arXiv subjects

Oren Becker

Publications and source records attributed to Oren Becker.

10 recordsLinked to original sources

arXiv2026

Uniform expansion in finite groups of Lie type

We prove that finite simple groups $G(p)$ of bounded rank and with $p$ prime have uniform expansion, that is, the family of all the Cayley graphs forms a family of (two-sided) expanders, except perhaps when $p$ belongs to a small family of exceptional primes. Furthermore, for all prime powers $q$, the set of possible exceptions to uniform expansion of $G(q)$ is shown to have ``dimension zero''. We also extend these results to semisimple and perfect algebraic groups.

math.GR↗

arXiv2025

Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks

We show that random walks on semisimple algebraic groups do not concentrate on proper algebraic subvarieties with uniform exponential rate of anti-concentration. This is achieved by proving a uniform spectral gap for quasi-regular representations of countable linear groups. The method makes key use of Diophantine heights and the Height Gap theorem. We also deduce a non-abelian version of the Littlewood--Offord inequalities and prove logarithmic bounds for escape from subvarieties. In a sequel to this paper, we will show how to transform this uniform gap into uniform expansion for Cayley graphs of finite simple groups of bounded rank $G(p)$ over almost all primes $p$.

math.GR↗

arXiv2022

Testability in group theory

This paper is a journal counterpart to our FOCS 2021 paper, in which we initiate the study of property testing problems concerning a finite system of relations $E$ between permutations, generalizing the study of stability in permutations. To every such system $E$, a group $\Gamma=\Gamma_E$ is associated and the testability of $E$ depends only on $\Gamma$ (just like in Galois theory, where the solvability of a polynomial is determined by the solvability of the associated group). This leads to the notion of testable groups, and, more generally, Benjamini-Schramm rigid groups. The paper presents an ensemble of tools to check if a given group $\Gamma$ is testable/BS-rigid or not.

math.GR↗

arXiv2020

Testability of relations between permutations

We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple $(\sigma_1,\dotsc,\sigma_d)$ of permutations on $\{1,\dotsc,n\}$, and one wishes to determine whether this tuple satisfies a certain system of relations $E$, or is far from every tuple that satisfies $E$. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that $E$ is testable. For example, when $d=2$ and $E$ consists of the single relation $\mathsf{XY=YX}$, this corresponds to testing whether $\sigma_1\sigma_2=\sigma_2\sigma_1$, where $\sigma_1\sigma_2$ and $\sigma_2\sigma_1$ denote composition of permutations. We define a collection of graphs, naturally associated with the system $E$, that encodes all the information relevant to the testability of $E$. We then prove two theorems that provide criteria for testability and non-testability in terms of expansion properties of these graphs. By virtue of a deep connection with group theory, both theorems are applicable to wide classes of systems of relations. In addition, we formulate the well-studied group-theoretic notion of stability in permutations as a special case of the testability notion above, interpret all previous works on stability as testability results, survey previous results on stability from a computational perspective, and describe many directions for future research on stability and testability.

cs.DS↗

arXiv2020

Stability of approximate group actions: uniform and probabilistic

We prove that every uniform approximate homomorphism from a discrete amenable group into a symmetric group is uniformly close to a homomorphism into a slightly larger symmetric group. That is, amenable groups are uniformly flexibly stable in permutations. This answers affirmatively a question of Kun and Thom and a slight variation of a question of Lubotzky. We also give a negative answer to Lubotzky's original question by showing that the group $\mathbb{Z}$ is not uniformly strictly stable. Furthermore, we show that $\text{SL}_{r}(\mathbb{Z})$, $r\geq3$, is uniformly flexibly stable, but the free group $F_{r}$, $r\geq 2$, is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.

math.GR↗

arXiv2018

Abelian groups are polynomially stable

In recent years, there has been a considerable amount of interest in stability of equations and their corresponding groups. Here, we initiate the systematic study of the quantitative aspect of this theory. We develop a novel method, inspired by the Ornstein-Weiss quasi-tiling technique, to prove that abelian groups are polynomially stable with respect to permutations, under the normalized Hamming metrics on the groups $\operatorname{Sym}(n)$. In particular, this means that there exists $D\geq 1$ such that for $A,B\in \operatorname{Sym}(n)$, if $AB$ is $\delta$-close to $BA$, then $A$ and $B$ are $\epsilon$-close to a commuting pair of permutations, where $\epsilon\leq O(\delta^{1/D})$. We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties.

math.GR↗

arXiv2018

Group stability and Property (T)

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group $\Gamma$ with respect to a sequence of groups $\left\{G_{n}\right\}_{n=1}^{\infty}$, equipped with bi-invariant metrics $\left\{d_{n}\right\}_{n=1}^{\infty}$. We consider the case $G_{n}=\operatorname{U}\left(n\right)$ (resp. $G_{n}=\operatorname{Sym}\left(n\right)$), equipped with the normalized Hilbert-Schmidt metric $d_{n}^{\operatorname{HS}}$ (resp. the normalized Hamming metric $d_{n}^{\operatorname{Hamming}}$). Our main result is that if $\Gamma$ is infinite, hyperlinear (resp. sofic) and has Property $\operatorname{(T)}$, then it is not stable with respect to $\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$ (resp. $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$). This answers a question of Hadwin and Shulman regarding the stability of $\operatorname{SL}_{3}\left(\mathbb{Z}\right)$. We also deduce that the mapping class group $\operatorname{MCG}\left(g\right)$, $g\geq 3$, and $\operatorname{Aut}\left(\mathbb{F}_n\right)$, $n\geq 3$, are not stable with respect to $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$. Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on $\operatorname{U}\left(n\right)$ and the (unnormalized) $p$-Schatten metrics, since many groups with Property $\operatorname{(T)}$ are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim. We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to $\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$ and $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$.

math.GR↗

arXiv2018

Stability and Invariant Random Subgroups

Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism $\rho_{n_k}\colon \Gamma\rightarrow\operatorname{Sym}(n_k)$ (i.e., for every $g,h\in\Gamma$, $\lim_{k\rightarrow\infty}d_{n_k}( \rho_{n_k}(gh),\rho_{n_k}(g)\rho_{n_k}(h))=0$) is close to an actual homomorphism $\varphi_{n_k} \colon\Gamma\rightarrow\operatorname{Sym}(n_k)$. Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and P\u{a}unescu showed the same for abelian groups and raised many questions, especially about P-stability of amenable groups. We develop P-stability in general, and in particular for amenable groups. Our main tool is the theory of invariant random subgroups (IRS), which enables us to give a characterization of P-stability among amenable groups, and to deduce stability and instability of various families of amenable groups.

math.GR↗

arXiv2016

Symmetric Unique Neighbor Expanders and Good LDPC Codes

An infinite family of bounded-degree 'unique-neighbor' expanders was constructed explicitly by Alon and Capalbo (2002). We present an infinite family F of bounded-degree unique-neighbor expanders with the additional property that every graph in the family F is a Cayley graph. This answers a question raised by Tali Kaufman. Using the same methods, we show that the symmetric LDPC codes constructed by Kaufman and Lubotzky (2012) are in fact symmetric under a simply transitive group action on coordinates.

math.CO↗

arXiv2012

The minimal degree of permutation representations of finite groups

In this thesis we study the following property of a finite group G: the minimal number n such that G embeds in Sn. We start with an explicit formula for the number n for abelian groups. Then, we study the behavior of this group property in respect to direct products. Finally, we define and explore the "compression ratio" of a finite group G which measures how much better the best embedding is relative to the embedding given by Cayley's theorem.

math.GR↗