Non-trivial Finite Normal Subgroups Research Articles

Equations in acylindrically hyperbolic groups and verbal closedness

Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system $S$ of equations with constants from $H$ is equivalent to a single equation. We also show that the algebraic set associated with $S$ is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such an equation has the form $w(x\_1,\ldots,x\_n)=h$, where $w\in F(X)$, $h\in H$). From this we deduce the following statement: Let $G$ be an arbitrary overgroup of the above group $H$. Then $H$ is verbally closed in $G$ if and only if it is algebraically closed in $G$. These statements have interesting implications; here we give only two of them: If $H$ is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from $H$ is equivalent to a single equation. We give a positive solution to Problem 5.2 from the paper \[J. Group Theory 17 (2014), 29–40] of Myasnikov and Roman’kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts. Moreover, we describe solutions of the equation $x^ny^m=a^nb^m$ in acylindrically hyperbolic groups (AH-groups), where $a$, $b$ are non-commensurable jointly special loxodromic elements and $n,m$ are integers with sufficiently large common divisor. We also prove the existence of special test words in AH-groups and give an application to endomorphisms of AH-groups.

Normalish Amenable Subgroups of the R. Thompson Groups

Results in [Formula: see text] algebras, of Matte Bon and Le Boudec, and of Haagerup and Olesen, apply to the R. Thompson groups [Formula: see text]. These results together show that [Formula: see text] is non-amenable if and only if [Formula: see text] has a simple reduced [Formula: see text]-algebra. In further investigations into the structure of [Formula: see text]-algebras, Breuillard, Kalantar, Kennedy, and Ozawa introduce the notion of a normalish subgroup of a group [Formula: see text]. They show that if a group [Formula: see text] admits no non-trivial finite normal subgroups and no normalish amenable subgroups then it has a simple reduced [Formula: see text]-algebra. Our chief result concerns the R. Thompson groups [Formula: see text]; we show that there is an elementary amenable group [Formula: see text] [where here, [Formula: see text]] with [Formula: see text] normalish in [Formula: see text]. The proof given uses a natural partial action of the group [Formula: see text] on a regular language determined by a synchronising automaton in order to verify a certain stability condition: once again highlighting the existence of interesting intersections of the theory of [Formula: see text] with various forms of formal language theory.

A topological zero-one law and elementary equivalence of finitely generated groups

Let G denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces S⊆G satisfying the following zero-one law: for any sentence σ in the infinitary logic Lω1,ω, the set of all models of σ in S is either meager or comeager. In particular, we prove that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, we show that generic torsion-free lacunary hyperbolic groups are elementarily equivalent; the same claim holds for lacunary hyperbolic groups without non-trivial finite normal subgroups. Our paper has a substantial expository component. We give streamlined proofs of some known results and survey ideas from topology, logic, and geometric group theory relevant to our work. We also discuss some open problems.

On invertible elements in reduced C⁎-algebras of acylindrically hyperbolic groups

Let G be an acylindrically hyperbolic group. We prove that if G has no non-trivial finite normal subgroups, then the set of invertible elements is dense in the reduced C⁎-algebra of G. The same result is obtained for finite direct products of acylindrically hyperbolic groups.

Virtually free finite-normal-subgroup-free groups are strongly verbally closed

Any virtually free group H containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing H as a verbally closed subgroup.

Property $$P_{naive}$$ P naive for acylindrically hyperbolic groups

We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1, \dots, h_k$, there exists another element $\gamma\neq 1$ such that for all $i$, $\langle h_i, \gamma \rangle = \langle h_i \rangle* \langle \gamma \rangle$. We also obtain that if a collection of subgroups $H_1, \dots, H_k$ is a hyperbolically embedded collection, then there is $\gamma \neq 1$ such that for all $i$, $\langle H_i, \gamma \rangle = H_i * \langle \gamma \rangle$.

Bounded cohomology with coefficients in uniformly convex Banach spaces

We show that for acylindrically hyperbolic groups $\Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $\rho$ of $\Gamma$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(\Gamma;\rho)$ is infinite dimensional. The result was known for the regular representations on $\ell^p(\Gamma)$ with $1<p<\infty$ by a different argument. But our result is new even for a non-abelian free group in this great generality for representations, and also the case for acylindrically hyperbolic groups follows as an application.

