Residual Finiteness Research Articles

Local permutation stability

We introduce a notion of “local stability in permutations” for finitely generated groups. If a group is sofic and locally stable in our sense, then it is also locally embeddable into finite groups. Our notion is weaker than the “permutation stability” introduced by Glebsky–Rivera and Arzhantseva–Paunescu, which allows one to upgrade soficity to residual finiteness. We prove a necessary and sufficient condition for a finitely generated amenable group to be locally permutation stable, in terms of invariant random subgroups (IRSs), inspired by a similar criterion for permutation stability due to Becker, Lubotzky, and Thom. We apply our criterion to prove that derived subgroups of topological full groups of Cantor minimal subshifts are locally stable, using Zheng’s classification of IRSs for these groups. This last result provides continuum-many groups which are locally stable, but not stable.

A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups

AbstractWe give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). In genus at least three, there are no known infinite hyperbolic quotients of mapping class groups. However, using the hierarchically hyperbolic quotients we construct, we show, under a residual finiteness assumption, that mapping class groups have many nonelementary hyperbolic quotients. Using these quotients, we relate questions of Reid and Bridson–Reid–Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.

Residual finiteness growth in virtually abelian groups

A group G is called residually finite if for every non-trivial element g∈G, there exists a finite quotient Q of G such that the element g is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient Q depending on the complexity of the element g, for example by using the word norm ‖g‖G if the group G is assumed to be finitely generated. The residual finiteness growth RFG:N→N is then defined as the smallest function such that if ‖g‖G≤r, there exists a morphism φ:G→Q to a finite group Q with |Q|≤RFG(r) and φ(g)≠eQ.Although upper bounds have been established for several classes of groups, exact asymptotics for the function RFG are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals logk for some k∈N, where the value k is given by an explicit expression. As an application, we show that for every m≥1 and every 1≤k≤m, there exists a group G containing a normal abelian subgroup of rank m and with RFG≈logk.

Homology Growth, Hyperbolization, and Fibering

We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and—for almost all primes p—\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{F} _{p}$\\end{document}-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic 7-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.

On the structure of lower bounded HNN extensions

AbstractThis paper studies the structure and preservational properties of lower bounded HNN extensions of inverse semigroups, as introduced by Jajcayová. We show that if $S^* = [ S;\; U_1,U_2;\; \phi ]$ is a lower bounded HNN extension then the maximal subgroups of $S^*$ may be described using Bass-Serre theory, as the fundamental groups of certain graphs of groups defined from the $\mathcal{D}$ -classes of $S$ , $U_1$ and $U_2$ . We then obtain a number of results concerning when inverse semigroup properties are preserved under the HNN extension construction. The properties considered are completely semisimpleness, having finite $\mathcal{R}$ -classes, residual finiteness, being $E$ -unitary, and $0$ - $E$ -unitary. Examples are given, such as an HNN extension of a polycylic inverse monoid.

Quantifying lawlessness in finitely generated groups

Abstract We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the lawlessness growth function A Γ : N → N \mathcal{A}_{\Gamma}\colon\mathbb{N}\to\mathbb{N} . We show that A Γ \mathcal{A}_{\Gamma} is bounded if and only if Γ has a non-abelian free subgroup. By contrast, we construct, for any non-decreasing unbounded function f : N → N f\colon\mathbb{N}\to\mathbb{N} , an elementary amenable lawless group for which A Γ \mathcal{A}_{\Gamma} grows more slowly than 𝑓. We produce torsion lawless groups for which A Γ \mathcal{A}_{\Gamma} is at least linear using Golod–Shafarevich theory and give some upper bounds on A Γ \mathcal{A}_{\Gamma} for Grigorchuk’s group and Thompson’s group 𝐅. We note some connections between A Γ \mathcal{A}_{\Gamma} and quantitative versions of residual finiteness. Finally, we also describe a function M Γ \mathcal{M}_{\Gamma} quantifying the property of Γ having no mixed identities and give bounds for non-abelian free groups. By contrast with A Γ \mathcal{A}_{\Gamma} , there are no groups for which M Γ \mathcal{M}_{\Gamma} is bounded: we prove a universal lower bound on M Γ ⁢ ( n ) \mathcal{M}_{\Gamma}(n) of the order of log ⁡ ( n ) \log(n) .

The Bounded Isomorphism Conjecture for Box Spaces of Residually Finite Groups

The Bounded Isomorphism Conjecture for Box Spaces of Residually Finite Groups

On the Residual Finiteness of a Class of Polycyclic Groups II

Let A be a free abelian group of rank n (n ≥ 2) and ϕ be an automorphism of A, then $$A\langle \phi \rangle $$ is a polycyclic group. This paper investigates the residual finiteness of $$A\langle \phi \rangle $$ , and the special case that ϕ is a cyclic permutation of the direct components twisted by a rational integer m.

