An explicit construction of locally testable codes of constant rate, constant distance and constant number of queries is given. Hence answering affirmatively the c^3-problem.

We prove two results on some special generators of finite simple groups and use them to prove that every non-abelian finite simple group S admits a non-congruence presentation (as conjectured by Chen, Lubotzky, and Tiep (2024)), and that if S has a non-trivial Schur multiplier, then it admits a smooth cover (as conjectured by Chen, Fan, Li, and Zhu (2024)).

Let s_n^ch(Γ ) denote the number of characteristic subgroups of index at most n in a finitely generated group Γ . In response to a question of I. Rivin, we show that if Γ = F_r is the free group on r ≥ 2 generators, then the growth type of s_n^ch(F_r) is n^log(n) . This is in contrast with the expectation of W. Thurston who predicted that there should be a difference between r = 2 and r > 2 . Along the way, we answer a question of Barnea and Schlage-Puchta (J Group Theory 23(1):1–15, 2020) on the normal subgroup growth of large groups.

In this paper, we describe how to explicitly construct infinitely many finite simple groups as characteristic quotients of the rank 2 free group F_{2} . This shows that a “baby” version of the Wiegold conjecture [in: Geometry, Rigidity, and Group Actions (2011), 609–643] fails for F_{2} and provides counterexamples to two conjectures in the theory of noncongruence subgroups of \mathrm{SL}_{2}(\mathbb{Z}) by Chen [Math. Ann. 371 (2018), 41–126]. Our main result explicitly produces, for every prime power q\ge 7 , the groups \mathrm{SL}_{3}(\mathbb{F}_{q}) and \mathrm{SU}_{3}(\mathbb{F}_{q}) as characteristic quotients of F_{2} . Our strategy is to study specializations of the Burau representation for the braid group B_{4} , exploiting an exceptional relationship between F_{2} and B_{4} first observed by Dyer, Formanek, and Grossman [Arch. Math. (Basel) 38 (1982), 404–409]. Weisfeiler’s strong approximation theorem guarantees that our specializations are surjective for infinitely many primes, but they are not effective. To make our result effective, we give another proof of surjectivity via a careful analysis of the maximal subgroup structures of \mathrm{SL}_{3}(\mathbb{F}_{q}) and \mathrm{SU}_{3}(\mathbb{F}_{q}) . These examples are minimal in the sense that no finite simple group of the form \mathrm{PSL}_{2}(\mathbb{F}_{q}) appears as a characteristic quotient of F_{2} .

We revisit the paper of Alexander Grothendieck where he introduced Grothendieck pairs and discuss the relation between profinite rigidity and left/right Grothendieck rigidity. We also show that various groups are left and/or right Grothendieck rigid and, in particular, all ascending HNNextensions of finitely generated free groups are right Grothendieck rigid. Along the way we present a number of questions and suggestions for further research.

Coboundary expansion (with $\mathbb{F}_2$ coefficients), and variations on it, have been the focus of intensive research in the last two decades. It was used to study random complexes, property testing, and above all Gromov's topological overlapping property. In part I of this paper, we extended the notion of coboundary expansion (and its variations) to cochains with permutation coefficients, equipped with the normalized Hamming distance. We showed that this gives a unified language for studying covering stability of complexes, as well as stability of group homomorphisms -- a topic that drew a lot of attention in recent years. In this part, we extend the theory to the permutation coefficients setting. This gives some new results, even for $\mathbb{F}_2$ coefficients, opens several new directions of research, and suggests a pattern to proving the existence of non-sofic groups. Along the way, we solve the dimension $2$ case of a problem of Gromov, exhibiting a family of bounded degree coboundary expanders with $\mathbb{F}_2$ coefficients.

This paper is motivated by recent developments in group stability, high dimensional expansion, local testability of error correcting codes and topological property testing. In Part I, we formulate and motivate three stability problems: 1. Homomorphism stability: Are almost homomorphisms close to homomorphisms? 2. Covering stability: Are almost coverings of a cell complex close to genuine coverings of it? 3. Cocycle stability: Are 1-cochains whose coboundary is small close to 1-cocycles? We then prove that these three problems are equivalent.

