Random Ratio Ergodic Theorem. From this we deduce the 'Hopf dichotomy' for many nonsingular Bernoulli actions.

math.DS↗

arXiv2024

Poissonian Actions of Polish Groups

We define and study Poissonian actions of Polish groups as a framework to Poisson suspensions, characterize them spectrally, and provide a complete characterization of their ergodicity. We further construct 'spatial' Poissonian actions, answering partially a question of Glasner, Tsirelson & Weiss about Lévy groups. We also construct for every diffeomorphism group an ergodic free spatial probability preserving actions. This constitutes a new class of Polish groups admitting non-essentially countable orbit equivalence relations, obtaining progress on a problem of Kechris.

math.DS↗

arXiv2024

Symmetric Stable Processes on Amenable Groups

We show that if $G$ is a countable amenable group, then every stationary non-Gaussian symmetric $α$-stable (S$α$S) process indexed by $G$ is ergodic if and only if it is weakly-mixing, and it is ergodic if and only if its Rosinski minimal spectral representation is null. This extends the results for $\mathbb{Z}^d$, and answers a question of P. Roy on discrete nilpotent groups to the extent of all countable amenable groups. As a result we construct on the Heisenberg group and on many Abelian groups, for all $α$ in (0,2), stationary S$α$S processes that are weakly-mixing but not strongly-mixing.

math.PR↗

arXiv2022

Equivalence-Singularity Dichotomy in Markov Measures

We establish an equivalence-singularity dichotomy for a large class of one-dimensional Markov measures. Our approach is new in that we deal with one-sided and two-sided chains simultaneously, and in that we do not appeal to any 0-1 law. In fact we deduce a new 0-1 law from the dichotomy.

math.PR↗

arXiv2022

On Absolutely Continuous Invariant Measures and Krieger-Type of Markov Subshifts

It is shown that for a non-singular conservative shift on a topologically mixing Markov subshift with Doeblin Condition the only possible absolutely continuous shift-invariant measure is a Markov measure. Moreover, if it is not equivalent to a homogeneous Markov measure then the shift is of Krieger-type $\mathrm{III}_1$. A criterion for equivalence of Markov measures is included.

math.DS↗