Residual finiteness of extensions of arithmetic subgroups of {{,mathrm{SU},}}(d,1) with cusps

Let Gamma be an arithmetic subgroup of {{,mathrm{SU},}}(d,1) with cusps, and let X_Gamma be the associated locally symmetric space. In this paper we investigate the pre-image of Gamma in the covering groups of {{,mathrm{SU},}}(d,1). Let H^bullet _!(X_Gamma ,mathbb {C}) be the inner cohomology, i.e. the image in H^bullet (X_Gamma ,mathbb {C}) of the compactly supported cohomology. We prove that if the first inner cohomology group H^1_!(X_Gamma ,mathbb {C}) is non-zero then the pre-image of Gamma in each connected cover of {{,mathrm{SU},}}(d,1) is residually finite. At the end of the paper we give an example of an arithmetic subgroup Gamma satisfying the criterion H^1_!(X_Gamma ,mathbb {C}) ne 0.

Residually Finite Lattices in PU(2,1)˜ and Fundamental Groups of Smooth Projective Surfaces

This paper studies residual finiteness of lattices in the universal cover of PU(2,1) and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in PU(2,1) or a finite covering of it. First, we prove that certain lattices in the universal cover of PU(2,1) are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in PU(2,1) to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in PU(2,1).

Free wreath products with amalgamation

We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name “free wreath products of pairs” in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition.

Residual finiteness of certain 2-dimensional Artin groups

We show that many 2-dimensional Artin groups are residually finite. This includes 3-generator Artin groups with labels ≥4 except for (2m+1,4,4) for any m≥2. As a first step towards residual finiteness we show that these Artin groups, and many more, split as free products with amalgamation or HNN extensions of finite rank free groups. Among others, this holds for all large type Artin groups with defining graph admitting an orientation, where each simple cycle is directed.

SEPARABILITY CONDITIONS IN ACTS OVER MONOIDS

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and strongly subact separable (and hence weakly subact separable); all acts over a commutative idempotent monoid are residually finite and strongly subact separable; and all acts over a Clifford monoid are strongly subact separable.

Quantifying local embeddings into finite groups

We study a function LΓ which quantifies the LEF (local embeddability into finite groups) property for a finitely generated group Γ. We compute this LEF growth function in some examples, including certain wreath products. We compare LEF growth with the analogous quantitative version of residual finiteness, and exhibit a family of finitely generated residually finite groups which nevertheless admit many more local embeddings into finite groups than they do finite quotients. Along the way, we give a new proof that B.H. Neumann's continuous family of 2-generated groups contains no finitely presented group, a result originally due to Baumslag and Miller. We compare LΓ with quantitative versions of soficity and other metric approximation properties of groups. Finally, we show that there exists a “universal” function which is an upper bound on the LEF growth of any group on a given number of generators, and that (for non-cyclic groups) any such function is non-computable.

On the Residual Finiteness of a Class of Infinite Soluble Groups

Let [Formula: see text] be a completely decomposable homogeneous torsion-free abelian group of rank [Formula: see text] ([Formula: see text]). Let [Formula: see text] be the split extension of [Formula: see text] by an automorphism [Formula: see text] which is a cyclic permutation of the direct components twisted by a rational integer [Formula: see text]. Then [Formula: see text] is an infinite soluble group. In this paper, the residual finiteness of [Formula: see text] is investigated.

Topology of leaves for minimal laminations by hyperbolic surfaces

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the second systole of the surface, which is done in the Appendix. In a companion paper, the case of other generic leaves is treated.

On uniform Hilbert Schmidt stability of groups

A group Γ \Gamma is said to be uniformly HS-stable if any map φ : Γ → U ( n ) \varphi : \Gamma \to U(n) that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We present a complete classification of uniformly HS-stable groups among finitely generated residually finite ones. Necessity of the residual finiteness assumption is discussed. A similar result is shown to hold assuming only amenability.

Residual finiteness for central extensions of lattices in $\operatorname{PU}(n,1)$ and negatively curved projective varieties

Residual finiteness for central extensions of lattices in $\operatorname{PU}(n,1)$ and negatively curved projective varieties

Residual finiteness and Abelian subgroup separability of some high dimensional graph manifolds

We generalize $3$-manifolds supporting non-positively curved metric to construct manifolds which have the following properties : (1) They are not locally $\mathrm{CAT}(0)$. (2) Their fundamental groups are residually finite. (3) They have subgroup separability for some abelian subgroups.

ON SEPARABILITY FINITENESS CONDITIONS IN SEMIGROUPS

AbstractTaking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroupS, these three separability conditions coincide and are equivalent to every$\mathcal {H}$-class ofSbeing finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many$\mathcal {H}$-classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.