We exhibit infinitely many pairs of non-isomorphic finitely presented, residually finite groups $Δ$ and $Γ$ with $Δ$ having Property FA, $Γ$ having a non-trivial action on a tree and $Δ$ and $Γ$ having isomorphic profinite completions.

A long standing problem asks whether every group is sofic, i.e., can be separated by almost-homomorphisms to the symmetric group Sym(n). Similar problems have been asked with respect to almost-homomorphisms to the unitary group U(n), equipped with various norms. One of these problems has been solved for the first time in [De Chiffre, Gelbsky, Lubotzky, Thom, 2020]: some central extensions Γ of arithmetic lattices Γ of Sp(2g,ℚ_p) were shown to be non-Frobenius approximated by almost homomorphisms to U(n). Right after, it was shown that similar results hold with respect to the p-Schatten norms in [Lubotzky, Oppenheim, 2020]. It is natural, and has already been suggested in [Chapman, Lubotzky, 2024] and [Gohla, Thom, 2024], to check whether the Γ are also non-sofic. In order to show that they are (also) non-sofic, it suffices: (a) To prove that the permutation Cheeger constant of the simplicial complex underlying Γ is positive, generalizing [Evra, Kaufman, 2016]. This would imply that Γ is stable. (b) To prove that the (flexible) stability of Γ implies the non-soficity of Γ. Clause (b) was proved by Gohla and Thom. Here we offer a more algebraic/combinatorial treatment to their theorem.

This paper, and its companion [BCV24], are devoted to a negative resolution of the Aldous–Lyons Conjecture [AL07, Ald07]. This conjecture, originated in probability theory, is well known (cf. [Gel18]) to be equivalent to the statement that every invariant random subgroup of the free group is co-sofic. We disprove this last statement. In this part we introduce subgroup tests. These tests are finite distributions over continuous functions from the space of subgroups of the free group to {0,1}. Subgroup tests provide a general framework in which one can study invariant random subgroups of the free group. Classical notions such as group soficity and group stability arise naturally in this framework. By the correspondence between subgroups of the free group and Schreier graphs, one can view subgroup tests as a property testing model for certain edge-labeled graphs. This correspondence also provides the connection to random networks. Subgroup tests have values, which are their asymptotic optimal expectations when integrated against co-sofic invariant random subgroups. Our first main result is that, if every invariant random subgroup of the free group is co-sofic, then one can approximate the value of a subgroup test up to any positive additive constant. Our second main result is an essentially value preserving correspondence between certain non-local games and subgroup tests. By composing this correspondence with a stronger variant of the reduction in MIP*=RE [JNV+21], proved in the companion paper [BCV24], we deduce that approximating the sofic value of a subgroup test is as hard as the Halting Problem, and in particular, undecidable. The combination of our two main results proves the existence of non co-sofic invariant random subgroups of the free group.

Property testing has been a major area of research in computer science in the last three decades. By property testing we refer to an ensemble of problems, results and algorithms which enable to deduce global information about some data by only reading small random parts of it. In recent years, this theory found its way into group theory, mainly via group stability. In this paper, we study the following problem: Devise a randomized algorithm that given a subgroup H of G, decides whether H is the whole group or a proper subgroup, by checking whether a single (random) element of G is in H. The search for such an algorithm boils down to the following purely group theoretic problem: For G of rank k, find a small as possible test subset A⊆ G such that for every proper subgroup H, |H∩ A|≤ (1-δ)|A| for some absolute constant δ>0, which we call the detection probability of A. It turns out that the search for sets A of size linear in k and constant detection probability is a non-commutative analogue of the classical search for families of good error correcting codes. This paper is devoted to proving that such test subsets exist, which implies good universal error correcting codes exist – providing a far reaching generalization of the classical result of Shannon. In addition, we study this problem in certain subclasses of groups – such as abelian, nilpotent, and finite solvable groups – providing different constructions of test subsets for these subclasses with various qualities. Finally, this generalized theory of non-commutative error correcting codes suggests a plethora of interesting problems and research directions.