The Atiyah Conjecture and Artinian Rings

Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of KG in U(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, and \alpha \in \Mat_n(KG). Then \alpha induces a bounded linear map \alpha: l^2(G)^n \to \l^2(G)^n, and \ker\alpha has a well-defined von Neumann dimension \dim_{N(G)} (\ker\alpha). This is a nonnegative real number, and one version of the Atiyah conjecture states that d \dim_{N(G)}(\ker\alpha) \in Z. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG,U(G)) is a d \times d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG.

Subgroups of finite index and the just infinite property

A residually finite (profinite) group G is just infinite if every non-trivial (closed) normal subgroup of G is of finite index. This paper considers the problem of determining which (closed) subgroups of finite index of a just infinite group are themselves just infinite. If G is just infinite and not virtually abelian, we show that G is hereditarily just infinite if and only if all maximal (closed) subgroups of finite index are just infinite. This result will be used to show that a finitely generated pro- p group G is just infinite if and only if G has no non-trivial finite normal subgroups and Φ ( G ) has a just infinite open subgroup.

Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups

We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. $\Gamma=SL(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ , or Γ=H×H’ with H an infinite Kazhdan group and H’ arbitrary), and $\mathcal{V}$ is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. $\mathcal{V}$ countable discrete, or separable compact), then any $\mathcal{V}$ -valued measurable cocycle for a measure preserving action $\Gamma\curvearrowright X$ of Γ on a probability space (X,μ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action $\Gamma\curvearrowright[0,1]^{\Gamma}$ ) is cohomologous to a group morphism of Γ into $\mathcal{V}$ . We use the case $\mathcal{V}$ discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between $\Gamma\curvearrowright X$ and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ≃Λ.

Relatively hyperbolic groups are [formula omitted]-simple

We characterize relatively hyperbolic groups whose reduced C ∗ -algebra is simple as those, which have no non-trivial finite normal subgroups.

On 3-dimensional Poincaré duality complexes and 2-knot groups

AbstractWe show that a PD3-complex P such that π = π1(P) is infinite and has a non-trivial finite normal subgroup must be homotopy equivalent to RP2 × S1. Hence if A is an abelian normal subgroup of a 2-knot group πK which is not contained in the commutator subgroup πK′ and πK′ is infinite then A is torsion free.

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Any virtually free group H containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing H as a verbally closed subgroup.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

On the cohomology of polycyclic-by-finite groups

Let k[ G] be the group algebra of a polycyclic-by-finite group G over a field k. We show that, for any finitely generated k[G]- module V, H ∗(G,V)=⊕ n≥0H n(G,V) is a Noetherian module over the cohomology algebra H ∗(G,k) . In particular, the power series P G(V; t)=Σ n≥0 dim kH n(G, V)t nϵ Z〚t〛 is a rational function in t of the form f( t)/ Π i=1 s (1− t k i ), where f(t)ϵ Z[t] . For example, if G has no non-trivial finite normal subgroups and contains a torsion-free normal subgroup of index 2 and char k=2, then we show that P G ( V; −1) is always defined and equals (−1) h ρ( V)/ ρ( k[ G]), where h= h( G) is the Hirsch number of G and ρ(·) denotes the Goldie rank.

Class groups and automorphism groups of group rings

This paper is a sequel to [2]. A polycyclic-by-finite group G was there called dihedral free if G contains no subgroup isomorphic to 〈b, a:ba = b-1 a2 = 1〉 whose normalizer has finite index in G. It was shown in [2, Theorem F] that, if R is a commutative Noetherian domain, the group ring RG is a prime Noetherian maximal order if and only if R is integrally closed, G is dihedral free, and G has no non-trivial finite normal subgroups. Throughout, R and G will be assumed to satisfy these hypotheses. The main aim of the paper is to study the class group of the maximal order RG.

The Coefficient Ring of a Primitive Group Ring

All rings are associative with unity. A ring R is prime if xRy ≠ 0 whenever x and y are nonzero. A ring R is (left) primitive if there exists a faithful irreducible left R-module.If the group ring R[G] is primitive, what can we say about R? First, since every primitive ring is prime, we know that R is prime, by the followingTHEOREM 1 (Connell [1, 675]). The group ring R[G] is prime if and only if R is prime and G has no non-trivial finite normal subgroup.