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Gábor Elek

Publications and source records attributed to Gábor Elek.

At least 19 recordsLinked to original sources
arXiv2026

Uniform Borel Amenability

We study a uniform, quantitative form of the amenability-hyperfiniteness paradigm for bounded-degree Borel graphs generating countable Borel equivalence relations. We introduce \emph{uniform Borel amenability} and prove that it is equivalent to \emph{randomized Borel hyperfiniteness}, a probabilistic version of hyperfiniteness. Consequences are three strengthenings of the Connes-Feldman-Weiss theorem. In the setting of uniformly Borel amenable Følner graphs (e.g. Borel graphs of not necessarily free actions of amenable groups or Borel graphs of subexponential growth), we establish an analogous equivalence to randomized Borel almost finiteness. We further obtain measure-theoretic structural results, including almost finiteness outside a $μ$-null invariant set extending a recent result of Conley et al. for free amenable actions, and an Ornstein--Weiss type packing theorem that is uniform over all invariant measures. Finally, we show that uniformly Borel amenable graphs are hyperfinite modulo a compressible invariant set, i.e., after removing a Borel invariant set that is of measure zero for every invariant probability measure.

math.DS↗
arXiv2025

Strong almost finiteness

A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many Følner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on the boundary of a tile is uniformly small. We give various equivalents for strong almost finiteness. In particular, we prove that Property A together with the Følner property is equivalent to strong almost finiteness. Using these characterizations, we show that graphs of subexponential growth and Schreier graphs of amenable groups are always strongly almost finite, generalizing the celebrated result of Downarowicz, Huczek and Zhang about amenable Cayley graphs, based on graph theoretic rather than group theoretic principles. We give various equivalents to Property A for graphs, and show that if a sequence of graphs of Property A (in a uniform sense) converges to a graph $G$ in the neighborhood distance (a purely combinatorial analogue of the classical Benjamini-Schramm distance), then their Laplacian spectra converge to the Laplacian spectrum of $G$ in the Hausdorff distance. We apply the previous theory to construct a new and rich class of classifiable $C^{\star}$-algebras. Namely, we show that for any minimal strong almost finite graph $G$ there are naturally associated simple, nuclear, stably finite $C^{\star}$-algebras that are classifiable by their Elliott invariants.

math.GR↗
arXiv2022

Planarity can be Verified by an Approximate Proof Labeling Scheme in Constant-Time

Approximate proof labeling schemes were introduced by \\Censor-Hillel, Paz and Perry \cite{CPP}. Roughly speaking, a graph property~$\cP$ can be verified by an approximate proof labeling scheme in constant-time if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are having the property $\cP$ or at least they are not far from being having the property $\cP$. The main result of this paper is that bounded-degree planar graphs (and also outer-planar graphs, bounded genus graphs, knotlessly embeddable graphs etc.) can be verified by an approximate proof labeling scheme in constant-time.

math.CO↗
arXiv2021

Convergence and limits of finite trees

Motivated by the work of Lovász and Szegedy on the convergence and limits of dense graph sequences, we investigate the convergence and limits of finite trees with respect to sampling in normalized distance. Based on separable real trees, we introduce the notion of a dendron and show that the limits of finite trees are exactly the dendrons. We also prove that the limit dendron is unique.

math.CO↗
arXiv2021

Uniform Local Amenability implies Property A

In this short note we answer a query of Brodzki, Niblo, Špakula, Willett and Wright by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser proved that if $Γ$ is a finitely generated group and $\{H_i\}^\infty_{i=1}$ is a Farber sequence of finite index subgroups, then the associated Schreier graph sequence is of Property A if and only if the group is amenable. We show however, that there exist a non-amenable group and a nested sequence of finite index subgroups $\{H_i\}^\infty_{i=1}$ such that $\cap H=\{e_Γ\}$, and the associated Schreier graph sequence is of Property A.

math.MG↗
arXiv2021

Almost commuting matrices with respect to the rank metric

We show that if A_1, A_2, ... , A_n are square matrices, each of them is either unitary or self-adjoint, and they almost commute with respect to the rank metric, then one can find commuting matrices B_1, B_2, ... , B_n that are close to the matrices A_i in the rank metric.