We solve the derandomized direct product testing question in the low acceptance regime, by constructing new high dimensional expanders that have no small connected covers. We show that our complexes have swap cocycle expansion, which allows us to deduce the agreement theorem by relying on previous work. Derandomized direct product testing, also known as agreement testing, is the following problem. Let X be a family of k-element subsets of [N] and let {f(s) : s -> Sigma vertical bar s is an element of X} be an ensemble of local functions, each defined over a subset s subset of [N]. Suppose that we run the following so-called agreement test: choose a random pair of sets s(1), s(2) is an element of X that intersect on root k elements, and accept if f(s1), f(s2) agree on the elements in s(1) boolean AND s(2). We denote the success probability of this test by Agree({f(s)}). Given that Agree({f(s)}) = epsilon > 0, is there a global function G : [N] -> Sigma such that f(s) = G vertical bar(s) for a non-negligible fraction of s is an element of X ? We construct a family X of k-subsets of [N] such that vertical bar X vertical bar = O(N) and such that it satisfies the low acceptance agreement theorem. Namely, Agree({f(s)}) > epsilon double right arrow there exists G : [N] -> Sigma, P-s[f(s) (0.99)approximate to G vertical bar s] >= poly(epsilon). A key idea is to replace the well-studied LSV complexes by symplectic high dimensional expanders (HDXs). The family X is just the k-faces of the new symplectic HDXs. The latter serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers. We also give a polynomial-time algorithm to construct this family of symplectic HDXs.

By a classical result of Neukirch, Ikeda, Iwaswa and Uchida, a number field K is determined by the structure of its absolute Galois group Gal(K). We show that K is not determined by the structure of the Sylow subgroups of Gal(K), answering a question raised by Florian Pop.

This paper is a journal counterpart to [5], 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 Γ = Γ E is associated and the testability of E depends only on Γ (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 Γ is testable/BS-rigid or not.

It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Namely, we show that for (most) high-rank lattices, every finite-dimensional unitary "almost-representation" of $\Gamma$ is a small deformation of a (true) unitary representation. This extends a result of Kazhdan (1983) for amenable groups and of Burger-Ozawa-Thom (2013) for SL(n,Z) (for n>2). Towards this goal, we first build an elaborate cohomological theory capturing the obstruction to such stability, and show that the vanishing of second cohomology implies uniform stability in this setting. This cohomology can be roughly thought of as an asymptotic version of bounded cohomology, and sheds light on a question raised in Monod (2006) about a possible connection between vanishing of second bounded cohomology and Ulam stability.

Using cohomological methods, we show that lattices in semisimple groups are typically stable with respect to the Frobenius norm but not with respect to the operator norm.

We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple (σ 1 , …, σ d ) of permutations on \{1, \ldots, 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 \mathrm{XY}= \mathrm{YX}, this corresponds to testing whether σ 1 σ 2 =σ 2 σ 1 , where σ 1 σ 2 and σ 2 σ 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 testa-bility 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. This is an extended abstract. The full version is available at https://arxiv.org/abs/2011.05234. All references beyond Sections I and II refer to the full version.

By work of Belyi [2], the absolute Galois group G(Q) = Gal((Q) over bar /Q) of the field Qof rational numbers can be embedded into A = Aut((F) over cap (2)), the automorphism group of the free profinite group (F) over cap (2) on two generators. The image of G(Q) lies inside (GT) over cap, the Grothendieck-Teichmuller group. While it is known that every abelian representation of G(Q) can be extended to (GT) over cap, Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of (GT) over cap. We do this virtually, namely by showing that a rich class of arithmetically defined representations of G(Q) can be extended to finite index subgroups of (GT) over cap. This is achieved, in fact, by extending these representations all the way to finite index subgroups of A = Aut((F) over cap (2)). We do this by developing a profinite version of the work of Grunewald and Lubotzky [7], which provided a rich collection of representations for the discrete group Aut(F-d). (c) 2021 Elsevier Inc. All rights reserved.

We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators d(Γ) of a lattice Γ in a high rank simple Lie group H grows sub-linearly with v = μ(H/Γ), the co-volume of Γ in H. We prove this for non-uniform lattices in a very strong form, showing that for 2−generic such H’s, d(Γ) = OH(log v/ log log v), which is essentially optimal. While we can not prove a new upper bound for uniform lattices, we will show that for such lattices one can not expect to achieve a better bound than d(Γ) = O(log v).