math.RA↗
arXiv2020

Uniform hyperfiniteness

Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for measurable graphed equivalence relations of bounded vertex degrees and prove that these two notions coincide as well. Roughly speaking, a measured graph $\cG$ is uniformly hyperfinite if for any $\eps>0$ there exists $K\geq 1$ such that not only $\cG$, but all of its subgraphs of positive measure are $(\eps,K)$-hyperfinite. We also show that this condition is equivalent to weighted hyperfiniteness and a strong version of fractional hyperfiniteness, a notion recently introduced by Lovász. As a corollary, we obtain a characterization of exactness of finitely generated groups via uniform hyperfiniteness.

math.DS↗
arXiv2019

Learning Very Large Graphs with Unknown Vertex Distributions

Recently, Goldreich introduced the notion of property testing of bounded-degree graphs with an unknown distribution. We propose a slight modification of his idea: the Radon-Nikodym Oracles. Using these oracles any reasonable graph property can be tested in constant-time against any reasonable unknown distribution in the category of planar graphs. We also discuss Randomized Local Distributed Algorithms, which work on very large graphs with unknown distributions. Finally, we discuss how can we learn graph properties using observations instead of samplings.

cs.DS↗
arXiv2019

Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms

The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and local-global graph limit theories for sparse graphs. The natural limit objects are continuous actions of finitely generated groups on totally disconnected compact metric spaces. We prove that the space of weak equivalent classes of free Cantor actions is compact and contains a smallest element, as in the measurable case. We will introduce and study various notions of almost finiteness, the qualitative analogue of hyperfiniteness, for classes of bounded degree graphs. We prove the almost finiteness of a new class of étale groupoids associated to Cantor actions and construct an example of a nonamenable, almost finite totally disconnected étale groupoid, answering a query of Suzuki. Motivated by the notions and results on qualitative graph limits, in the second part of our paper we give a precise definition of constant-time distributed algorithms on sparse graphs. We construct such constant-time algorithms for various approximation problems for hyperfinite and almost finite graph classes. We also prove the Hausdorff convergence of the spectra of convergent graph sequences in the strongly almost finite category.

math.DS↗
arXiv2018

On universal continuous actions on the Cantor set

Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $Γ$, there exists a free continuous action $ζ: Γ\curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $α: Γ\curvearrowright X$ on the standard Borel space, there exists an injective Borel map $Θ_α: X\to C$ such that $Θ_α\circ α=ζ\circ Θ_α$. We extend our theorem for (nonfree) Borel $(Γ,Z)$-actions, where $Z$ is a uniformly recurrent subgroup.

math.DS↗
arXiv2014

Hyperlinearity, essentially free actions and $L^2$-invariants. The sofic property

We prove that Connes' Embedding Conjecture holds for the von Neumann algebras of sofic groups, that is sofic groups are hyperlinear. Hence we provide some new examples of hyperlinearity. We also show that the Determinant Conjecture holds for sofic groups as well. We introduce the notion of essentially free actions and amenable actions and study their properties.

math.GR↗
arXiv2011

Dynamical properties of profinite actions

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($τ$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to answering the question of Lubotzky and Zuk, whether for families of subgroups, property ($τ$) is inherited to the lattice of subgroups generated by the family. On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicite estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expander covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

math.GR↗
arXiv2009

Sofic equivalence relations

We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of Lück, Sauer and Wegner hold for treeable equivalence relations.

math.FA↗
arXiv2008

A measure-theoretic approach to the theory of dense hypergraphs

In this paper we develop a measure-theoretic method to treat problems in hypergraph theory. Our central theorem is a correspondence principle between three objects: An increasing hypergraph sequence, a measurable set in an ultraproduct space and a measurable set in a finite dimensional Lebesgue space. Using this correspondence principle we build up the theory of dense hypergraphs from scratch. Along these lines we give new proofs for the Hypergraph Removal Lemma, the Hypergraph Regularity Lemma, the Counting Lemma and the Testability of Hereditary Hypergraph Properties. We prove various new results including a strengthening of the Regularity Lemma and an Inverse Counting Lemma. We also prove the equivalence of various notions for convergence of hypergraphs and we construct limit objects for such sequences. We prove that the limit objects are unique up to a certain family of measure preserving transformations. As our main tool we study the integral and measure theory on the ultraproduct of finite measure spaces which is interesting on its own right.

math.CO↗
arXiv2006

The combinatorial cost

We study the combinatorial analogues of the classical invariants of measurable equivalence relations. We introduce the notion of cost and $β$-invariants (the analogue of the first $L^2$-Betti number introduced by Gaboriau) for sequences of finite graphs with uniformly bounded vertex degrees and examine the relation of these invariants and the rank gradient resp. mod $p$ homology gradient invariants introduced by Lackenby for residually finite groups.

math.GR↗